Teoría Total simplificada, Revista Chilena de Ingeniería, Vol. 16, Nº1

Transcription

ISSN 0718-3291 Versión impresa
ISSN 0718-3305 Versión en línea
INGENIARE
Revista Chilena de Ingeniería
INGENIARE
Revista Chilena de Ingeniería
ISSN 0718-3291 Printed version
ISSN 0718-3305 On line version
Volume 16, N° 1
January - March 2008
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Í N D I C E
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EDITORIAL
EDITORIAL
MODERN ELECTROMAGNETIC ENGINEERING
Carlos Villarroel González
4
CHIRAL ELECTRODYNAMIC: CONNECTION FOR THE UNIFICATION OF ELECTROMAGNETISM AND GRAVITATION
H. Torres-Silva
NEW INTERPRETATION OF THE ATOMIC SPECTRA OF THE HYDROGEN ATOM: A MIXED MECHANISM
OF CLASSICAL LC CIRCUITS AND QUANTUM WAVE-PARTICLE DUALITY
H. Torres-Silva
MAXWELL’S THEORY WITH CHIRAL CURRENTS
H. Torres-Silva
CHIRAL FIELD IDEAS FOR A THEORY OF MATTER
H. Torres-Silva
THE CLOSE RELATION BETWEEN THE MAXWELL SYSTEM AND THE DIRAC EQUATION WHEN
THE ELECTRIC FIELD IS PARALLEL TO THE MAGNETIC FIELD
H. Torres-Silva
DIRAC MATRICES IN CHIRAL REPRESENTATION AND THE CONNECTION WITH THE ELECTRIC FIELD
PARALLEL TO THE MAGNETIC FIELD
H. Torres-Silva
6
ELECTRODINÁMICA QUIRAL: ESLABÓN PARA LA UNIFICACIÓN DEL ELECTROMAGNETISMO Y LA GRAVITACIÓN
H. Torres-Silva
NUEVA INTERPRETACIÓN DEL ESPECTRO ATÓMICO DEL ÁTOMO DE HIDRÓGENO: UN MECANISMO MIXTO
DE CIRCUITOS LC Y LA DUALIDAD ONDA CUÁNTICA-PARTÍCULA
2
6
24
H. Torres-Silva
24
31
TEORÍA DE MAXWELL CON CORRIENTES QUIRALES
H. Torres-Silva
31
36
IDEAS DE CAMPO QUIRAL PARA UNA TEORÍA DE LA MATERIA
H. Torres-Silva
36
43
LA ESTRECHA RELACIÓN ENTRE EL SISTEMA DE MAXWELL Y LA ECUACIÓN DE DIRAC, CUANDO EL CAMPO
ELÉCTRICO ES PARALELO AL CAMPO MAGNÉTICO
H. Torres-Silva
43
48
MATRICES DE DIRAC EN REPRESENTACIÓN QUIRAL Y LA CONEXIÓN CON EL CAMPO ELÉCTRICO PARALELO
AL CAMPO MAGNÉTICO
H. Torres-Silva
48
ECUACIONES DE MAXWELL PARA UNA FUNCIONAL DE LAGRANGE GENERALIZADA
H. Torres-Silva
53
CALIBRE QUIRAL PARA AUMENTAR EL COEFICIENTE DE RENDIMIENTO DE MOTORES MAGNÉTICOS
H. Torres-Silva
60
ELECTRODINÁMICA DE PODOLSKY BAJO UN ENFOQUE QUIRAL
H. Torres-Silva
65
ESPÍN Y RELATIVIDAD: UN MODELO SEMICLÁSICO PARA EL ESPÍN DEL ELECTRÓN
H. Torres-Silva
72
TEORÍA EXTENDIDA DE ONDAS DE EINSTEIN EN LA PRESENCIA DE TENSIONES EN EL ESPACIO-TIEMPO
H. Torres-Silva
78
MAXWELL EQUATIONS FOR A GENERALISED LAGRANGIAN FUNCTIONAL
H. Torres-Silva
53
ASYMMETRICAL CHIRAL GAUGING TO INCREASE THE COEFFICIENT OF PERFORMANCE OF MAGNETIC MOTORS
H. Torres-Silva
60
PODOLSKY'S ELECTRODYNAMICS UNDER A CHIRAL APPROACH
H. Torres-Silva
65
SPIN AND RELATIVITY: A SEMICLASSICAL MODEL FOR ELECTRON SPIN
H. Torres-Silva
72
EXTENDED EINSTEIN`S THEORY OF WAVES IN THE PRESENCE OF SPACE-TIME TENSIONS
H. Torres-Silva
78
EINSTEIN EQUATIONS FOR TETRAD FIELDS
H. Torres-Silva
85
A METRIC FOR A CHIRAL POTENTIAL FIELD
H. Torres-Silva
91
CHIRAL UNIVERSES AND QUANTUM EFFECTS PRODUCED BY ELECTROMAGNETIC FIELDS
H. Torres-Silva
99
CHIRAL WAVES IN A METAMATERIAL MEDIUM
H. Torres-Silva
LA INGENIERÍA ELECTROMAGNÉTICA MODERNA
ARTÍCULOS
ARTICLES
A NEW RELATIVISTIC FIELD THEORY OF THE ELECTRON
H. Torres-Silva
VO L U M E N 1 6 - N º 1 E N E RO - M A R Z O 2 0 0 8
ECUACIONES DE EINSTEIN PARA CAMPOS TETRADOS
H. Torres-Silva
85
UNA MÉTRICA PARA UN CAMPO POTENCIAL QUIRAL
H. Torres-Silva
91
UNIVERSOS QUIRALES Y EFECTOS CUÁNTICOS PRODUCIDOS POR CAMPOS ELECTROMAGNÉTICOS
H. Torres-Silva
99
UNA NUEVA TEORÍA RELATIVÍSTICA DE CAMPO PARA EL ELECTRÓN
111
H. Torres-Silva
111
119
ONDAS QUIRALES EN UN MEDIO METAMATERIAL
H. Torres-Silva
119
UNIVERSIDAD DE TARAPACÁ
ARICA-CHILE
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INGENIARE. REVISTA CHILENA DE INGENIERÍA1
VOLUMEN 16 Nº 1, ENERO – MARZO 2008
VOLUME 16 Nº 1, JANUARY – MARCH 2008
2
COMITÉ EDITOR ASESOR
ADVISORY EDITOR COMMITTEE
EDITOR
Enrique Fuentes Heinrich (President)
Raúl Borjas Montero
Jaime Gómez Douzet
Ingrid Guillén Figueroa
Ernesto Ponce López
Héctor Valdés González
COEDITOR
Adelheid Mahla Álvarez
Universidad de Santiago de Chile
PRODUCCIÓN EDITORIAL
EDITORIAL PRODUCTION
Carolina Cautín Barría
COMITÉ EDITOR
EDITOR COMMITTEE
Yurilev Chalco Cano
Universidad de Tarapacá, Chile
Eva María Navarro
Instituto Mexicano del Petróleo, México
Luis Cifuentes Seves
Universidad de Chile, Chile
Liliana Pedraja Rejas
Gerardo Espinosa Pérez
Universidad Autónoma de México, México
Manuel Recuero López
Universidad Politécnica de Madrid, España
Sylviane Gentil
Institut National Polytechnique de Grenoble, France
Miguel Ríos Ojeda
Pontificia Universidad Católica de Chile, Chile
Hugo Hernández Figueroa
Universidade de Campinas, Brasil
Marko Rojas Medar
Cynthia Junqueira
General-Command of Aerospace Technology,
Institute of Aeronautics and Space, Brazil
Heriberto Román Flores
Andre Koch Torres Assis
Mario Letelier Sotomayor
Universidad de Santiago de Chile, Chile
Orestes Llanes Santiago
Instituto Superior Politécnico José Antonio Echeverría, Cuba
Sebastián Lorca Pizarro
1
Fideromo Saavedra Guzmán
Osvaldo Saavedra Mendez
Universidade Federal do Maranhão, Brasil
Mario Salgado Brocal
Universidad Técnica Federico Santa María, Chile
Salah S. A. Obayya
Brunel University, United Kingdom
Linda Madsen
European Journal of Engineering Education, Denmark
Héctor Torres Silva
Adelheid Mahla Álvarez
Miguel Villablanca Martínez
João Marcos Romano
Andrés Weintraub Pohorille
Universidad de Chile, Chile
Antonio Martins Soares
Universidade de Brasilia, Brasil
Juan Zolezzi Cid
Nelson Moraga Benavides
Ernesto Zumelzu D.
Universidad Austral de Chile, Chile
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Ingeniare. Revista chilena de ingeniería, vol. 16
15 Nº 1,
3, 2008,
2007 pp. 2-3
EDITORIAL
LA INGENIERÍA ELECTROMAGNÉTICA MODERNA
La ingeniería electromagnética es una rama de la física aplicada, con tal velocidad de desarrollo que en un futuro
inmediato los ingenieros electromagnéticos serán indispensables en una nueva e importante área emergente: la ahora
denominada Estructura de Onda de la Materia (Wave Structure of Matter, WSM).
La razón principal es la capacidad de penetración de la tecnología electromagnética, donde ingenieros
especializados serán necesarios para el diseño de sistemas relacionados con la tecnología WSM. Por ejemplo, en muy
altas frecuencias, en sistemas tales como, redes inalámbricas de comunicaciones, chips de computadores, redes ópticas,
antenas, y en frecuencias muy bajas, en extracción de energía, en dispositivos de almacenamiento de energía y en
sistemas relacionados con un Enfoque Métrico de la Ingeniería (Metric Engineering Approach, MEA), en propulsión
con campos electromagnéticos.
La dificultad y la complejidad de las leyes que gobiernan el diseño de sistemas relacionados con la ingeniería
electromagnética indican que la teoría y el análisis del electromagnetismo es una ciencia en continua evolución y es un
área activa de investigación que ha atraído el interés de matemáticos, científicos de la computación y de los ingenieros.
Sin embargo, un buen entendimiento del análisis electromagnético moderno requiere de un profundo conocimiento de la
física, habilidad para el análisis matemático y del conocimiento de los algoritmos numéricos utilizados en computación.
Aunque algunas universidades enfatizan en el análisis computacional del electromagnetismo, tenemos que ser conscientes
de que un estudiante de ingeniería electromagnética debe entender los conceptos de física involucrados y desarrollar
intuición y entendimiento de los problemas a resolver. Estas habilidades son importantes tanto para el análisis como
para el diseño. Por lo tanto, es importante formar a los estudiantes de postgrado en los métodos modernos del análisis
electromagnético, y en las nuevas teorías tales como: metamateriales, electrodinámica quiral y electrogravedad. Por
ejemplo, el análisis electromagnético quiral debe incluir, entre otros, los conceptos de ondas polarizadas circularmente,
ondas superficiales, ondas que se arrastran (creeping waves), ondas laterales, modos guiados, modos evanescentes,
modos radiantes y los modos filtrados (leaky modes). Todo esto en la física de altas frecuencias donde la dualidad
onda/partícula emerge como un nuevo enfoque físico de las interacciones electromagnéticas de la WSM.
Recientemente se han producido avances en la WSM, por ejemplo, en microcircuitos industriales y en
electrodinámica, donde existen corrientes de lazos cerrados de ondas de electrones, siendo el electrón no una partícula
puntual sino una estructura de onda. Aquí la mayoría de las aplicaciones, como ser nanotubos quirales y sustratos de
metamateriales para uso en microcircuitos, requiere de la comprensión del comportamiento de la materia en “dimensiones
muy pequeñas”, donde la aproximación de la partícula falla y la WSM se hace necesaria para entender qué ocurre
cuando interactúan diferentes sustratos, a nivel químico, eléctrico o biológico. A nivel de microestructuras, empresas
como Intel están empezando a utilizar biología y genética en las técnicas de fabricación de dispositivos orgánicos,
usando partes biológicas para sintetizar filamentos quirales de DNA (Deoxyribonucleic acid), donde las ondas que se
propagan son equivalentes a las WSM. Por otra parte a nivel macroscópico, para entender adecuadamente la naturaleza
de la interacción entre un campo electromagnético de muy alta frecuencia con la materia, debemos considerar la
electrodinámica quiral relacionada con la relatividad. Un ejemplo relevante es el diseño de nuevos sistemas GPS
(Global Positioning System) con distinta polarización circular, que son más exactos, con la finalidad de mejorar los
sistemas actuales.
Un estudiante de electromagnetismo debe estar consciente de la metamorfosis, que ocurre en la física, cuando
trabajamos en distintas longitudes de onda o en distintas frecuencias. Cuando la longitud de onda es muy larga, nos
encontramos en el dominio de la electro estática y de la magneto estática; aquí se aplica la teoría de circuitos, un ejemplo
es el área de los dispositivos de almacenamiento de energía, desde baterías comunes a sofisticados dispositivos híbridos
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Villarroel: La ingeniería electromagnética moderna
utilizados para almacenamiento de energía. Concretamente, en los automóviles modernos, dichos elementos están
hechos de mezclas químicas cuyas energías vinculantes son diferentes. Si se conoce la forma en que los elementos de
la mezcla se unen, se podrían diseñar baterías para fines específicos, con cálculos basados en la WSM. En el futuro,
la WSM requerirá de nuevas técnicas de aplicación, cálculo y diseño de la ingeniería electromagnética. Por otro lado,
la mayoría de las aleaciones más valiosas que se utilizan ampliamente en las aplicaciones industriales, como ser el
acero, el bronce y el duraluminio, son mezclas simples de elementos básicos, esto es posible gracias a que las uniones
de las aleaciones son del tipo Estructuras de Onda.
En relación con todo esto tenemos la MEA, enfoque que será muy importante en las próximas décadas. Esta
metodología, para tratar los cambios métricos, ha surgido a través de años de estudio de las teorías electro gravitacionales.
Este enfoque es isomórfico con la representación general de la relatividad del vacío, tratando el vacío como un medio
polarizable con cambios métricos internos, en términos de la permisividad y la permeabilidad consideradas constantes
en el vacío. Este enfoque es básico para obtener energía a partir del vacío (motores magnéticos). Aquí, las ecuaciones
de Maxwell en el espacio curvo se modelan como un medio polarizable de índice de refracción variable en el espacio
plano, donde la curvatura de un rayo de luz y la reducción de la velocidad de la luz en un potencial gravitacional se
representan por un aumento efectivo del índice de refracción. Con este método es posible estudiar los Sistemas de
Propulsión de Campos Electromagnéticos, donde la propagación de fotones posee momentum producido por los campos
magnéticos y eléctricos ortogonales entre sí (vector de Poynting).
Estos desafíos tecnológicos nos hacen ver que es importante atraer para este campo a personas más calificadas
y creativas, reclutando los mejores estudiantes y estimulando su creatividad. En esta perspectiva de la enseñanza de la
ingeniería electromagnética, la gente joven siempre puede generar buenas ideas, forjar nuevas fronteras, crear nuevas
áreas de trabajo y cultivar el pensamiento independiente, estimulados por el profesor en el desafío de pensar.
Puesto que el análisis electromagnético ha sido usado como una importante herramienta de predicción en muchas
ramas de la ingeniería eléctrica, seguirá siendo aún más importante en las nuevas tecnologías. La larga y rica historia
del electromagnetismo nos ofrece un desafío sobre cómo debemos educar a nuestros estudiantes de postgrado en esta
área. El total del conocimiento requerido no se puede entregar dentro del corto período de su enseñanza universitaria.
Por lo tanto, es fundamental educarlos en los conocimientos básicos, ya que aprender todo lo pertinente a la tecnología
electromagnética requiere de toda una vida de aprendizaje. Asimismo, es importante educar a dichos estudiantes como
pensadores, en vez de adquirir los conocimientos en forma mecánica, siendo esta forma de enseñar un importante
aporte para nuestra sociedad.
Es así como, en este número, presentamos el aporte del doctor Héctor Torres-Silva en esta área del
electromagnetismo moderno, mediante la electrodinámica vinculada a la mecánica cuántica y a la gravitación. Este
trabajo incluye aspectos fundamentales de la WSM, al unificar al electromagnetismo y a la gravitación a través de
la electrodinámica quiral, mostrando en este estudio, en forma rigurosa, que la mecánica cuántica de Dirac es una
consecuencia lógica de dicha unificación.
Editor
Arica, Chile
3
16 Nº 3,
1, 2008,
2007 pp. 4-5
EDITORIAL
MODERN ELECTROMAGNETIC ENGINEERING
Electromagnetic engineering is a branch of applied physics which is developing so fast that in the near future
electromagnetic engineers will be indispensable in an important emerging area, that which is now known as Wave
Structure Matter (WSM).
The main reason for this is the penetration capacity of electromagnetic technology where specialized engineers
will be needed for the design of systems related to WSM technology. Examples of such systems with very high frequencies
are wireless communication networks, computer chips, optical networks, antennae and at low frequencies, energy
storage devices and systems related to a Metric Engineering Approach (MEA) with electromagnetic fields.
The difficulty and complexity of the laws that govern the design of systems related with electromagnetic
engineering indicate that the theory and analysis of electromagnetism is a continually-evolving science and an area of
active research that has attracted the interest of mathematicians, computer scientists and engineers. However, a good
understanding of modern electromagnetic analysis requires a deep knowledge of physics, a capacity for mathematical
analysis and knowledge of the numerical algorithms used in computing. Even though some universities emphasize the
computational analysis of electromagnetism, we must be aware that a student of electromagnetic engineering should
understand the concepts of physics involved and develop intuition and understanding of the problems to be solved.
These abilities are as important in design as they are in analysis. Therefore, it is important to train postgraduate students
in modern methods of electromagnetic analysis and new theories such as: metamaterials, electrodynamism chiral
electrogravity. For example, chiral electromagnetic analysis ought to include, amongst other things, the concepts of
circular-polarized waves, superficial waves, creeping waves, lateral waves, guided modes, evanescent modes, radiant
modes and leaky modes. All of this in high-frequency physics where the wave-particle duality is emerging as a new
physical focus of the electromagnetic interactions of WSM.
Recently, advances have been made in WSM, for example in industrial micro-circuits and electrodynamics
where there are currents in closed loop of real electron waves, since the electron is not a point particle but rather a
wave structure. Here the majority of applications, such as chiral nanotubes and metamaterial substrates for use in
microcircuits, require the understanding of material in “very small dimensions” where an approximation of the particle
fails and WSM makes it necessary to understand what happens when different substrates interact at the chemical,
biological and physical level. A the microstructural level, companies such as Intel are beginning to use biology and
genetics in the production techniques for organic devices, using living elements to synthesize chiral filaments of DNA
(deoxyribonucleic acid) where the propagating waves are equivalent to WSM. Moreover, at the microscopic level, in
order to adequately understand the nature of the interaction between a very high frequency electromagnetic field and
the material, we should consider the chiral electrodynamic related to the relativity. A pertinent example is that of the
design of new GPS (Global Positioning Systems) with distinct circular polarization that are more accurate with the
aim of improving the current systems.
A student of electromagnetism should be aware of the metamorphosis that occurs in physics when we work
in different wavelengths or frequencies. When wavelength is very long, we find ourselves in the electrostatic and
magnostatic domain, where circuit theory applies. An example from this area is that of energy storage devices that
range from ordinary batteries to sophisticated hybrid devices used for storing energy. To give a concrete example, in
modern automobiles, these elements are made from chemicals with different bond energies. If we know the way in
which the elements of the mixture bond, batteries can be designed for specific purposes with calculations based on
WSM. In the future, WSM will require new techniques for application, calculation and design from electromagnetic
engineering. Furthermore, the majority of the most costly alloys that are widely used in industrial applications, such
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Villarroel: La ingeniería electromagnética moderna
as steel, bronze and hard aluminium are simple mixtures of basic elements. They serve their purposes thanks to the
bonds between the alloys that have wave structure.
Connected to all of this is MEA, an area that will be extremely significant during the next few decades. This
methodology for treating metric changes has arisen through years of study of electrogravitational theories. This focus
is isomorphic with the general representation of vacuums, treating them as polarizable media with internal metric
changes in terms of the permittivity and permeability considered to be constant in the vacuum. This focus is the basis
for obtaining energy from vacuum (magnetic motors). Here, Maxwell’s equations in curved space are modelled as a
polarizable medium of the variable refraction index in flat space where the curvature of a ray of light and the reduction
in the speed of life in a gravitational potential are represented by an increase in the refractive index. With this method
it is possible to study Electromagnetic Field Propulsion Systems, where the propagation of photons has a momentum
produced by the crossed electric and magnetic fields (Poynting’s vector).
These technical challenges enable us to see that for this field it is important to attract qualified and creative
people, to recruit the best students and stimulate their creativity. From the perspective of electromagnetic engineering
teaching, young people can generate good ideas, stretch boundaries, create new areas for study and cultivate independent
thought, stimulated by teachers that challenge them to think.
Given that electromagnetic analysis has been used as an important tool in prediction in many areas of electric
engineering, it will continue to be one of the most important tools in new technologies. The long and rich history
of electromagnetism makes the question of how to train our postgraduate students in this area challenging. All the
knowledge required cannot be conveyed during the short period of university education. Thus, it is fundamental to
provide the most essential knowledge; learning about everything related to electromagnetic technology would take a
whole life-time of learning. Moreover, it is important to train these students to be thinkers, rather than mechanically
acquiring knowledge, and thus contribute significantly to our society.
It is for these reasons, that in this issue, we present Dr. Héctor Silva-Torres’s contribution to the area of modern
electromagnetism through electrodynamics linked to quantum mechanics and gravity. This work includes fundamental
aspects of WSM, to unify the electromagnetism and gravity through electrodynamics chiral showing rigorously in this
study, that the Dirac’s quantum mechanics is a logical consequence of this unification.
Editor
Arica, Chile
5
vol. 16 Nº 1,
Nº 1, 2008,
2008 pp. 6-23
ELECTRODINÁMICA QUIRAL: ESLABÓN PARA LA UNIFICACIÓN
DEL ELECTROMAGNETISMO Y LA GRAVITACIÓN
CHIRAL ELECTRODYNAMIC: CONNECTION FOR THE UNIFICATION
OF ELECTROMAGNETISM AND GRAVITATION
H. Torres-Silva1
Recibido el 5 de septiembre de 2007, aceptado el 12 de diciembre de 2007
Received: September 5, 2007 Accepted: December 12, 2007
RESUMEN
Una alternativa a la teoría cuántica de la gravedad, aún no descubierta, es la Teoría Total Simplificada (TTS) aquí propuesta,
que postula unificar la gravedad con el electromagnetismo (EM) teniendo como corolario fundamental la ecuación cuántica
de Dirac.
Con ello, aquí se propone todo un programa de unificación en el cual el electromagnetismo quiral juega el rol central. La
TTS se deriva de las ecuaciones originales de Einstein-Hilbert G µv = kTµv, donde el tensor de Einstein no se modifica.
El tensor EM en cambio es quiral y la masa de las partículas es de naturaleza electromagnética. Para el caso del electrón
se tiene como consecuencia que, por primera vez, se obtiene la ecuación de Dirac a partir de ondas EM con el campo
eléctrico paralelo espacialmente al campo magnético. Como modelo del universo se propone una interfaz o membrana
de separación donde ocurren solamente eventos cuánticos. Hay dos regiones enantioméricas de un universo cerrado, o un
universo derecho y un universo izquierdo, relacionados por un elemento de simetría PCT (paridad, carga, tiempo) a lo largo
de la interfaz. Las ecuaciones de Einstein-Hilbert son estudiadas bajo el enfoque quiral y se discute la electrodinámica
quiral y la gravedad en la era de Planck.
Palabras clave: Unificación, electrodinámica quiral, era de Planck.
ABSTRACT
An alternative to the theory of quantum gravity, not yet discovered, is the Theory Simplified Total (TTS) proposal, which
aims to unify gravity with the EM taking as a corollary essential quantum Dirac equation.
Thus, this article proposes a whole program of unification in which electromagnetism chiral plays the main role. TTS
is derived from the original equations of Einstein-Hilbert G µv = kTµv, where the Einstein tensor is unchanged. The EM
tensor instead is chiral and the mass of the particles is electromagnetic nature. In the case of the electron the consequence
of this is that, for the first time, Dirac’s equation is obtained from EM waves with the electric field spatially parallel to
the magnetic field. As a model of the universe an interface or membrane separation is proposed as the only location for
quantum events. There are two enanciometrics regions in a closed universe, or right and left universe, connected by an
element of PCT (parity, charge, time) symmetry along the interface. Einstein-Hilbert equations are studied under the
chiral approach and discusses the chiral electrodynamics and gravity in Planch’s era are discussed.
Keywords: Unification, chiral electrodynamics, age Planck.
INTRODUCCIÓN
No es difícil mostrar, sin lugar a dudas, que la física se
lleva el honor de ser la disciplina con los mayores adelantos
teóricos y con las más grandes aplicaciones tecnológicas.
Hoy, a comienzos del siglo XXI, la física sigue el rumbo
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y la impronta marcada por los grandes progresos logrados
el siglo pasado. Como un logro no alcanzado se destaca la
formulación de una teoría unificada de todas las fuerzas de
la naturaleza. Físicos teóricos de todo el mundo, en solitario
o en equipo, han dedicado y dedican una enorme cantidad
de tiempo y esfuerzo para la consecución de ese sueño.
Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: [email protected]
Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008
H. Torres-Silva: Electrodinámica quiral: eslabón para la unificación del electromagnetismo y la gravitación
Al mencionar este esfuerzo, en la búsqueda de la unificación,
siempre conviene recordar los trabajos pioneros de Albert
Einstein (1879-1955), quien se convirtió en el líder de un
sueño singular de hacerlo realidad: la teoría geométrica
para la unificación de los campos gravitacionales y
electromagnéticos (EM).
El programa de Einstein para la unificación no tuvo
éxito por el hecho de no considerar la existencia de
campos eléctricos y magnéticos paralelos en el espacio
y desfasados en el tiempo pues la existencia teórica y
experimental de dichas ondas se ha verificado solo en
los últimos años.
A fin de proporcionar una visión global, se ha dividido
este trabajo en tres partes. En la primera, se hace una
introducción a la electrodinámica quiral (EQ), en la
segunda se describen sintéticamente las ideas acerca de
la Teoría General de la Relatividad (TGR), los trabajos
de Einstein y otros autores para la unificación de los
campos, y en la tercera parte se propone una teoría de
unificación basada en la electrodinámica quiral para un
universo de cuatro dimensiones que da lugar a la Teoría
Total Simplificada (TTS).
INTRODUCCIÓN A LA QUIRALIDAD
Con el avance en la construcción de compuestos artificiales,
los materiales quirales han asumido una gran importancia
tecnológica en antenas, circuitos de alta frecuencia y en
fibras ópticas. Tales materiales, que no tienen la simetría
tipo espejo, también se encuentran en la naturaleza desde
las galaxias en espiral a moléculas tipo hélices como el
DNA, que son ópticamente activas, las cuales muestran
birrefringencia a frecuencias ópticas y de microondas.
Ya que la quiralidad es un concepto geométrico, es
posible concebir la fabricación de materiales quirales
artificiales y metamateriales, con aplicaciones en ingeniería
electromagnética.
El fenómeno de la actividad óptica, en ciertas substancias
biológicas y materiales, fue descubierto por Pasteur
(1848-1850) interpretando las observaciones de arreglos
asimétricos de átomos dentro de un material ópticamente
activo, por lo que se tiene una imagen espejo no superpuesta,
arreglo definido como “maniobrable” o quiral.
Con los avances en la teoría de campos electromagnéticos
el campo de la esteroquímica se expande fuertemente
y se conocen detalles de la estructura de las moléculas
y, además, de la naturaleza de estructuras moleculares
indispensables para la vida.
Un medio quiral está caracterizado por su maniobrabilidad
en su microestructura, ya sea a la izquierda o a la derecha.
Esto resulta en un medio quiral polarizado circularmente
a la izquierda (LCP) o a la derecha (RCP) y los campos
se propagan a diferentes velocidades de fase: el campo
con esta última polarización viaja, a través de un medio
manipulado a la derecha, más rápido que un campo
circularmente polarizado a la izquierda, y viceversa.
La actividad óptica, la cual se encuentra en una serie
de moléculas orgánicas a frecuencias ópticas, es una
manifestación de la quiralidad nativa de estas moléculas.
Se observan fenómenos similares al dicroísmo circular
(CD) y a la dispersión óptica rotatoria (ORD), con absorción
diferencial de las ondas polarizadas circularmente a la
izquierda o a la derecha al interior del medio quiral.
Estudios giroscópicos tienen tal riqueza de información
que se puede decir que “la actividad óptica entrega una
ventana hacia el interior de la fábrica del universo”. Los
átomos son ahora considerados quirales debido a la débil
violación de paridad de la corriente neutral de interacción
entre el núcleo y los electrones; la pequeña actividad óptica
resultante para los átomos ha hecho más concordante la
teoría con la práctica. Los avances tecnológicos en las
décadas de los 80-90 han hecho posible la detección de
asimetría quiral en dispersión Raman.
Aunque han sido estudiados muchos aspectos de mecánica
cuántica, la quiralidad ha sido poco estudiada, falta
un estudio sistemático de la teoría clásica de campos
electromagnéticos considerando la quiralidad. Con los
avances en la ciencia de los polímeros (dieléctricos quirales
activos a frecuencias milimétricas, por ejemplo), hacen
necesario revisar todos los aspectos relacionales de la
teoría de campos electromagnéticos.
Además, a causa de que la quiralidad es un atributo
geométrico específico, el conocimiento recogido del estudio
de la estructura molecular debe trasladarse al diseño y
manufactura de medios quirales artificiales, los cuales deben
exhibir CD y ORD a frecuencias de telecomunicaciones.
Sólo en el rango intermedio de frecuencia la quiralidad
molecular no desaparece cuando se efectúa una transición
desde escala microscópica a macroscópica en teoría
electromagnética. Lo anterior significa que es posible la
construcción de medios artificiales introduciendo objetos
quirales micrométricos en un medio huésped aquiral. En
un rango de frecuencias intermedias, la microestructura
podría tener una dimensión adecuada (2-5%) respecto
de la longitud de onda del material huésped (substrato);
consecuentemente, el medio compuesto podría comportarse
como efectivamente quiral. La factibilidad de esta idea
ha significado un intenso estudio sobre la propagación de
ondas electromagnéticas en medios quirales.
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A modo de ejemplo, el ojo humano contiene dos tipos de
estructura activa para distinguir entre proceso fotópico
(umbral alto) y proceso escotópico (umbral bajo) que son:
el bastón y los conos, respectivamente. Varias partes en
el ojo son anisotrópicas; en particular las fibras de la
retina son algo uniaxiales. Se puede usar la anisotropía
estructural para explicar la diferencia en la sensibilidad de
los ojos a la luz polarizada circularmente a la derecha y a la
izquierda. Así, cualquier tratamiento de la física del ojo no
sólo debe incluir anisotropía debido a la microestructura,
bastones, conos, ganglios, etc., sino que debería también
considerar la macroestructura, es decir, la helicidad de los
componentes moleculares cuyas dimensiones pueden ser
una fracción significativa de la longitud de onda óptica.
Esto será relevante en futuros sistemas ópticos.
recientes indican que valores de T. para compuestos
quirales artificiales, en el rango de 8-40 GHz, estarán
luego disponibles. Nuestro trabajo específico sobre teoría
y simulaciones de la electrodinámica quiral aplicada a
solitones, fibras ópticas, sistemas de microondas, sistemas
biológicos, etc. están relacionados en [1-18].
La actividad óptica puede ser explicada por la sustitución
directa de nuevas relaciones constitutivas en las
ecuaciones de Maxwell, es decir, D = εE + Tε∇xE y
B = µΗ + T µ∇ × Η. Aquí ε y µ son la permitividad
y permeabilidad respectivamente, mientras T es el
parámetro con dimensión de longitud y es el resultado
directo de cualquier quiralidad en la manoestructura
del medio. Como tal, las ecuaciones constitutivas
quirales son aplicables a cualquier región del espectro
electromagnético como radiación. En óptica, hasta muy
recientemente, sólo fue posible realizar mediciones de
intensidad, es decir, de la magnitud pero no de la fase.
Así la literatura sobre actividad óptica está relacionada,
generalmente, sólo con la diferencia en la intensidad de
la luz dispersada cuando un volumen quiral es irradiado
ya sea por una onda plana LCP o RCP. Esto significa que
sólo mediciones de (nL –nR) están disponibles, donde nL
y nR son los índices de refracción para las ondas LCP y
RCP respectivamente. Aunque cada uno de estos índices
de refracción puede estar muy relacionado a ε, µ , T, el
conocimiento de (nL –nR) no es suficiente para inferir
los valores de los parámetros constitutivos.
Con el surgimiento de la Relatividad Especial en 1905,
casi de inmediato los físicos llegaron a reconocer la
invariancia lorentziana en la teoría de Maxwell; y dada
la geometría de Minkowski se tornó claro para Einstein y
D. Hilbert que la unificación implicaba, de algún modo, la
unificación del espacio tridimensional y el tiempo en un
“espacio-tiempo continuo” cuadridimensional. En 1915,
Hilbert presentó por primera vez una teoría de campo
unificado basado en los primeros trabajos de Einstein
(1914) sobre la teoría relativista de la gravitación y en
los artículos de G. Mie (1912) sobre la electrodinámica
no lineal de la materia.
Cuando uno considera este fenómeno a un rango de
frecuencias mucho más bajo, 0.5-100 GHz, se podría
estar más interesado en ε, µ, T que en nL y nR. Con el
advenimiento de los analizadores vectoriales de redes
ahora ha llegado a ser posible realizar mediciones muy
exactas de magnitud y fase, pero la generación de ondas
circularmente polarizadas requiere de tecnología de
punta y se prevén aplicaciones en antenas de polarización
circular en sistemas GPS de última generación y en
comunicaciones satelitales.
El uso del analizador vectorial de redes, en conjunto
con experimentos canónicos deseablemente definidos,
puede entonces facilitar la medición de T. Investigaciones
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EL PROGRAMA DE EINSTEIN PARA
LA UNIFICACIÓN
La historia muestra que la idea de la unificación de las
fuerzas de la naturaleza no se origina con los trabajos de
Einstein; ni la propuesta de unificación a altas dimensiones
tampoco es original de Kaluza [19].
En el trabajo de Hilbert se obtienen las ecuaciones de
Euler-Lagrange, derivadas de un principio variacional.
Cinco días después de la conferencia de Hilbert sobre su
teoría de unificación, Einstein publicó su TGR verificable
y verificada que vinculara directamente la distribución
y movimiento de materia a la geometría del espaciotiempo.
En la TGR, Einstein geometriza la gravitación en el sentido
de que toda la información acerca de las interacciones
gravitacionales está contenida en el elemento de línea
del espacio-tiempo. La poderosa y bella descripción de
la gravitación entusiasmó a los físicos y matemáticos a
intentar una geometrización para la unificación con el
electromagnetismo (EM).
La búsqueda de una teoría geométrica y unificada de los
campos gravitacionales y electromagnéticos ocupó un
rol dominante en los últimos veinte años de la actividad
científica de Einstein.
Las ideas principales de Einstein para la unificación
clásica de las interacciones eran: geometrizar el
electromagnetismo, unificar las variables básicas de la
gravitación y el electromagnetismo en un único objeto
geométrico y obtener las ecuaciones de campos unificados
a partir de un principio variacional.
Einstein esperaba que las propiedades fundamentales de las
partículas elementales y sus respectivos comportamientos
cuánticos pudieran ser de algún modo descritos y
explicados en el marco de una teoría clásica puramente
geométrica.
La teoría general de la relatividad (TGR 1915) está
basada en dos objetos geométricos fundamentales: un
tensor métrico g y una conexión lineal Γ. La métrica es
necesaria para medir distancias, intervalos de tiempos,
velocidades relativas y ángulos. La conexión basada en
la noción de transporte paralelo de Levi-Civita sirve a
su vez para comparar direcciones, fuerzas y campos en
puntos separados en el espacio-tiempo de Riemann.
Todos los intentos iniciales en unificación estaban
basados en los objetos geométricos antes mencionados.
La idea básica es obtener nuevos grados de libertad en
la TGR para describir el electromagnetismo relajando o
imponiendo restricciones sobre el tensor métrico (g) y/o
Γ, o incrementar el número de dimensiones de la variedad
riemanniana.
Dos años después de que Einstein postulara la TGR, H.
Weyl (discípulo de Hilbert) propone un modelo geométrico
de la gravitación y del electromagnetismo. Weyl consideró
que la geometrización podría ser generalizada a otras
fuerzas de la naturaleza. Así, propuso una conexión
general dependiente de la trayectoria cuando se compara
la longitud de vectores en diferentes puntos del espaciotiempo. En otras palabras, él notó que la TGR está basada
en la “relatividad de la dirección” y propuso extenderla a
fin de tomar en cuenta la “relatividad de las magnitudes”
al permitir una transformación conforme de la métrica.
Esta idea, que llegó a ser conocida como teoría de calibre
o de “gauge”, no prosperó porque llegaba a contradecir
la escala absoluta de masas del mundo real. A pesar del
fracaso Weyl reinterpretó los calibres (gauge en el contexto
de la física cuántica) al indicar que podrían actuar en las
funciones de onda de las partículas cargadas más bien que
sobre g. Esta idea inspiró a las teorías de calibre no abelianas
y a la interpretación de los potenciales electromagnéticos
y de Yang-Mills como conexiones en fibrados principales.
También llegó a constituirse en el germen para el desarrollo
de las llamadas “teorías gauge de la gravedad”.
En 1919, T. Kaluza propuso una teoría de la gravitación de
cinco dimensiones. Esta idea fue trabajada por Einstein y
colaboradores en 1923. Einstein vuelve sobre esta teoría
pentadimensional en un trabajo publicado en 1927; también
lo hace en cuatro trabajos (1931, 1932, 1938 y 1944) con
sus colaboradores, pero sin llegar a geometrizar el EM
ni a la unificación de los campos gravitacional y EM.
A. Eddington propuso considerar a Γ como la cantidad
básica de la TGR y derivar de ella tanto el tensor métrico
como el campo electromagnético al dividir el tensor de
Ricci, R µν, en sus partes simétricas y antisimétricas.
Einstein, en cinco trabajos (cuatro en 1923 y uno en
1925), desarrolló esta idea al postular que la densidad
lagrangiana debería ser proporcional a la cantidad det
Rµν1/2. Desafortunadamente, todos estos intentos llevaron
a ecuaciones incompatibles con los experimentos y Einstein
se vio obligado a abandonar este camino. En 1925 Einstein
consideró una teoría basada en la conexión Γ y una gµν
no simétrica e identificaba a g[µν] (la parte antisimétrica
de gµν ) con el campo electromagnético; volvió sobre esta
idea en los últimos años de su vida trabajando en una
“teoría asimétrica” fundamentada en la métrica y en la
conexión. Sobre esta línea de investigación publicó 11
trabajos entre 1925 y 1955. Bajo la influencia de Cartan,
Einstein genera una nueva línea de investigación en la
que se dota a la variedad del espacio-tiempo de una
nueva entidad geométrica llamada torsión inventada por
Cartan en 1922.
Este esquema fue transformado por un nuevo y poderoso
concepto geométrico llamado teleparalelismo, también
desarrollado por Cartan. Teleparalelismo significa que la
curvatura total es cero, o una suposición más débil: que el
tensor total de Ricci es cero. Estas ideas fueron trabajadas
por Einstein y publicadas en tres trabajos (uno en 1929 y dos
en 1930). En esa área tampoco logró la ansiada unificación.
Las teorías de unificación basadas en teleparalelismo han
sido reconsideradas en años recientes siguiendo el enfoque
de la geometría diferencial moderna [20].
En resumen, Einstein se embarcó en un programa
geométrico de unificación de las interacciones clásicas
gravitacionales y electromagnéticas en más de cuarenta
trabajos. A pesar del fracaso, aun así entreabrió nuevas
sendas hacia la búsqueda de la unificación de las fuerzas
de la naturaleza, en cuya tarea se han ocupado importantes
físicos durante el siglo XX.
Pero la mayoría de estos esfuerzos están bajo el enfoque
de la teoría cuántica de campos. De hecho, las tres cuartas
partes de las fuerzas de la naturaleza conocidas son
estudiadas en el marco de la mecánica cuántica; y ya se
ha logrado la unificación de las fuerzas débiles con las
electromagnéticas. La unificación con la fuerte en este
esquema no debe tardar en concretarse. La gravitación, por
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su naturaleza misma, se resiste a entrar en ese esquema.
Existen por lo menos cuatro programas de cuantización
del campo gravitatorio [21], todos los cuales parecen
están destinados al fracaso. El más popular de estos
fracasados esfuerzos es el programa de las “variables de
Asthekar”, físico hindú que pretende cuantizar el campo
gravitatorio “a la canónica”, es decir, siguiendo el mismo
procedimiento que llevó a la cuantización del campo EM.
El punto crucial es que si deseamos cuantizar el campo
gravitacional deberíamos, como bien han señalado R.
Penrose y R. Wald [21], reconstruir la mecánica cuántica
sobre nuevos fundamentos.
El otro programa de unificación vía geometría diferencial
es también extraordinariamente difícil. Es pertinente
mencionar que en esta área de trabajo se distinguen dos
líneas de investigación. Una en la que se mezclan conceptos
de calibres de la mecánica cuántica con conceptos y
herramientas de la topología diferencial. Un ejemplo de
ello son las llamadas teorías de Kaluza-Klein, en cinco
o más dimensiones. La otra línea de acción proviene de
los trabajos de Wheeler, quien partiendo del enfoque
de una geometría diferencial pura ha publicado la más
elaborada geometrización del electromagnetismo de toda
la literatura [20].
MODELO QUIRAL DEL UNIVERSO
Einstein en su visión del universo y en su programa de
unificación, aun teniendo presente el origen cuántico de
la materia, no pudo concretar la unificación GEM. Tal vez
el recorrido zigzagueante de Einstein en su programa de
unificación fue producto de las numerosas tentativas de
modificar el lado izquierdo de su ecuación GµvkTµv, dejando
el tensor de materia Tµv sin alterar. En la TTS lo que cambia
es el tensor Tµv. Conviene aquí decir algo al respecto de
la relatividad general y la mecánica cuántica.
En la actualidad, no hay duda de que la teoría de la
relatividad y el modelo del Big-Bang son exitosos a la hora
de presentarnos un panorama general de cómo el universo
que hoy disfrutamos es consecuencia de la evolución bajo
ciertas condiciones iniciales del universo que había luego
de unas cuantas fracciones de segundo y de la aplicación
de leyes conocidas de la física. Insistimos, no es que se
conozcan todas las respuestas ni todos los detalles, sino
que el modelo brinda la plataforma sobre la cual estas
preguntas y estos detalles pueden ser bien planteados y
abordados con la estrategia de las ciencias físicas.
La relatividad de Einstein permanecerá como una portentosa
contribución de la ciencia del siglo XX y un formidable
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tributo al ingenio humano. Tanto en su versión especial
como en la general cuando haya materia que curve el
espacio-tiempo, la relatividad será una poderosa herramienta
de interpretación de una parte de la realidad física.
La revolución iniciada por la relatividad cambió de
manera contundente la forma como debemos entender
al espacio, al tiempo y a la materia. Nos brinda una
imagen más coherente y unificada del mundo físico:
la manera por la que brillan las estrellas tiene que ver
con el retraso de relojes en movimiento. Entendemos
mejor por qué cierta escala el sistema newtoniano
da tan buenos resultados. Una buena parte de sus
predicciones han sido corroboradas dándole sentido
a las observaciones.
Otras, como la existencia de ondas gravitatorias, nos
permitirán ‘mirar’ el universo con otra mirada, más
profunda, que habrá de revelarnos mucho acerca del
universo en que vivimos. La ‘flexibilidad’ del tiempo y
el espacio permite considerar las seductoras posibilidades
de desaparición del tiempo como en los agujeros negros,
la aparición del tiempo en el Big Bang, la expansión del
espacio a escala cosmológica, que en la rígida perspectiva
newtoniana eran impensables.
Sin embargo, sabemos que algo importante está faltando.
Las dos grandes revoluciones del siglo XX, la relatividad
general y la cuántica, son incompatibles ente sí. Cada una
es exitosa en su ámbito: la teoría cuántica describiendo
el micromundo y la relatividad general, el cosmos a gran
escala. Usan estrategias diferentes, imágenes de la realidad
diferentes, metáforas diferentes y métodos matemáticos
diferentes. La relatividad elude la naturaleza cuántica y la
teoría cuántica elude el espacio-tiempo curvo. La primera
no acepta el principio de incertidumbre y la segunda no
acepta el principio de equivalencia. Para la relatividad
general, la constante de Planck h es igual a cero; para la
teoría cuántica, la constante, gravitacional de Newton G
es igual a cero.
Obviamente ambas son aproximaciones. La construcción de
una teoría cuántica de la gravitación de la cual obtengamos
casos límites apropiados, a la teoría cuántica de campos
y a la relatividad, es la parte faltante de la revolución de
la física del siglo XX, y es tarea pendiente para la física
del nuevo milenio. Únicamente con esta teoría en la mano
podremos entender qué ocurre cuando lo muy pequeño
pero muy pesado aparecen en la misma situación física.
Tan sólo con una teoría cuántica de la gravedad podremos
hablar con propiedad de la naturaleza del Big Bang o de
la singularidad escondida en el centro de los agujeros
negros.
Los intentos y acercamientos a esa(s) teoría(s) sugieren
que en la llamada escala de Planck en tiempo y espacio
(L P ≈ 10–33 cm, tp ≈ 10 –43 seg), característica de los
fenómenos cuánticos gravitacionales, la naturaleza del
espacio y el tiempo es radicalmente distinta de lo que
observamos, tal vez cambie el número de dimensiones
del espacio. Lo importante, como siempre, estará en las
consecuencias y predicciones que una presunta teoría
cuántica de la gravedad proponga, y que nos permita
entender un poco mejor el universo que nos alberga, y
tal vez un poco mejor a nosotros mismos.
Una alternativa a la teoría cuántica de la gravedad, aún
no descubierta, es la teoría propuesta en los artículos en
esta edición de Ingeniare, la Teoría Total Simplificada
(TTS) que postula unificar la gravedad con el EM teniendo
como corolario fundamental la ecuación cuántica de
Dirac. Ver figura 1.
Con ello, aquí se propone todo un programa de unificación
en el cual el electromagnetismo quiral juega el rol central
[22]. La TTS se deriva de las ecuaciones originales de
Einstein-Hilbert Gµv = kTµv, donde el tensor de Einstein no
se modifica. El tensor EM en cambio es quiral y la masa
de las partículas es de naturaleza electromagnética. Para
el caso del electrón se tiene como consecuencia que por
primera vez se obtiene la ecuación de Dirac a partir de
ondas EM con el campo eléctrico paralelo espacialmente
al campo magnético [22-26].
En la figura 1 se muestra una interfaz o membrana de
separación donde ocurren solamente eventos y sucesos
cuánticos.
Hay dos regiones enantioméricas de un universo cerrado, o
un universo derecho y un universo izquierdo, relacionados
por un elemento de simetría PCT (paridad, carga, tiempo)
a lo largo de la interfaz. Las características principales
de ambas regiones enantioméricas están definidas en la
figura y representan un modelo con todos los atributos
requeridos por un vacío teórico. Lejos de la membrana
de separación son válidas las ecuaciones de Einstein-
Universo derecho: plasma de partículas
espacio-tiempo >0, Electrón: espín ±h/2,
masa me, carga –e¯, tiempo t>0
Rµv = Λclgµn
Radiación quiral EM y
ondas/partículas producidas
por la curvatura del factor T
líneas de tiempo:
Universo diestro
líneas de espacio
Membrana Espacio Tiempo
(cuerdas quirales) µ(ε) [1+ T∇x]
interfaz de vacío: Rµv = Λcgµv
Λc: constante cosmológica cuántica
Superficie espacio-tiempo
Universo izquierdo: plasma de
antipartículas espacio-tiempo <0,
Positron: espín h/2, masa mp,
carga +e+, tiempo t<0
Figura 1. Modelo del universo con dos regiones enantioméricas separadas por una membrana cuántica con Λc/ Λcl del
orden de 10120.
11
Hilbert G µv = kTµv, que son de naturaleza clásica. En la
vecindad de la capa de separación hay efectos cuánticos
y es válida la ecuación de Dirac para partículas en la
región enantiomérica diestra y para antipartículas en la
otra región [22-26].
Cada región puede ser considerada como un vacío, tiene
una constante cosmológica Λcl, que tiende a cero, la
membrana posee una constante cosmológica de naturaleza
cuántica tal que Λc / Λcl~10120.
ECUACIONES EINSTEIN-HILBERT
BAJO EL ENFOQUE QUIRAL
Descubierta por Hilbert, que desempeña un papel clave
para la obtención de las ecuaciones de gravitación en
el marco del principio de mínima acción [32]. Pero fue
Einstein sobre la base de la idea de la equivalencia de la
aceleración y la gravedad que formula la ley de conservación
general de la energía-impulso [31].
A través del principio de mínima acción (Axioma I
de Hilbert) y de la teoría de invariantes es que en este
artículo se considera un modelo quiral para el universo
considerando no la electrodinámica de Mie sino la
EQ.
Vamos a examinar con atención el enfoque de Hilbert
[32]. El Axioma I es tal que:
Desde los estudios de J. Earman y C. Glymour [27] se
hizo evidente que las ecuaciones de la relatividad general
de Einstein se descubrieron casi simultáneamente, pero
con diferentes métodos, por D. Hilbert y A. Einstein
[28-30].
Según el actual punto de vista Einstein y Hilbert, de
forma independiente uno de otro y de diferentes maneras,
descubrieron las ecuaciones del campo gravitacional [30].
En el trabajo de Einstein con Λcl = 0 las ecuaciones de
campo gravitacional son dadas por:
Las leyes de la física son los eventos definidos por la
función mundo H cuyos argumentos son
gµν , gµν l =
qs , qsl =
∂gµν
∂x l
∂qs
∂x l
, gµν lk =
∂2 gµν
∂x l ∂x k
, ( l , s = 1, 2, 3, 4)
(4)
(5)
Siendo que la variación de la integral
− gRµν = −κ (Tµν − gµν T ) (1)
Donde, como de costumbre, gµv es un tensor métrico;
R µv es el tensor de Ricci, k es la constante gravitacional
de acoplamiento, Tµ v es el tensor de densidad de
energía-momento para la materia, T es la traza de Tµv:
T = gµvTµv.
1
Hilbert, habiendo visto el término “traza” gµν T , también
2
lo introdujo en sus ecuaciones,
g ( Rµν − gµν R) = −
∂ gL
∂g µν
(2)
el término traza (en este caso 1 g R , donde la traza
2 µν
R = g µν Rµν ).
El enfoque de Hilbert es exacto donde todo es definido
por la función lagrangiana
12
H = R+ L (3)
∫H
g dω (6)
con
( g =| gµν |, dω = dx1dx 2 dx3 dx 4 )
(7)
es cero para cualquiera de los 14 potenciales gµv, qS. Bajo
el enfoque quiral la variación temporal ∂/∂t pasa a ser
∂/∂t
(1+T∇x)∂/∂t.
En cuanto a la función mundo, de acuerdo a Hilbert,
axiomas adicionales son necesarios para su definición
no ambigua. Si sólo las segundas derivadas de los
potenciales pueden entrar en las ecuaciones de la
gravitación, esto no cambia con EQ la función tiene
que tener la forma
H = R + LQ (8)
Donde R es un invariante del tensor de Riemann (curvatura
escalar de un sistema múltiple de cuatro dimensiones):
R = g µν Rµν , (9)
Rµν = ∂ν Γαµα − ∂α Γαµν + Γ λµα Γαλν − Γ λµν Γαλα , (10)
L Q es una lagrangiana quiral que es función de las
variables g µν , gµν , qs , qsk , T El factor quiral T > o
corresponde a la región enantiomérica diestra o nuestro
universo de materia, T < 0 a la región izquierda (ver
figura 1 del modelo para el universo) y Rµv es el tensor
de Ricci. Además de eso, suponemos más adelante que
LQ no depende de gµν .
A partir de la variación en los 10 potenciales gravitacionales
se tienen 10 ecuaciones diferenciales de Lagrange:
Luego, sobre la base del Teorema II, se tiene que la función
de Lagrange depende de las derivadas del potencial qv,
sólo a través del tensor Fµv, es decir,
LQ ( Fµν ) (15)
Fµν = ∂ µ qν − ∂ν qµ . (16)
Donde
Según el teorema II las cuatro identidades tienen lugar
en el invariante LQ:
Q
∂ gR
∂g µν
− ∂k
∂ gR
∂gkµν
+ ∂ k ∂l
∂ gR
∂gklµν
=−
∂ gLQ
∂g µν
(11)
Es fácil de ver de las ecuaciones anteriores que R y Rµv
que son derivadas de segundo orden entran en la métrica
linealmente. Todos los demás tensores se obtienen
como combinaciones de estos tensores con similares
propiedades.
Así, la ecuación de campo gravitacional sometido al
tensor electromagnético quiral es
µ
Se desprende de la identidad (17) que, en caso de las
ecuaciones de movimiento de un sistema material (14) se
sostienen, entonces la ley de la conservación covariante
se lleva a cabo:
Q
∇ µ Tνµ = 0  g R  = −κ T µν .

 µν
(12)
(18)
Si uno hace uso de las ecuaciones de la gravitación para
la identidad (17) se obtiene de la misma manera que de
las ecuaciones de Hilbert [31, 32]:
Q
µ
∇ µ Tνµ = Fµν  g LQ  + qν ∂ µ  g LQ  (17)




µ
µ
Fµν  g LQ  + qν ∂ µ  g LQ  = 0 



(19)
De aquí la ley de conservación covariante de la energíaimpulso se deduce naturalmente. El tensor densidad
 g R  contiene por construcción las derivadas

 µν
lineales de segundo orden.
Ecuaciones (19) tienen que ser compatibles con las
ecuaciones, que se deriven del principio de mínima
acción con la misma lagrangeana LQ. Sólo es posible en
el caso de que “ecuaciones generalizadas de Maxwell”
se autosustentan:
Las ecuaciones de Lagrange, de Hilbert bajo el enfoque
quiral son consecuencia del principio de mínima acción
(Axioma I de Hilbert). Así las ecuaciones de la gravitación
tienen la forma:
Q
1
g ( Rµν − gµν R) =  g R  = −κ T µν .

 µν
2
(13)
Se elige el invariante L Q en función de las variables
g µν , qσ , ∂ν qσ , T , por lo que se obtienen las ecuaciones
generalizadas de Maxwell
ν
 g L  = 0
Q

(14)
ν
 g L  = 0
Q

(20)
En el caso particular de que L Q = αQI. El segundo
término en Eq. (19) desaparece idénticamente y se llega
a las ecuaciones
µ
Fµν  g LQ  = 0 

(21)
De ello se deduce, por lo tanto, que si el determinante
Fµv no es cero, se tienen las ecuaciones de Maxwell
µ
 g L  = 0 . Esto está plenamente de acuerdo con
Q

el principio de mínima acción (Axioma I de Hilbert). De
esta forma, la ecuación de Maxwell son consecuencia de
las ecuaciones de gravedad y de las cuatro identidades
13
(21). Cabe señalar que originalmente Hilbert obtuvo la
ecuación de campo gravitatorio “no de un arbitrario
sistema material, sino que en base a la teoría de Mie [33]
con un lagrangeano en la forma L = α Q + f (q) donde
α es una constante y f(q) es el término no covariante
de Mie con
Q = Fµν Fλσ g µσ gνλ , q = qµ qν g µν , (22)
Este hecho hace que de dicha teoría no se obtiene una teoría
sustentable para el electrón, pero el método de Hilbert es
correcto en general, y es un excelente punto de partida
para la unificación propuesta en este trabajo.
En las referencias [22-26] se muestra específicamente
que la unificación de la gravedad con el EM de Maxwell
conocido en aquella época no era sustentable, por
cuanto el determinante del tensor de campo de Maxwell
siempre es cero. El hecho de que las ecuaciones de
la gravitación implican cuatro ecuaciones para el
sistema material, hace muy atractivo el método de
Hilbert por la sencillez y potencia del mismo, por lo
que se presta muy bien al enfoque de la gravitación
con electrodinámica quiral.
Basado en los argumentos expuestos en [34-36] y
suponiendo que el electromagnetismo quiral nace en la
interfaz de vacío (membrana de separación de las dos
regiones enantioméricas), y recordando que ∂ / ∂t se
transforma en (1+ T ∇×)∂ / ∂t , la ecuación (5) de Hilbert
reformulada es (1+ T ∇×)∂qs / ∂t .
Las tensiones electromagnéticas espaciales [34] son de
la forma
ε → ε 0 (1 + T ε µlν
∂
∂
), µ → µ0 (1 + T ε µlν
) (23)
∂xl
∂xl
Además se tiene que el potencial qs tiene las componentes
vectoriales q = A +T ∇ × A , donde A es el potencial
vector de Maxwell y el campo magnético en un espacio
curvado es B = ∇ × q .
El factor quiral T > 0 corresponde a la región enanciométrica
diestra donde se tiene un plasma de partículas, T < 0
a la región izquierda, donde existe un plasma de
antipartículas (ver figura 1 del modelo para el universo).
En el apéndice, las ecuaciones de Maxwell derivadas
plenamente en forma relativística de LQ son presentadas
además de las ecuaciones de onda en régimen quiral. En
la siguiente sección se discute esta teoría en los inicios
del Big-Bang.
14
LA ELECTRODINÁMICA QUIRAL Y LA
GRAVEDAD EN LA ERA DE PLANCK
Se sabe que el universo se está expandiendo debido
a la oscuridad de la noche. La dinámica dominante
del cosmos es, al parecer, una expansión repulsiva de
“antigravedad” a gran escala en contraste con la de
corto alcance de atracción de la materia en las galaxias
y las estrellas.
Este fenómeno es la respuesta a la paradoja de Olber, es
decir, el hecho de que el cielo, que se llena de un infinito
campo de estrellas y galaxias, no brilla como una estrella
sólida, sino que es predominantemente oscuro y tiene sólo
la radiación de temperatura de 2,7 ºK. El hecho de que el
universo se está expandiendo significa que las estrellas y
galaxias se alejan entre sí y hay un corrimiento al rojo,
por lo que el cielo de la noche es de por sí oscuro y frío,
en lugar de ser un campo brillante y caliente de polvo
de estrellas. Este hecho permite un universo de baja
temperatura donde la vida pueda florecer. Por lo tanto,
la expansión del universo puede ser vista como esencial
para la vida.
El fenómeno que provoca la expansión acelerada del
universo se conoce como “energía oscura” y se puede
decir que la causa el vacío. La densidad de la energía
oscura puede ser identificada como el término “constante
cosmológica” en las ecuaciones de la relatividad general.
Este término puede ser entendido a través del concepto de
un universo de plasma, donde la electrodinámica cósmica
desempeña un papel de igualdad con la gravitación que
es la conformación del cosmos y de sus estructuras. De
hecho, la gravitación ahora puede ser entendida como una
manifestación de la electrodinámica de un gran número
de partículas cargadas.
Este término cosmológico puede estudiarse en el contexto
de esta teoría TTS que determina el valor de la constante
de la gravitación.
La principal hipótesis del universo de plasma es que la
electrodinámica desempeña un papel igualitario con
la gravedad en la configuración de las estructuras del
cosmos. Es posible ampliar este principio incluso a la
microescala del cosmos, y considerar la posibilidad de
que incluso el vacío en sí mismo puede ser analizado
como un “plasma virtual” de partículas cargadas.
De esta manera es posible desarrollar un modelo del
universo que va desde la longitud de Planck al radio de
Hubble para el universo, como un tejido continuo de la
electrodinámica, y con los dos límites de longitud que
se correlacionan entre sí.
Esta unificación se basa en dos postulados: el EM se
unifica con la gravedad a la longitud de Planck, donde
el campo eléctrico Ep es paralelo al campo magnético
Bp. El segundo postulado establece que en el tiempo de
Planck la gravedad y los campos EM están unificados y
que se separaran, y se convierten en diferentes variables
a nivel de mesoescala. Aquí encontramos diferencias con
[37]. Esta teoría puede predecir campos de gravedad y el
valor de la constante de gravedad. Asimismo, se puede
predecir el valor del tiempo de Hubble y temperatura de
la radiación cósmica de fondo. Según esta teoría, la razón
de masa de electrones a protones Rm asume un rol central.
Sin embargo, es evidente que el valor de Rm depende
de la fuerza fuerte, y los nucleones, que constituyen la
mayor parte de la masa visible del universo, se derivan
de los quarks. Esto es esencial para cualquier teoría de
unificación [38]. Aquí, en la TTS, la EQ es como una
extensión de la teoría de Sahkarov para la gravedad y el
origen tanto de protones y electrones, pero la unificación
nace en la era de Planck con un plasma cosmológico en
nuestro universo diestro.
La teoría TTS es un intento de crear una teoría
geométrica para resolver el problema de Einstein-Dirac
de la unificación de la gravedad y el EM. La teoría por
ahora se limita a los protones y electrones. La teoría
está todavía en un estado temprano de desarrollo, es
decir, que se describe como un “modelo de Bohr” de
la unificación, por analogía con la mecánica cuántica
y el modelo del átomo de hidrógeno, y se basa en una
extensión de los trabajos de Einstein y Kaluza. El
aspecto separado de la gravedad y el EM relativos a
los protones y electrones proviene desde la escala de
Planck y se produce con la aparición de una dimensión
quiral como un equivalente de la quinta dimensión de
Kaluza-Klein. La teoría TTS comienza con el principio
de acción de Hilbert, que permite la obtención de las
ecuaciones fundamentales de los campos de vacío con
la extremización de la acción integral
H = (16π G )−1 ∫ ( R − 2 Λ) − gdx 4 24)
Donde es la curvatura escalar de Ricci de la relatividad
general, Λ es la constante cosmológica, G es la constante
de la gravitación de Newton. Aquí, la acción integral para
un campo cuántico de partículas con espín y masa, m
cuando LEM procede de una EQ y Ep = iBp. Este campo
produce un tensor de tensión de la forma p = pc2, es decir,
una presión negativa, que a su vez impulsa una explosión
del cosmos [39, 40]. Si suponemos que el cosmos es una
entidad electrodinámica, entonces sería natural que, en
un universo en expansión rápida, una especie de reacción
de la ley de Lenz se debe haber producido para frenar la
expansión cósmica. Esto también puede ser considerado como
un “principio cósmico de la mínima acción”. Esta reacción
sería la aparición de campos EM y de materia que presentan
una densidad de energía positiva que dramáticamente frena
la explosión inicial del universo. Este escenario es, de
hecho, el escenario inflacionario, que se considera ahora
el principal modelo de la cosmología. Sin embargo, puesto
que nuestro objetivo final es la modificación tecnológica
práctica de la gravedad, vamos a examinar ahora la relación
de la gravedad y el EM en detalle.
En la teoría de la relatividad general (RG) de Einstein,
la gravedad surge de la geometría del espacio tiempo
determinado por las propiedades del tensor métrico gij.
El límite newtoniano se recupera donde el espacio no es
muy curvado y el potencial de Newton φ es en realidad
parte del elemento diagonal principal del tensor métrico
gtt = −1 − 2φ / c 2 . Desde el punto de vista de la TTS, la
gravedad surge de la electrodinámica quiral. Sobre la
base de esto, parece ser que una buena generalización
para modelar la gravedad es que el tensor métrico de la
RG es en realidad un tensor EM normalizado. Es decir,
los campos EM no sólo son la curvatura de la métrica,
sino que la métrica misma. Esto significa que el espacio
tiempo es en realidad un vasto mar de radiación ultrapoderoso de EM.
Con el fin de satisfacer los postulados de la TTS debemos
tener una generalización covariante y física de campo
para el vacío
g ij = 4 Fik Fjk / F µν Fµν = 4 Fik Fjk / T0 (25)
Donde Fik es el tensor de Faraday y T0, es la covariante
generalización normalizada del escalar de tensión escalar.
La forma de esta expresión para el tensor métrico es
determinada por el primer postulado TTS, esto es, para
que un campo EM ultrafuerte llene el cosmos el tensor
de Maxwell es
Tij = 1 / 4π ( Fij Fjk − 1 / 4 gij F µν Fµν ) (26)
que debe desaparecer en todas partes. Así, en la TTS, un
poderoso campo EM determina la geometría del espacio,
pero en sí no es detectable directamente, sino que por
ser tan poderoso se anula a sí mismo. Cabe señalar que
Fjk , la forma mixta del tensor de Faraday, es a menudo
escrito como gij F ik. Físicamente, sin embargo, el objetivo
principal de la TTS es demostrar que la gravedad, los
campos EM, y, por tanto, la geometría, son unificadas
y son partes de una relación cíclica. Esto significa que
en la TTS, EQ implica la geometría y viceversa, de
15
manera que para empezar con uno en lugar del otro es
una cuestión de convención, y para reclamar que uno
es superior y más fundamental que el otro no parece
razonable. La EQ es dada por
LQ = ∫ κ gµν − g EQ ⋅ BQ dx 4 ,
EQ ( BQ ) = (1 + T ∇×) E ( B )
(27)
la densidad de energía neta del espacio puede ser ligeramente
negativa con este lagrangeano si T tiene un valor apropiado.
Un mecanismo similar fue propuesto por primera vez en
[40] y se denomina “Zeldovich reacción”. En este artículo
se deriva el valor de G como la “elasticidad de la métrica
del espacio” con el supuesto de que la longitud de Planck
Tp = (Gh/c3)1/2, donde Tp es el factor quiral de nuestro
R-Universo,  es la constante de Planck. El valor de la
integral se determina por la frecuencia de corte cerca de la
frecuencia de Planck ωp = c/Tp. Numéricamente este valor
de T es igual al radio de Planck [39, 40]. Esta conexión
entre la gravedad y el campo EQ alienta la posibilidad de
que la gravedad pueda un día ser modificada directamente
por medios externos.
La manera más sencilla de obtener una energía oscura
o la constante cosmológica es permitir que E→iB y
esto genera una constante cosmológica y, por tanto, un
universo en expansión.
Teniendo en cuenta la hipótesis de un universo de plasma
donde domina la electrodinámica quiral, suponemos la
existencia de modos quirales que componen la energía
oscura. En la escala de Planck, la incertidumbre de
Heisenberg permite que se generen masas Mp, que tienen
una longitud de onda de Compton igual a su radio de
Schwartzchild
GM P
/c
2
= rP = TP =  / 2 M P c = G  / 2c3 (28)
A la longitud de Planck, los horizontes de evento de los
agujeros negros aparecen y desaparecen en un período
de Planck, y la distinción topológica entre estar dentro
y fuera de un evento horizonte desaparece. Por breves
instantes de un período de Planck, las partículas de
antimateria pueden interaccionar con las partículas
ordinarias. Esto produce la cuantización de la carga
e = khc/2πT. Sin embargo, el tamaño de T debe ser
del orden de la longitud de Planck, para dar el correcto
valor de la carga y las masas son del orden de la masa de
Planck. Sin embargo, la existencia de la escala de masas
de protones y electrones significa que el tamaño efectivo
de la dimensión compacta debe ser mucho mayor, es
16
decir, del orden de la longitud de onda de Compton de
un protón o el radio clásico electrónico. Esto puede ser
entendido conceptualmente como un “renormalization”,
efecto debido a la energía negativa de la formación de
la dimensión quiral.
La teoría de TTS así permite, por un pequeño aumento de
la dimensionalidad, la aparición explícita de los campos
EM, y las partículas con carga y masa, para protones
y electrones, junto con la gravedad, desde un principio
variacional con σ = mp/me)1/2 = 42.85003.
Esto constituye una descripción muy básica del cosmos
como un todo, cuyos principales componentes más
conocidos son los protones y electrones. Si se concibe
que el principio de la acción asume un campo EQ no
masivo, entonces la aparición de la quiralidad permite
la captura o la dispersión de quantas no masivos que
crean masa en reposo, carga, espín y por lo tanto las
partículas. La captura de la energía EM quirales en una
dimensión compacta puede ser concebida como una
imagen compactada del espacio-tiempo. El tamaño de
la dimensión compactada puede considerarse que sea del
orden del radio clásico del electrón, re = e2/4πε0 m 0 c 2,
que es el tamaño aproximado del protón. El carácter de
la dimensión compacta debe ser una imagen global del
espacio-tiempo, esto significa que el protón, es decir, el
espacio, como partícula, debe tener tres subdimensiones
para satisfacer la condición de radio clásico y también
para la neutralidad del cosmos. Esta combinación es
satisfecha por la actual teoría de los quarks, donde
el protón está formado por tres quarks que satisfacen
qx = q y = 2e/3 y qz = –e/3, es decir,
qx + q y + qz = e(2 / 3 + 2 / 3 − 1 / 3) = e (29)
qx 2 + q y 2 + qz 2 = e 2 (4 / 9 + 4 / 9 + 1 / 9) = e 2 (30)
Donde qx, qx, qz son las cargas de los quarks que componen
los protones. Por lo tanto, una imagen global del espaciotiempo, sujeta a la simetría de rotación, significa que la
dimensión compacta (T) tiene el carácter de un radio
o un intervalo de tiempo y, por tanto, actúa como un
escalar, pero puede tener tres espacios internos –como
grados de libertad. Las condiciones: la suma de las cargas
de los quarks y la de suma de sus cuadrados significa
que deben satisfacer una simetría de rotación SO (3).
Los protones, por tanto, compuestos de quarks, son,
pues, fundamentales en la TTS para nuestro universo
diestro.
Si uno se concentra por un momento en la acción de
Hilbert-Einstein, sobre la expansión de la EQ de Lagrange
y de la selección de aquellas partes que desaparecen en el
marco de la caída libre en el modelo de TTS, obtenemos
(en unidades ESU).
R
E 2 − B2
+
16π G
8π
2
2
S
g
=−
+ E = iB
2π G u c 2
(16π G )−1 K field =
Donde Fαβ es el tensor de campo electromagnético y jα es
la densidad de corriente en 4 dimensiones. Claramente,
en un sistema inercial local las ecuaciones anteriores se
reducen a las ecuaciones estándares en el vacío. Para
simplificarlas, se introduce un sistema de coordenadas
ortogonal-temporal en el cual: gλ4 = 0.
Se introducen los tensores antisimétricos Hαβ y Bαβ tal
E = iB
(31)
que F αβ = H αβ / − g44 = Bαβ y los vectores D α y E α
por F α 4 = − Dα / − g44 = E α / g44 .
0
donde S = 0, se asocia a los campos de la gravedad. Esta
expresión cumple el principio de equivalencia, porque ambos
términos desaparecen en un sistema de caída libre.
Las expresiones covariantes correspondientes son:
Fαβ = gαγ gβδ F γδ = gαγ gβδ Bγδ =
H αβ
= Bαβ − g44
APÉNDICE: ELECTRODINÁMICA QUIRAL
Comúnmente, el electromagnetismo de Maxwell es
tratado en una aproximación lineal que en muchos
casos es suficiente para explicar los resultados de
los experimentos. La premisa que en este trabajo de
unificación se plantea es que una onda electromagnética
propagándose a través de un medio es influenciada por
este último y viceversa.
La idea anterior se parece al principio de Mach. Se le
recuerda al lector que dicho principio sugiere que la
masa de un cuerpo es debida a la influencia de todas
las estrellas del universo en ese cuerpo. Expresada en
términos matemáticos, esta dependencia toma la forma
m(x) = ξ[1/T(x)] donde m(x) es la masa de partícula, ξ
una constante de acoplamiento y 1/T es la masa generada
por el campo a través de la quiralidad. La idea central
entonces es no considerar la masa de una partícula como
una cantidad fija e intrínseca, sino más bien pensarla
como una cantidad variable dependiente del campo en
el cual se mueve.
En este apéndice se construye la electrodinámica en espacio
curvo. Para ello se parte con las ecuaciones de Maxwell
en un Sistema Arbitrario de Coordenadas.
∂γ Fαβ + ∂α Fβγ + ∂β Fγα = 0 1
g
∂β
(( g ) F ) = J αβ
α
γδ
β
Fα 4 = gαγ g4δ F = gαβ − g44 D = − g44 Dα = Eα Esto permite escribir las ecuaciones de Maxwell en la
forma
1
−g
∂β
(
∂γ Bαβ + ∂α Bβγ + ∂β Bγα = 0
)
1
−g
1
− gH αβ −
∂α
−g
(
∂t
(
)
− gDα = ρuα
)
− gDα = ρ
Una manera de escribirlas en una forma más familiar,
se introducen los vectores duales correspondientes a los
tensores respectivos por la prescripción estándar, dando
como resultado
B1 =
H1 =
1
−g
1
−g
1
B23 , B 2 =
−g
H 23 , H 2 =
B3 1, B3 =
1
−g
H 31 , H3 =
1
−g
B12
1
−g
H 12
(1)
Finalmente haciendo las sustituciones respectivas se tienen
las ecuaciones vectoriales de Maxwell sin cargas
(2)
∇×E =−
1
−g
∂t
(
)
−gB , ∇ ⋅ B = 0 (3)
17
∇×H =
1
−g
∂t
(
)
−g D , ∇ ⋅ D = ρ
(4)
La conexión entre las ondas electromagnéticas en un
medio y las mismas ondas en el vacío pero en un espacio
curvo emerge claramente. Si g no depende del tiempo y si
se tiene µ = ε = –(–g44)-1, entonces el campo, responsable
por la curvatura del espacio, se comporta como un
medio con la permitividad dieléctrica y susceptibilidad
magnética dada por la ecuación anterior. La conjetura
que se plantea en esta investigación es que ε, 0 y µ0 en
el espacio curvado son definidos como los operadores:
ε 0 (1 + T ∇×) y µ0 (1 + T ∇×) donde T es el factor quiral
que permite la torsión del campo dando lugar a la creación
de las partículas.
0)
J ( 0 ) = ε T (−2 B,(00
− T ∇ 2 E,(00 ) ) (11)
Las ecuaciones de Maxwell rescritas anteriormente son
formalmente similares a las ecuaciones de Maxwell en
un medio normal con densidad de corriente J(0) y densidad
de carga ρ(0) = 0 que obedece la ecuación de continuidad
de la carga ∇ ⋅ J ( 0 ) + ρ,0 ( 0 ) = 0 . Se llamará J(0) a la
corriente quiral en el marco de referencia en reposo. En
un sistema arbitrario S, las ecuaciones de Maxwell pueden
ser escritas ∂a = a , [41].
,α
∂x α
F,ααλ − ( µε − 1) F,γλα uα uγ = µ J λ (12)
Las ecuaciones de campo
Fαβ ,γ + Fγα ,β + Fβγ ,α = 0
(13)
En un sistema de referencia, donde el medio está en
reposo, las relaciones constitutivas quirales de BornFederov son:
respectivamente. Aquí u α es la velocidad uniforme del
medio, Fαβ es el tensor del campo electromagnético
con componentes F0 i = Ei Fij = −ε ijk Bk . También se
introduce el dual del tensor de campo
D(0) = ε E (0) + ε T ∇ × E (0)
B ( 0 ) = µ H ( 0 ) + µT ∇ × H ( 0 )
(5)
en el sistema en reposo So del medio. Se restringe el
estudio a medios homogéneos y no dispersivos donde ε,
µ y T son constantes. Las ecuaciones de Maxwell en este
marco, en ausencia de cargas son
∇ ⋅ D(0) = 0
∇ ⋅ B (0) = 0
∇ × H ( 0 ) = D,(00 ) ∇ × E ( 0 ) = − B,(00 ) (6)
(7)
∂a
donde a,0 =
. Las ecuaciones de Maxwell pueden ser
∂t
escritas en términos de los campos E(0) y B(0)
∇ ⋅ E ( 0 ) = 0, ∇ × B ( 0 ) = µ J ( 0 ) + µε E,(00 ) (8)
∇ ⋅ B ( 0 ) = 0, ∇xE ( 0 ) = – B,0(0) (9)
0 ) (10)
K ( 0 ) = 2 E,(00 ) + T ∇ × E,(00 ) = 2 E,(00 ) − T B,(00
También se tiene
18
1
Gαβ = ε αβγδ Fγδ
2
con componentes G0 i = Bi Gij = +ε ijk Ek , de modo que
la corriente quiral puede ser escrita como
J λ = ε T ε αλρσ uα K ρ ,σ .
(14)
..
con ε 0123 = +1 y K ρ = 2 F ρ − T G ρ .
Aquí usamos la definición: F ρ = F ρα uα ; G ρ = G ρα uα y
donde la operación punto (.) es definida por a = uα a ,α , la
cual se reduce a la derivada temporal ordinaria en el medio
en reposo. En este marco Fi se reduce a las componentes
del campo eléctrico (y F0 desaparece) y Gi a las del campo
magnético (y Go desaparece).
La corriente quiral se puede escribir en la forma
..
donde J ( 0 ) = ε T ∇ × K ( 0 ) con
.
J α = ε T (−2 Gα + Th µν F,αµν ) (15)
donde h µv está relacionado al tensor métrico gµv por
h µυ = g µυ − u µ uυ
Se hace notar que la ecuación de continuidad es considerada.
Jλ,λ = 0. Efectuando una contracción con uλ y notando que
uλ Jλ = 0 se tiene Fα ,α = Fαλ ,α uλ = 0. Este resultado no es
nada más que ∇. E ( 0 ) = 0, ∇ × B ( 0 ) = µ J ( 0 ) + εµ E,0 ( 0 )
en el medio en reposo.
1
F αλ = − ε αλβδ Gβδ 2
λα
Si se introduce esta relación en Fαλ , α − (εµ − 1) F ,
γ
λ y haciendo la contracción con ε
ρλσγ se
γ uα u = µ J
.
.
α
εT 2 G α = Th µν F,µν
En un marco de referencia en reposo Sº, esta ecuación
se simplifica a la ecuación de Beltrami si hacemos

T≡
,
mc
obtiene
..
α
J α = εT ( −2 G α + Th µν F,µν
) =0
..
El inverso de G βδ es
o sea
∇ × E (0) +
2 (0)
E =0
T
(18)
.
λ
λ
Gσγ , ρ + Gρσ ,γ + Gγρ ,σ − ( µε − 1)ε ρλσγ F = µε ρλσγ J donde se ha usado la identidad
ε λαβµ ε λρσγ = −δ ρα δ βσ δγµ + δσα δ βρδγµ − δγα δ βρδσµ






c on σγ  = σγ − γσ . L a e cu a c ión homogéne a
Fαβ ,γ + Fγα ,β + Fβγ ,α = 0 se transforma en G,γσ
γ = 0 que
inmediatamente sigue de la contracción de la ecuación
homogénea con ελσαβ. Para obtener la ecuación de onda
del tensor de campo se diferencia Fαβ ,γ + Fγα ,β + Fβγ ,α = 0
con respecto a γ, y se usa F αλ ,α − (εµ − 1) F λα ,γ uα uγ = µ J λ
para obtener
..
γ
Fαβ
,γ − ( µε − 1) F αβ = − µ ( Jα ,β − J β ,α ) (16)
en. este result ado f ina l se ut il iza la relación
Fαβ = uγ Fαβ ,γ = Fα.. ,β − Fβ ,α . La ecuación de onda
Fαβ ,γ γ + (εµ − 1) Fαβ = − µ ( Jα ,β − Jβ ,α ) es el resultado
fundamental de este trabajo de investigación que
permite la unificación del electromagnetismo con la
gravitación. Esta ecuación de onda de segundo orden
en tiempo y espacio permite obtener la propagación
de gravitones si ( Jα ,β − Jβ ,α ) = 0 y de fotones si T ≡ 0
respectivamente. Para examinar el caso de gravitones
con spin 2 el primer miembro de la ecuación de onda
(16) es
..
Fαβ ,γ γ + (εµ − 1) Fαβ = 0 (17)
que corresponde al caso de corriente quiral igual a cero,
esto es
es trivial obtener de la ecuación de onda (17) con ∂ / ∂t = iω ,
−1/ 2
y la velocidad de la luz dada por c = (εµ )
la expresión
ω
k0 = tal que mcT = 2 . Esto corresponde a partículas
c
con spin 2. Se observa además que de la ecuación
∇ × E ( 0 ) = − B,(00 ) = −iω B ( 0 ) 1 se tiene entonces que
E ( 0 ) = iω B ( 0 ) , o sea los campos son paralelos en el espacio
tridimensional con el vector de Poynting E ( 0 ) × B ( 0 ) = 0.
Esto implica una gran dificultad en detectar este tipo de
partículas con detectores usuales de radiación.
El caso de ondas electromagnéticas normales es analizado
con la ecuación
∇ × ∇ × (1 − k02T 2 ) E − 2 k02T ∇ × E − k02 E = 0 (19)
Si T = 0 se obtiene la usual ecuación de onda en un medio
normal y homogéneo que se encuentra en los textos de
electromagnetismo, ∇ × ∇ × E − k02 E = 0 , ondas que
al ser tratadas como partículas se tiene que el spin es
igual a uno.
Si k0 T ≤ 1 se obtienen las ondas quirales circularmente
polarizadas que se propagan en medios electromagnéticos
complejos y en medios biológicos (por ejemplo, ondas en
el tejido cerebral debido a las microondas de teléfonos
celulares).
Si k0 T ≥ 1 o mucho mayor que uno, se obtiene una ecuación
de tipo Beltrami que entre otras situaciones físicas puede
modelar las ondas en una estrella de neutrones, explicar
las llamaradas solares donde la corriente es paralela al
campo magnético. Esta relación también permite obtener
el radio del universo si T =  / mc y µ es la masa del fotón
en el espacio curvado de Einstein.
19
Usando un modelo de fluido de fotones, permite obtener
rotB = λ B que puede calcular la distancia del Sol a cada
uno de los planetas del Sistema Solar.
Además puede explicar: la generación de bolas de luz, la
formación de galaxias en espiral y las ondas EM quirales
como autoestados de moléculas de ADN.
Se puede demostrar rigurosamente que con ondas
electromagnéticas donde E es perpendicular a B,
( E ⊥ B ) la ecuación tensorial de Einstein, con el tensor
de Maxwell Tµv es
Rµν
1
− gµν R = −κ Tµν 2
(21)
Esta es la razón del porqué Einstein no pudo obtener
la anhelada unificación del electromagnetismo y la
gravitación.
Esta formulación es un enlace entre la Teoría Cuántica y
la Relatividad General, siendo clave el concepto de campo
Beltrami como fundamental en la creación de partículas.
En la Física actual, ambas teorías están profundamente
cimentadas en marcos espacio-tiempo distintos. La primera
en el espacio-tiempo de Minkowski, y la segunda en el
espacio-tiempo curvado.
Tal como A. Wheeler lo hizo notar, los intentos de
unificación y el desafío han sido la introducción de
la mecánica cuántica con spin 1/2 en la Relatividad
General, por un lado, y la introducción de la curvatura
en Mecánica Cuántica
por otro. La ecuación de onda
..
Fαβ ,γ γ (T ) + (εµ − 1) Fαβ (T ) = − µ ( Jα ,β (T ) − Jβ ,α (T )) que
es la generalización de la ecuación de Klein-Gordon
permite esta conexión, si la transformamos poniendo en
evidencia el factor de spin 1/2.
20
(22)
.
..
λ
Jep
= ε ( 1 − O  F λ − 2T G λ ) (23)
O ≡ 1 + T 2uα (u,αα ),α (24)
donde
manipulando las ecuaciones (22), (23) y (24), se puede
obtener la ecuación de onda
gµν = gµν + Fµν ds 2 = gµν dx µ dxν = gµν dx µ dxν λ
O( F,ααλ − ( µε − 1) F,γλα uα uγ ) = µ Jep
λ
donde el tensor corriente, Jep
, corresponde a la corriente
quiral del electrón (T>0) o positrón (T<0)
(20)
Ya el d et e r m i n a nt e d e F µ v e s ig u a l a c e r o
( det Fµν = 2 ( E ⋅ B )2 = 0) se tiene que
Es transparente al campo de manera que estos modos de
propagación no permiten la unificación ya que al hacer lo
hecho por primera vez por Einstein para el caso usual de
campos de Maxwell ( E ⊥ B ) , o sea (E·B = 0)
L a e c u a c i ó n t e n s o r i a l d e M a x w e l l (1 2)
F,ααλ − ( µε − 1) F,γλα uα uγ = µ J λ , se puede transformar a
..
OFαβ ,γ γ (T ) + O(εµ − 1) Fαβ (T ) =
= − µ ( J(ep)α ,β (T ) − J(ep) β ,α (T ))
(25)
Explícitamente la ecuación de onda puede ser linealizada
siguiendo la genial línea de raciocinio de P. Dirac (que a
partir de la ecuación de segundo orden de Klein Gordon
desarrolla las matrices α, β para obtener la ecuación
de primer orden de Dirac, donde el spin ½ aparece en
forma precisa).
Siguiendo el mismo raciocinio de Dirac, hacemos O=0,
y al integrar una vez la ecuación (25) se tiene
.
..
F λ = 2T G λ (26)
que en el sistema de referencia del electrón o positrón
corresponde a
∇ × E (0) +
1 (0)
E =0
2T
donde se ha hecho k0 T = ±1. Aquí se considera el principio
de incertidumbre de Heissenberg pT = ±  / 2 = mcT . Se
hace notar que este resultado general se obtiene también
de la ecuación (19) que permite obtener relaciones
isomórficas que conducen a una manera inédita de obtener
la ecuación de Dirac para una partícula elemental como el
electrón o positrón a partir de la obtención de la ecuación
de Beltrami (free force). La onda inicialmente con spin
1 y energía ω es transformada en campos Beltrami
de spin +1/2 y –1/2. En otras palabras, si el sistema de
ecuaciones de Maxwell es multiplicado por la matriz
de Pauli permite obtener en forma rigurosa y original
la Ecuación de Dirac. Esta deducción se muestra en el
artículo correspondiente.
Usando álgebra cuaterniónica en un sistema de referencia
fijo a la partícula (onda de luz curvada sobre sí misma),
de la ecuación anterior se obtiene sin aproximaciones la
ecuación de Dirac. En ella aparece en forma explícita el
spin de la partícula, cuando k0T = 1. O sea si T =  / 2mc
entonces
k0 T = ω T / c = 1 = ω / 2mc 2 ⇒ E = 2mc 2 .
De esta forma, a partir del escalar T, se obtiene la ecuación
fundamental de Einstein entre materia y energía. El escalar
T se encuentra implícito en
universo derecho y uno universo izquierdo, relacionados
por un elemento de simetría PCT (paridad, carga, tiempo)
a lo largo de la interfaz. Las ecuaciones de EinsteinHilbert fueron estudiadas bajo el enfoque quiral y se
han analizado la electrodinámica quiral y la gravedad
en la era de Planck.
AGRADECIMIENTOS
Se agradecen las fructíferas discusiones sobre el tema
con los colegas del Instituto de Alta Investigación y de la
Escuela Universitaria de Ingeniería Eléctrica - Electrónica
de la Universidad de Tarapacá, Arica, Chile.
REFERENCIAS
..
Fαβ ,γ γ (T ) + (εµ − 1) Fαβ (T ) = − µ ( Jα ,β (T ) − Jβ ,α (T )) [1]
H. Torres-Silva. “Chiroplasma surface wave”.
Electromagnetic of Chiral Bi-Isotropic and BiAnisotropic Media. CHIRAL ’96. Proceeding of
Nato Series, Moscú. 1996.
[2]
H. Torres-Silva, P. Sakanaka and N. Reggiani.
“The Effect of Chirality on a Plasma Media”. Rev.
Mex. de Física. Vol. 42, pp. 989-1000. 1996.
[3]
H. Torres-Silva and C. Villarroel González.
“Electromagnetic properties of a Chiral-Plasma
Medium”. Pramana-Journal of Physics. Vol. 49,
pp. 431-442. 1997.
[4] H. Torres-Silva. “Chiroplasma surface wave”.
A. Pr iou, editor: Advances in Complex
Electromagnetic. Materials. Vol. 28, pp. 249-258.
Kluwer Academic Publishers. 1997.
[5]
H. Torres-Silva. “Electromagnetic Waves in a
Chiral Plasma”. Journal of the Physical Soc. of
Japan. Vol. 67, pp. 850-857. 1998.
[6]
H. Torres-Silva. “Propagación de ondas pulsadas
en un chiroplasma magnetizado”. Rev. Mex. de
Física. Vol. 44, pp. 53-58. 1998.
[7]
H. Torres-Silva. “Convective Instabilities of
Transverse Wave In Magnetized Chiral Media”.
J. Plasma Res. Vol. 1, p. 395. 1999.
[8]
H. Torres-Silva y M. Zamorano Lucero. “Ecuación
de Onda de Schrödinger para una fibra Óptica
Chiral”. Revista Mexicana de Física Vol. 46,
pp. 62-66. 2000.
Si T = 0 se tiene el espacio plano de cuatro dimensiones
de Minkowski, base del electromagnetismo de Maxwell
y de la Relatividad Especial.
Recientes modelos que incluyen supercuerdas cuánticas son
caminos alternativos que consideran partículas extendidas,
es decir, no puntuales que conducen a operadores espaciotiempo no diferenciables y a geometría no conmutativa,
pero la teoría propuesta aquí es más económica, más
simple en dimensiones y en la potente idea de que los
campos electromagnéticos con E  B son la verdadera
fuente para la gravitación
CONCLUSIONES
Como una alternativa a la teoría cuántica de la gravedad,
aún no descubierta, la TTS ha sido propuesta, que postula
unificar la gravedad con el EM teniendo como corolario
fundamental la ecuación cuántica de Dirac.
En este programa de unificación en el cual el
electromagnetismo quiral juega el rol central, la TTS ha
sido derivada de las ecuaciones originales de Einstein
-Hilbert Gµν = κ Tµν , donde el tensor de Einstein no ha
sido modificado. El tensor EM en cambio es quiral y la
masa de las partículas es de naturaleza electromagnética.
Para el caso del electrón se tiene como consecuencia que
por primera vez se obtiene la ecuación de Dirac a partir de
ondas EM con el campo eléctrico paralelo espacialmente
al campo magnético. Como modelo del universo se
propuso una interfaz o membrana de separación entre
dos regiones enantioméricas de un universo cerrado, o un
21
[9] A. Assis and H. Torres-Silva. “Relation between
Maxwell Equations and Weber Force”. Pranama
Journal of Physics. Vol. 54 Nº 6, pp. 1-12. 2000.
[10]
H. Torres-Silva and M. Zamorano Lucero. “Polarized
spatial solitons in cubic chiral materials”. PIER
2002. Progress In Electromagnetics Research.
Cambridge, Massachusetts, USA. July 1-5.
2002.
[20] C.W. Misner, K.S. Thorne and J.A. Wheeler.
Gravitation, Ch. 13, Freeman, pp. 310-311. 1973.
[21]
[11] C. Villarroel González y H. Torres-Silva. “Difracción
en el borde de un semiplano inmerso en un medio
quiral bianisotrópico”. Rev. Mex. de Física. Vol. 47,
pp. 136-141. 2001.
[12]
[13]
H. Torres-Silva and M. Zamorano Lucero. “Chiral
effects on optical solitons”. Mathematics and
Computers in Simulations. Vol. 62, pp. 149-161.
2003.
M. Zamorano Lucero y H. Torres-Silva. “Efecto
de la quiralidad sobre solitones polarizados en un
medio anisotrópico”. Revista Mexicana de Física.
Vol. 49 Nº 1, pp. 20-27. 2003.
[14] H. Torres-Silva and M. Zamorano Lucero. “Nonlinear polarization and chiral effects in birefringent
solitons”. Pramana Journal of Physics. Vol. 62
Nº 1, p. 37. 2004.
[15]
M. Zamorano Lucero y H. Torres-Silva. “Sar
inducido en un modelo bioplasmático quiral por
radiación de teléfonos celulares”. Revista Mexicana
de Física. Vol. 51 Nº 2, pp. 209-216. 2005.
[16] H. Torres-Silva. “FDTD chiral brain tissue model
for specific absorption rate determination under
radiation from mobile phones at 900 and 1800
MHz”. Phy in Med. and Biol. Vol. 51, pp. 16611672. 2006.
[17]
[18]
[19]
22
L.A. Ambrosio, H.E. Hernández and H. Torres-Silva.
“Guided modes in metamaterial slabs”. Ingeniare.
Rev. chil. ing. Vol. 14 Nº 3, pp. 291-298. 2006.
H. Torres-Silva, C. Villarroel González and F.
Jiménez-Muñoz. “Electromagnetic waves at
the plane boundary between two chiral media”.
Ingeniare. Rev. chil. ing. Vol. 15 Nº 1, pp. 101-110.
2007.
T. Appelquist. Modern Kaluza-Klein Theories.
Frontier in Physics. Addisson-Wesley. 1987.
R. Wald, General Relativity. Cap. 14. Ed. University
Chicago Press. 1984.
[22] H. Torres-Silva. “A new relativistic field theory
of the electron”. Ingeniare. Rev. chil. ing. Vol. 16
Nº 1, pp. 111-118. 2008.
[23] H. Torres-Silva. “Spin and relativity: a semiclassical
model for electron spin”. Ingeniare. Rev. chil. ing.
Vol. 16 Nº 1, pp. 72-77. 2008.
[24] H. Torres-Silva. “A metric for a chiral potential
field”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 91-98. 2008.
[25] H. Torres-Silva. “Maxwell’s theory with chiral
currents”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 31-35. 2008.
[26] H. Torres-Silva. “The close relation between the
Maxwell system and the Dirac equation when the
electric field is parallel to the magnetic field”.
2008.
[27] J. Earman and C. Glymour. “Einstein and Hilbert:
Two Months in the History of General Relativity”.
Archive for History of Exact Sciences. Vol. 19,
p. 291. 1978.
[28] L. Corry, J. Renn and J. Stachel. “Belated Decision
in the Hilbert-Einstein Priority Dispute”. Science.
Vol. 278, p. 1270. 1997.
[29] J. Renn and J. Stachel. Hilbert’s Foundation
of Physics: From a Theory of Everything to a
Constituent of General Relativity. Preprint of
Max-Planck-Institut für Wissenschaftsgeschichte.
Nº 118. 1999.
[30]
V.P. Vizgin. “On the discovery of the gravitational field
equations by Einstein and Hilbert: new materials”.
Physics-Uspekhi. Vol. 44 Nº 12, p. 1283. 2001.
[31]
A. Einstein. “The Collected Papers of Albert
Einstein”. Eds. R. Schulmann. Princeton, N.Y.
Princeton Univ. Press. Nº 8. 1998.
[32]
T. Sauer. “The Relativity of Discovery: Hilbert’s
First Note on the Foundations of Physics”. Archive
for History of Exact Sciences. Vol. 53, pp. 529-575.
1999.
[33]
A. Einstein. “Do Gravitational Fields Play an
Essential Part in the Structure of the Elementary
Particles of Matter? The Principle of Relativity”.
Dover, pp. 191-198. 1952.
[34] H. Torres-Silva. “Extended Einstein’s theory of
waves in the presence of space-time tensions”.
2008.
[35] H. Torres-Silva. “Einstein equations for tetrad
fields”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 85-90. 2008.
[36] H. Torres-Silva. “Chiral universes and quantum
effects produced by electromagnetic fields”.
2008.
[37] J.E. Brandenburg. “A model cosmology based
on gravity - electromagnetism unification,
Astrophys”. Space Sci. Vol. 227 Nº 1/2, pp. 133144. 1995.
[38] H.E. Puthoff. “Gravity as a zero-point fluctuation
force”. Phys. Rev. A. Gen. Phys. Vol. 39 Nº 5,
pp. 2333-2342. 1989.
[39] A.D. Sakharov. Vacuum quantum fluctuations
in curved space and the theory of gravitation.
Sov. Phys. Dokl. Vol. 12 Nº 2, pp. 1040-1041.
1967.
[40] Y. B. Zel’dovich. Cosmological constant and
elementary particles. Sov. Phys. JETP Lett. Vol. 6,
pp. 316-317. 1967.
[41]
S. Ragusa. “First-order conservation Laxs in Chiral
medium”. Brazilian Journal of Phycs. Vol. 26,
pp. 411-418. 1996.
23
vol. 16 Nº 1,
Nº 1, 2008,
2008 pp. 24-30
NEW INTERPRETATION OF THE ATOMIC SPECTRA OF THE HYDROGEN
ATOM: A MIXED MECHANISM OF CLASSICAL LC CIRCUITS AND
QUANTUM WAVE-PARTICLE DUALITY
NUEVA INTERPRETACIÓN DEL ESPECTRO ATÓMICO DEL ÁTOMO
DE HIDRÓGENO: UN MECANISMO MIXTO DE CIRCUITOS LC Y
LA DUALIDAD ONDA CUÁNTICA-PARTÍCULA
H. Torres-Silva1
RESUMEN
En este trabajo se presenta un estudio de las leyes macroscópicas de conversión de energía del oscilador armónico LC, la
onda electromagnética (fotones) y el átomo de hidrógeno. Como nuestro análisis indica, las energías de estos aparentemente
diferentes sistemas obedecen exactamente la misma ley de conversión de la energía. Sobre la base de nuestros resultados
y de la dualidad onda-partícula del electrón, nos encontramos con que el átomo de hidrógeno, de hecho, es un oscilador
LC microscópico naturalmente quiral.
En el marco de la teoría clásica de campos electromagnéticos se obtiene analíticamente, para el átomo de hidrógeno, el
radio cuantizado de la órbita electrónica rn=aon2 y la energía cuantizada En=–R Hhc/n2, (n=1, 2, 3..), donde a 0 es el radio
de Bohr y R H es la constante de Rydberg. Sin la adaptación de otros principios fundamentales de la mecánica cuántica,
se presenta una explicación razonable de la polarización de los fotones, las reglas de selección y principio de exclusión
de Pauli. Los resultados también ponen de manifiesto una conexión esencial entre el espín de electrón y el movimiento
helicoidal intrínseco de los electrones e indican que el espín es el efecto de un confinamiento cuántico.
Palabras clave: Átomo de Bohr, quiralidad, oscilador LC.
ABSTRACT
In this paper we study the energy conversion laws of the macroscopic harmonic LC oscillator, the electromagnetic wave
(photon) and the hydrogen atom. As our analysis indicates, the energies of these apparently different systems obey exactly
the same energy conversion law. Based on our results and the wave-particle duality of electrons, we find that the hydrogen
atom is, in fact, a natural chiral microscopic LC oscillator.
In the framework of classical electromagnetic field theory we analytically obtain, for the hydrogen atom, the quantized
electron orbit radiusr n=aon2, and quantized energy En=–R Hhc/n2, (n = 1, 2, 3, · · ·), where a 0 is the Bohr radius and R H
is the Rydberg constant. Without the adaptation of any other fundamental principles of quantum mechanics, we present
a reasonable explanation of the polarization of photon, selection rules and Pauli exclusion principle. Our results also
reveal an essential connection between electron spin and the intrinsic helical movement of electrons and indicate that
the spin itself is the effect of quantum confinement.
Keywords: Bohr atom, chirality, LC oscillator.
INTRODUCTION
No one doubt that twentieth century is the century of
quantum theory [1-10]. After 100 years of development
quantum physics is no longer just a field, it is the bedrock
1
of all of modern physics. Although the modern quantum
theory has provided a beautiful and consistent theory
for describing the myriad baffling microphenomena
which had previously defied explanation [3], one should
not neglect a curious fact that quantum mechanics
24
H. Torres-Silva: New interpretation of the atomic spectra of the hydrogen atom: a mixed mechanism of classical lc circuits…
never take into account the deep structures of atoms.
In fact, at the heart of quantum mechanics lies only
the Schrödinger equation [5], which is the fundamental
equation governing the electron. According to quantum
theory, it is the electromagnetic interaction (by the
exchange of photons) which hold electrons and nuclei
together in the atoms.
But, up to now, quantum theory never provides a practical
model of how electron and nuclei can absorb and emit
photons.
In this paper, we investigate the energy relationship of
electron in the hydrogen atom. Significantly, we find
a process of perfect transformation of two forms of
energy (kinetic and field energy) inside the atom and
the conservation of energy in the system. By applying
the principle of wave-particle duality and comparing
to known results of the macroscopic harmonic LC
oscillator and microscopic photon, we are assured that
electron kinetic energy in fact is a kind of magnetic
energy and the atom is a natural microscopic LC
oscillator. Moreover, the mixed mechanism (classical
LC circuits / quantum wave particle duality) turns
out to have remarkably rich and physical properties
which can used to describe some important quantum
principles and phenomena, for instance, polarization of
photon, Zeeman effect, Selection rules, the electron’s
mass and spin, zero point energy (ZPE), the Pauli
exclusion principle.
ENERGY TRANSFORMATION AND
CONVERSION IN HYDROGEN ATOM
Classically, as shown in figure 1, the hydrogen atom
consists of one electron in orbit around one proton with
the electron being held in place via the electric Coulomb
force. Equation of motion is
e
2
2
4πε 0 r
2
=
me u
r
(1)
where me is mass of electron. Eq. 1 can be rewritten in
the form of kinetic energy Ek and field energy Ef (stored
in the capacitor of hydrogen atom) as follows:
2
e
1
= m u2 2Cr 2 e
(2)
u
-e
me
r
+e
Cr=4πε0r
Figure 1. The diagram illustrating the hydrogen atom.
where Cr =4πε0 r is the capacitance of the hydrogen
system. Thus the total energy of the hydrogen system
is given by
Ttotal =
1
e2
e2
me u 2 −
=
2
4πε 0 r 2Cr
(3)
It should be pointed out that Eq. 2 and 3 are the foundation
of our study. These two equations together indicate a
process of perfect periodically transformation of two
1
forms of energy (kinetic energy Ek = me u 2 and field
2
e2
energy E f =
inside the atom and the conservation
2Cr
of energy in the system
Etotal = E f = Ek (4)
Recall the macroscopic harmonic LC oscillator where two
Q2
forms of energy, the maximum field energy E f = 0 of
2C
the capacitor C (carrying a charge Qo) and the maximum
I 02
of the inductor L, are
2L
mutually exactly interconvertible ( Etotal = E f = Ek ) with
magnetic energy Em =
a exchange periodic T = 2π LC . And for a microscopic
photon (electromagnetic wave), the maximum field
1
energy E f = ε 0 E02 and the maximum magnetic energy
2
1
2
Em = µ0 H 0 also satisfy Etotal = E f = Ek (See appendix
2
A about E f = − Em ).
Based on the above energy relationship for three totally
different systems and the requirement of the electromagnetic
interaction (by exchanging photon) between electron and
nuclei, we assure that the kinetic energy of electron (Eq. 2)
is a kind of magnetic energy and the hydrogen atom is a
natural microscopic LC oscillator.
25
In 2000, a multinational team of physicists had observed
for the first time a process of internal conversion between
bound atomic states when the binding energy of the
converted electron becomes larger than the nuclear
transition energy [11, 12]. This observation indicate
that energy can pass resonantly between the nuclear
and electronic parts of the atom by a resonant process
similar to that which operates between an inductor and
a capacitor in an LC circuit. These experimental results
can be considered a conclusive evidence of reliability of
our LC mechanism.
Here raise an important question: how can the electron
function as an excellent microscopic inductor?
CHIRALITY AND “INDUCTON”
OF FREE ELECTRON
In 1923, Broglie suggested that all particles, not just
photons, have both wave and particle properties [5]. The
momentum wavelength relationship for any material
particles was given by
λ = h / p
(5)
where λ is called de Broglie wavelength, h is Planck’s
constant [1] and p the momentum of the particle. The
subsequent experiments established the wave nature
of the electron [9, 10]. Eq. 5 implies that, for a particle
moving at high speed, the momentum is large and the
wavelength is small. In other words, the faster a particle
moves, the shorter is its wavelength. Furthermore, it should
be noted that any confinement of the studied particle will
shorten the λ and help to enhance the so-called quantum
confinement effects.
As shown in figure 2 (a) and (b), based on Eq. 5 and the
demanding that the electron would be a microscopic
inductor, we propose that a free electron can move along
a helical orbit (the helical pitch is de Broglie wavelength
λe) of left-handed or right-handed. In this paper, the
corresponding electrons are called “Left-hand” and
“Right-hand” electron which are denoted by Chirality
Indexes S = 1 and S = −1, respectively. Hence, the electron
can now be considered as a periodic-motion quantized
inductive particle which is called “inducton” (see figure 2).
Moreover, the particle-like kinetic energy of electron
can be replaced with a dual magnetic energy carried by
a “inducton”. Therefore, we have
Ek =
1
1
me u 2 = Le I 2 2
2
(6)
where u is the axial velocity of the helical moving electron
and Le is the inductance of the quantized “inducton”.
Figure 2. A free electron moving along a helical orbit with
a helical pitch of de Broglie wavelength λe.
The above relation indicates that the mass of electron
is associated with an amount of magnetic energy. From
figure 2, the electric current, for one de Broglie wavelength,
is given by
The answer lies in the intrinsic wave-particle duality nature
of electron. In our opinion, the wave-particle nature [7]
of electron is only a macroscopic behavior of the intrinsic
helical motion of electron within its world.
26
I=
eu
λe
(7)
From Eq. 7, it is important to note that the electric current
should be defined within an integral number of de Broglie
eu
(where
2π r
r is the electronic orbital radius in the hydrogen atom),
which was widely used in the semiclassical Bohr model,
may be physically invalid. Collecting Eq. 6 and 7 together,
we have the inductance of single “inducton”
wavelength. Hence, the electric current I =
Le =
me λe2
e2
(8)
Then the dual nature of electron can be uniquely determined
1 u
by Le, the periodic T (or frequency f = =
), the initial
T λe
phase ϕ0 and the chirality (S = 1 or S = −1).
ATOMIC SPECTRA OF HYDROGEN ATOM
Quantized radius and energy by the application of
helical electron orbit to the hydrogen atom (figure
2), we can explain the stability of the atom but also
give a theoretical interpretation of the atomic spectra.
Figure 3 shows four possible kinds of stable helical
electron orbits in hydrogen atom, and each subgraph
corresponds to a electron of different motion manner
within the atom. The electrons can be distinguished by
the following two aspects. First consider the chirality
of electron orbits, as shown in figure 3, the electrons
of figure 3(a) and (c) are “Left-hand” labelled by S = 1,
while electrons of figure 3(b) and (d) are “Right-hand”
labelled by S = −1. Secondly consider the direction of
electron orbital magnetic moment µ L, figure 3(a) and (b)
show that the µ L are in the Z direction (Up) while (c)
and (d) in the −Z direction (Down), the corresponding
electrons are labelled by J = 1 and J = −1, respectively,
here J is called Magnetic Index. Hence, the electrons of
different physical properties become distinguishable,
they are Up “Left-hand” (ULH) electron (J = 1, S = 1),
Up “Right-hand” (URH) electron (J = 1, S = −1), Down
“Left-hand” (DLH) electron (J = −1, S = 1) and Down
“Right-hand” (DRH) electron (J = −1, S = −1).
As shown in figure 3(a), the helical moving electron
around the orbit mean radius r can now be regarded as a
quantized “inducton” of λr, thus the hydrogen atom is a
natural microscopic LC oscillator.
We consider that the physical properties of the hydrogen
atom can be uniquely determined by these natural LC
parameters. To prove that our theory is valid in explaining
the structure of atomic spectra, we study the quantized
Figure 3. The quadruple degenerate stable helical electron
or-bits in hydrogen atom.
orbit radius and the quantized energy of hydrogen atom
and make a comparison between our results of LC
mechanism and the known results of quantum theory.
For the system of λr, the LC parameters of the hydrogen
atom is illustrated in figure 3. Then the LC resonant
frequency is
νr =
1
2π Lr Cr
(9)
Recall the well-known relationship E=hvr, we have
E = hν r =
e2 8πε 0 r
(10)
Combining Eq. 9 and Eq. 10 gives
λr =
2h
πε 0 r / me e
(11)
Then the stable electron orbits are determined by
2π r
= n , (n = 1, 2, 3 ···),
λr
(12)
where n is called Principal oscillator number. The
integer n shows that the orbital allow integer number
of “induction” of the de Broglie wavelength λr. From
Eq. 11 and Eq. 12, the quantized electron orbit mean
radius is given by
27
rn =
ε0 h2
π me e 2
n 2 = a0 n 2 (13)
where α0 is the Bohr radius. And the quantized energy
is
En = −
m e4 1
e2
hc
= − e2 2 2 = − RH 2 8πε 0 rn
n
8ε 0 h n
(14)
where R H is the Rydberg constant. Surprisingly, the
results of Eq. 13 and 14 are in excellent agreement with
Bohr model [3]. Besides, taking figure 3 into account, we
can conclude that the quantized energies of Eq. 14 are
quadruple degenerate.
CONCLUDING REMARKS
In conclusion, we have found a process of perfect
transformation of two forms of energy (kinetic and field
energy) inside the hydrogen atom and the conservation of
energy in the system. Then, we have shown that the helical
moving electron can be regarded as a inductive particle
(“inducton”) while atom is regarded as a microscopic LC
oscillator, then the indeterministic quantum phenomena
can be well explained by the deterministic classical theory.
For a microscopic photon (electromagnetic wave), the
1
maximum field energy E f = ε 0 E02 and the maximum
2
1
2
magnetic energy Em = µ0 H 0 are connected so E = iH
2
(see equation (A11) of appendix A). The vector Poyting
vanishes and the Hydrogen atom does no radiate and it
is stable. In particular, with this approach we can show
another phenomena such how a pairing Pauli electron
can move closely and steadily in a DNA-like double
helical electron orbit. Moreover, we can have pointed
out that the mass of electron, the ZPE and what has
been called the intrinsic “electron spin” are all really the
quantum confinement effects of the intrinsic chirality
of the electron of helical motion.
We have shown that the quantum mechanism is nothing
but an electromagnetic theory (with the radius of the
helical orbit re → 0 ) of the LC/wave-particle duality
mixed mechanism. Our mixed mechanics force us to
rethink the nature and the nature of physical world.
We believe all elementary particles, similar to photon
and electron, are only some different types of energy
representation.
28
Though, the standard quantum mechanics nature is
intrinsically probabilistic, permitting only predictions
about probabilities of the occurrence of an event.
Nevertheless, one century after its birth, it still presents
many unclarified issues at its very foundations. Starting
from an Einstein’s work [13], many attempts have been
devoted to build a deterministic theory reproducing all
the results of quantum mechanics. The latter include
the de Broglie-Bohm’s hidden variable theory, the most
successful attempt in this sense [14]. Recently, a first
experimental test of de Broglie-Bohm theory against
standard quantum mechanics was reported [15]. In our
study, it has been shown definitely that the electron
follows a perfectly defined trajectory in its motion,
which confirms the de Broglie-Bohm’s prediction. Also
in our work, it is found that the known wave-particle
duality can be best manifested by showing that the wave
motion associated with a electron is just the phenomenon
of its complex helical motion in real space. Therefore,
the wave-particle duality should lie at the heart of the
quantum universe.
We are now more and more convinced that the universe
was built in the simplest manner and all things in it are
unique and definitive. As Albert Einstein one said, “God
does not play dice with the universe”. Of course, a more
clear understanding of microscopic world is still of the
greatest challenge.
APPENDIX A
It is generally believed that in transverse electromagnetic
waves electric field E and magnetic field B are always
perpendicular to each other. In this Letter we show that,
however, a general class of transverse electromagnetic
waves with E||B exists in a chiral media. We show how to
obtain these waves in general and give example in vacuum
and plasma . All these waves carry magnetic helicity.
In a cold collision less chiro-plasma, the magnetostatic
mode [16, 17] of this class becomes the more familiar
force-free field ∇ × B = k B, k = k0 2 + µT where T is the
chiral parameter. We consider transverse electromagnetic
waves in a uniform medium. These transverse waves can
be described by
B = ∇ × F, (A1)
E=−
1 ∂F
,
c ∂t
(A2)
in which the vector potential F satisfies ∇⋅ F=0 and the
wave equation
∇×∇×F+
1 ∂2F 4π
=
( j + β∇ × J ) c
c 2 ∂t 2
E = (ω A / c) sin k0 z , cos k0 z , 0 sin ω t ,
and

j = σ ⋅ E, (A4)

where σ is the conductivity tensor operator of the
medium under consideration, After Fourier analysis in
time, we have
F = A sin k0 z , cos k0 z , 0 cos ω t ,
(A3)
Here
(
)

∇ × ∇ × F − ω 2 / c 2 K (ω ) ⋅ F = 0 )
(
)
(
)
(
B = k0 A sin k0 z , cos k0 z , 0 cos ω t .
This solution corresponds to two circularly polarized
waves [16] propagating opposite to each other in such a
way that their Poynting vectors are cancelled out, so
(A5)
E = icB = iηH (A10)
with the dielectric tensor

K


(ω ) = I − 4πσ (ω ) / iω . (A6)
For simplicity, we consider only cases where K (ω ) is
independent of wavelength. We first look at the Hydrogen
atom in vacuum σ=0, T= ω /c and j + β ∇ × J = 0 and
Eq (A3) becomes

(∇2 + k02 ) Fk = 0 (A7)
with ω 2 = k 2c 2 . This waves equation allows the well
0
known linear polarized plane waves with E  B  F [18].
For every solution of Eq. (A5), it is straightforward to
show that
Fk = A k
-1
+ k 0 ∇x A
k
(A8)
satisfies not only Eq. (A5) but also
∇ × Fk = k Fk . Therefore, a single helical photon with energy ω carries
a magnetic helicity of hc, and
ACKNOWLEDGEMENT
The author would like to thank to Instituto de Alta
Investigación (IAI) for the support of this work.
REFERENCES
[1]
M. Planck. Ann. Phys. Vol. 1, p. 69. 1900.
[2]
A. Einstein. Ann. Phys. Vol. 17, p. 132. 1905.
[3]
N. Bohr, Phil. Mag. Vol. 26, p. 576. 1913.
[4]
O. Stern, Z. Phys. Vol. 2, p. 49. 1920.
[5]
E. Schrodinger. Ann. Phys. Vol. 79, p. 361.
1923.
[6]
L. de Broglie, Phil. Mag. Vol. 47, p. 446. 1924.
[7]
W. Pauli. Z. Phys. Vol. 31, p. 373. 1924.
[8]
W. Heisenberg Z. Phys. Vol. 43, p. 172. 1927.
(A9)
For those vector potentials F,Tk0 = 1 satisfying Eq. (A7),
the electric field E and magnetic field B are parallel to each
other and both are perpendicular to the vector k0. Therefore,
for every plane wave solution, a wave solution with E  B
can be constructed with k = k0 ( 0, 0, 1) , so
1
1
E f = E E = ε 0 E02 = − µ0 H o2 = E H = Em (A11)
2
2
29
[9]
G.P. Thomson, Proc. Roy. Soc. Vol. 117, p. 600.
1928.
[10]
C.N. Yang, Selection Rules for the Dematerialization
of a Particle into Two Photons, Phys. Rev. Vol. 77,
pp. 242-245. 1950.
[11]
T. Carreyre. Phys. Rev. C 62. 2000.
[12]
S. Kishimoto. Phys. Rev. Lett. Vol. 85, p. 1831.
2000.
[13]
F. Reines and W. H. Sobel. “Test of the Pauli
Exclusion Principle for Atomic Electrons”. Phys.
Rev. Lett. Vol. 32, p. 954. 1974.
30
[14]
D. Bohm. Phys. Rev. Vol. 85, p. 166. 1952.
[15]
G. Brida. J. Phys. B 35, p. 4751. 2002.
[16]
C. Chu and T. Ohkawa. Phys Rev. Lett. 48, p. 837.
1982.
[17]
H. Torres-Silva. “Chiro-plasma surface waves”.
A. Priou et al editors: Advances in Complex
Electromagnetics Materials, Kruwer Academic
Pub. Vol. 28, pp. 249-258. 1997.
[18]
H. Torres-Silva. Pramana Journal of Physics.
Vol. 48, p. 67. 1997.
Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008, pp. 31-35
H. Torres-Silva: Maxwell’s theory with chiral currents
MAXWELL’S THEORY WITH CHIRAL CURRENTS
TEORÍA DE MAXWELL CON CORRIENTES QUIRALES
H. Torres-Silva1
Recibido el 5 de septiembre de 2007, aceptado el 28 de noviembre de 2007
Received: September 5, 2007 Accepted: November 28, 2007
RESUMEN
El contenido de energía y momento de un campo electromagnético puede ser expresado enteramente, en términos de los
campos a través del tensor energía momento, sin mención de las fuentes que crean los campos. Este tensor es definido
introduciendo corrientes quirales. En el caso de sin fuerza se tiene T 00 = 0 y E x B = 0. Este método permite una muy
simétrica derivación del contenido de energía y momento de los campos con E||B. Esta configuración es esencial para
la unificación del electromagnetismo y la gravedad, obteniendo una configuración de fuerza cero para el electrón. Para
obtener esta unificación se discute la geometrización de Rainich bajo condiciones quirales.
Palabras clave: Corrientes quirales, geometrización de Rainich, tensor energía momento, unificación.
ABSTRACT
The energy and momentum content of an electromagnetic field can be expressed entirely in terms of the fields through
the energy-momentum tensor with no mention of the sources creating the fields. This tensor is defined such that chiral
currents are introduced. In the case of free force we have T 00 = 0 and E x B = 0. This approach allows for a very symmetric
derivation of the energy and momentum content of the fields with E||B. This configuration is essential to the unification of
electromagnetism and gravity, obtaining a force-free configuration for the electron. To obtain this unification the Rainich
geometrization under chiral conditions is discussed.
Keywords: Chiral currents, Rainich geometrization, energy-momentum tensor, unification.
INTRODUCTION
Although it’s existence in this region of the universe
has yet to be confirmed, magnetic charge has a strong
theoretical and pedagogical history from Gilbert’s
initial magnetic theory to present day unified theories.
Maxwell’s equations for electromagnetic theory have
source terms for electric charges and currents, but none
for their magnetic counterparts. This, of course, reflects
the experimental fact that magnetic monopoles have
never been discovered [1]. Students however, should
not be sheltered from the possible existence of magnetic
monopoles. For example, grand unified theories, by
definition, admit the existence of magnetic monopoles,
and their absence represents a challenge for particle
physicists and cosmologists alike [2].
1
Probably the most famous theoretical use of magnetic
monopoles is the Dirac quantization condition [3]. The
absence of magnetic source terms from Maxwell’s equations
allows the introduction of the electromagnetic potential,
which takes on a fundamental role in the quantum theory
of electrodynamics. Dirac’s argument then proceeds by
requiring the potential to be well defined even in a theory
with magnetic monopoles, leading to the quantization
of the product of the fundamental electric and magnetic
charges.
The classical theory of electromagnetism can be formulated
using the fields themselves as the fundamental objects
and there is no need to invoke the potential formalism.
This then leaves the obvious lack of symmetry between
the dynamical and non dynamical Maxwell equations
31
)
that can naturally be filled by postulating the existence
of magnetic source terms, thus making all of Maxwell’s
equations dynamical [4, 5].
Where Jeµ = ( cρe , Je is the chiral electric 4-current. The
electric continuity equation follows from the antisymmetry
of Fµv.
This short note is intended to show a symmetric derivation
of the electromagnetic energy-momentum tensor from the
Lorentz force law and Maxwell’s equations, extended to include
chiral magnetic as well as chiral electric source terms.
The second pair of Maxwell’s equations can be written
in 4-vector notation by defining the pseudotensor F µν ,
the dual of Fµv
In section 2 we briefly review Maxwell’s theory of
electromagnetism with both electric and magnetic charges
and currents displaying it’s full theoretical symmetry. The
energy momentum tensor is defined in section 3 and it’s usual
form is shown to follow naturally from a theory with both
electric and magnetic chiral currents. In section 4, we give
the Rainich geometrization under chiral conditions. We
close with some discussion of our derivation in connection
with unification of electromagnetism and gravity.
We will use Gaussian units and a diagonal space-time metric
gµv with − g 00 = g11 = g 22 = g33 = 1. Greek indices will take
the values 0 through 3 and Roman indices 1 through 3.
MAXWELL’S THEORY WITH CHIRAL
ELECTRIC AND MAGNETIC CURRENT
The equations of electrodynamics can be extended to include
chiral magnetic and electric current into Ampère’s law
and Faraday’s law respectively. I will use the subscripts
e and m to distinguish between the electric and magnetic
charges and currents. In 3-vector notation Maxwell’s
equations for the case of chiral approach [9] without
charges and monopoles (ρe, Je, ρm, Jm = 0 ) are:
∇⋅ E = 0 1 ∂E
imc
∇×B−
=−
E
c ∂t

∇⋅B = 0 ∇×E+
1 ∂B imc =
B
c ∂t

(1a)
(1b)
(1c)
(1d)
The Lorentz invariance of the theory can be made manifest
by combining the fields into the usual electromagnetic
field tensor Fµv
F µν = − F νµ , F 0 i = Ei , F ij = ε ijk Bk
32
∂F µν
∂x
ν
=
imc µ 4π µ
J =−
E
c e
 e
(2)
(3)
εµvαβ is the completely antisymmetric pseudotensor, = +1 ~,1
or 0, if αβµν is an even, odd, or no, permutation of 0123.
Equations (1c) and (1d) then read:
∂F µν
∂x
ν
=
µ
W here Jm = ( cρm , Jm
4-current.
4π µ imc µ
J =
B c m
 m
)
(4)
is t he ch i ra l mag net ic
Thus the specification of the divergence of an antisymmetric
tensor and the divergence of it’s dual completely
determines the tensor (and hence in this case, the fields)
is a generalization of Helmholtz’s theorem to four
dimensional space time [8], the divergence of the dual
playing the role of the curl.
The Lorentz force per unit volume on an assembly of
charges is given by:
fµ =
)
(
1
J F µν + Jmν F µν c eν
(5)
Before we dive into the derivation of the full energymomentum tensor, we will take a moment to derive
Poynting’s theorem from (5) using 3-vector notation.
The ’zeroth’ component of (5) is:
cf 0 = J e ⋅ E + J m ⋅ B
which is the work done by the fields on the charges per
unit volume per unit time. Using (1b) and (1d) to eliminate
the currents leads to:
(εijk is the totally antisymmetric Levi-Civita symbol.) The first
pair of Maxwell’s equations (1a) and (1b) then become
1
F µν = ε αβµν Fαβ 2
(
)
 c

 ∂ c
cf 0 = −∇ ⋅ 
E × B − 
E 2 + B2 

 4π
 ∂t  8π
which has the interpretation of: the energy per unit volume
per unit time gained by the charges is equal to the energy
lost by the fields through the divergence of the Poynting
vector and the time rate of change of the energy density.
Note that the electric and magnetic currents were treated
on equal footing, as were equations (1b) and (1d).
We now turn to the derivation of the full energy-momentum
tensor. We will use 4-vector notation, which may hide
some of the details. If so, the reader is encouraged to
mimic the above calculation using the 3-vector part of
(5) to derive the Maxwell stress tensor.
THE ENERGY-MOMENTUM TENSOR
fµ ≡
∂T µν
∂x
ν
(6)
In this note, however, I wish to emphasize the fact that,
with the introduction of magnetic source terms, each of
Maxwell’s equations is treated on equal footing, and the
symmetric form for T µv follows naturally. In a theory with
no magnetic charges only the first term in equation (5)
exists, and Maxwell’s dynamical equations (2) are used
to write the source terms in terms of the derivatives of
the fields. The remaining Maxwell equations (4) are then
only used as no more than mathematical relations during
the derivation.
If magnetic charges are admitted to the theory the
electric and magnetic source terms are removed from
the Lorentz force law (5) using Maxwell’s equations (2)
and (4), giving
fµ =
− g ργ gσδ g µα − g ρδ gσα g µγ − g ρα gσγ g µδ
)
gives
µ
1  µ ∂F βα ∂F β βα 1 αµ γδ ∂Fγδ 
+ α F + g F
f =
F

4π  β ∂x α
2
∂x α 
∂x
µ
which can then be written as a total divergence
A frequent approach to defining the energy-momentum
tensor for the electromagnetic field is to generalize the
definition of the Hamiltonian density to a covariant form
[7]. This leads to what is called the canonical energymomentum tensor. This has a number of draw backs for
our current purpose. Firstly, the Hamiltonian approach
requires the definition of the potential, which we do not
wish to make and secondly, the canonical form is not
symmetric (nor gauge invariant). An alternate method
[8], and the approach taken here, of determining the
energy-momentum tensor T µv, is to define it such that the
Lorentz force per unit volume (5) is the 4-divergence of
the energy-momentum tensor
(
gνβ ε ρσµν ε γδαβ = − g ρδ gσγ g µα + g ργ gσα g µδ + g ρα gσδ g µγ −
1  µ ∂F βα  µ ∂F βα  + Fβ
F

4π  β ∂x α
∂x α 
Substituting in the definition of the dual field tensor (3)
and using the identity
fµ =
1 ∂  µ βα 1 αµ γδ 
F F + g F Fγδ  4π ∂x α  β
4

By comparing this with (6), the symmetric energymomentum tensor is obtained
T αµ =
1  µ βα 1 αµ γδ 
F F + g F Fγδ  4π  β
4

This ends the calculation, but it is instructive to write
out the components of this tensor in the more familiar
3-vector forms whose physical interpretation are given
by integrating (6):
Energy density: −T 00 =
(
1
E 2 + B2
8π
0i
i0
Poynting vector: −cT = −cT =
)
c
(E × B
4π
)i
Maxwell stress tensor:
T ij = T ji =
(
)

1 
1
Ei E j + Bi B j − δ ij E 2 + B 2 

4π 
2

In the special case where E=iB, we have –T 00 = 0 and
c
−cT 0 i = −cT i 0 =
(E × B i = 0 .
4π
)
THE RAINICH GEOMETRIZATION
UNDER CHIRAL CONDITIONS
In the literature, the algebraic Rainich conditions are
obtained using special methods as spinors, duality rotations,
eigenvalue problem for certain 4 x 4 matrices or artificial
tensors of 4th order. Here we show an elementary procedure
for to deduce an identity satisfied by determined class
of second order tensors in arbitrary R4, from which the
Rainich expressions are immediate. This result is applied
to chiral conditions.
33
Rainich [10-15] proposed a unified field theory for the
geometrization of the electromagnetic field, whose basic
relations can be obtained from the Einstein-Maxwell field
equations under the Einstein notation:
Gµν = Rµν −
Tµν =
1
Rg = Tµν
2 µν
1
g F F στ − Fµσ Fντ g σς
4 µν στ
(7)
If in (1) we contract µ with ν we find that:
R = 0
the Maxwell mixed tensor is
T µν Maxwell ≡ −
(8)
1
Rµν = − gµν Fστ F στ + Fµσ Fντ gσς 4
(9)
Rµτ Rντ =
1
( R R ab ) gµν 4 ab
If Fστ is known, then (9) is an equation for gµv and our
situation belongs to general relativity. The Rainich theory
represents the inverse process: To search a solution of (8)
and (10) (plus certain differential restrictions), and after
with (9) to construct the corresponding electromagnetic
field; from this point of view Fστ is a consequence of the
space time geometry.
The essence of the chiral argument advanced here is
that real world-space is not euclidean and that space is
generally curved into the time dimension, consistent with
the theory of general relativity. The curvature may not be
sufficient to become obvious in a local context. However,
it is sufficient to break the time-reversal symmetry that
seems to characterize the laws of physics. Not only does
it cause perpetual time with respect to all mass, but
actually identifies a fixed direction for this It creates an
arrow of time and thereby eliminates an inconsistency
in the logic of physics: how reversible microscopic
laws can underpin an irreversible macroscopic world.
General curvature of space breaks the time-reversal
symmetry and produces chiral space, manifest in the
right-hand force rule of electromagnetism. The presence
34

 uσ
1 µ τσ
 F Fνσ − 4 δ ν F Fτσ  (11a)


1  uσ
F Fνσ + * F uσ * Fνσ  
8π 
(11b)
with iF uσ = * F uσ and iFνσ = * Fνσ . In this case we have
T µv Maxwell = 0, then equation (4) is
Rµτ Rντ =
(10)
1
4π
T µν Maxwell ≡ −
used by several authors [10-15] to show the identity:
∂Fνσ
∂F
∂ * Fνσ
= 0 , νσ =
=0
∂xσ
∂xσ
∂xσ
with, µ , σ = 4, Fνσ = * Fνσ and ∂ / ∂t → ∂ / ∂t (1 + T ∇×) ,
then (1) adopts the form:
The fact that most other fundamental laws of physics do
not refers the chirality of space, nor the arrow of time,
confirms that the curvature on a local scale is barely
detectable.
Now under chiral conditions
where Rµv, R = Rµµ and Fµσ are the Ricci tensor, scalar
curvature and Faraday tensor [10], respectively.
of matter causes space to curl up and curvature of space
generates matter.
1
1
( Rab R ab ) gµν ⇔ Fµτ Fντ = ( Fab F ab ) gµν (12)
4
4
Only in this case we have a complete unification, ie,
a unified field theory between the gravity and the
electromagnetism
DISCUSSION
The energy and momentum content of an electromagnetic
field can be expressed entirely in terms of the fields
through the energy-momentum tensor with no mention
of the sources creating the fields. This tensor is defined
such that it’s divergence gives the Lorentz force. That
is, any change in the energy and momentum of a charge
distribution is given by the (negative of the) change in the
energy and momentum of the fields. In the case of free
force we have T 00 = 0 and E x B = 0. Here there is no
existence of magnetic charges, because they have never
been found in nature. This approach allows for a very
symmetric derivation of the energy and momentum content
of the fields with E||B. This configuration is essential to
the unification of electromagnetism and gravity, obtaining
a force free configuration for the electron [16]. To obtain
this unification, the Rainich geometrization under chiral
conditions is discussed.
REFERENCES
[1] [2]
[3]
[8]
M.A. Heald and J.B. Ma r ion. Classical
Electromagnetic Radiation. 3rd ed. Section 14.12.
Saunders College Publishing. Fort Worth. 1995.
[9]
H. Torres-Silva and M. Zamorano Lucero. “Chiral
electrodynamic”. URLs: http://www.chiral.cl
A.S. Goldhaber and W.P. Trower. “Resource letter
MM-1: Magnetic monopoles”. Am. J. Phys. Vol. 58,
pp. 429-439. 1990.
[10]
G.Y. Rainich. Electrodynamics in the general
relativity theory. Trans. Amer. Math. Soc. Vol. 27,
p. 106. 1925.
P.A.M. Dirac. “Quantized singularities in the
electromagnetic field”. Proc. R. Soc. London A133,
pp. 60-72. 1931.
[11]
C.W. Misner and J.A. Wheeler. Classical Physics
as geometry. Ann. Phys. Vol. 2, p. 525. 1957.
[12]
J. López-Bonilla, G. Ovando and J. Rivera. Intrinsic
geometry of curves and the Bonnor’s equation, Proc.
Indian Acad. Sci. Math. Sci. Vol. 107, p. 43. 1997.
[13]
L. Witten. Geometry of gravitation and electromagnetism,
Phys. Rev. Vol. 115, p. 206. 1959.
[14]
D. Lovelock. The algebraic Rainich conditions.
Gen. Rel. Grav. Vol. 4, p. 149. 1973.
B. Cabrera. “First results from a superconductive
device for moving magnetic monopoles”. Phys.
Rev. Lett. Vol. 48, pp. 1378-1380. 1982.
[4]
J.A. Heras. “Jefimenko’s, Formulas with Magnetic
Monopoles and the Liénard-Wiechert Fields of
a Dual-Charged Particle”. Am. J. Phys. Vol. 62,
pp. 525-531. 1994.
[5]
W.B. Zeleny. “Symmetry in electrodynamics: A
classical approach to magnetic monopoles”. Am.
J. Phys. Vol. 59, pp. 412-415. 1991.
[6]
D.H. Kobe. “Helmholtz theorem for antisymmetric
second-rank tensor fields and electromagnetism
with magnetic monopoles”. Am. J. Phys. Vol. 52,
pp. 354-358. 1984.
[15] R. Penney. Duality invariance and Riemannian
geometry. J. Math. Phys. Vol. 5, p. 1431. 1964.
[16]
[7]
J.D. Jackson. Classical Electrodynamics. 3rd ed.,
Section 12.10. John Wiley & Sons. New York.
1999.
H. Torres-Silva. “Electrodinámica quiral: eslabón
para la unificación del electromagnetismo y la
gravitación”. Ingeniare. Rev. chil. ing. Vol. 16
Nº 1, pp. 6-23. 2008.
35
vol. 16 Nº 1,
Nº 1, 2008,
2008 pp. 36-42
CHIRAL FIELD IDEAS FOR A THEORY OF MATTER
IDEAS DE CAMPO QUIRAL PARA UNA TEORÍA DE LA MATERIA
H. Torres-Silva1
RESUMEN
En este trabajo, para el desarrollo de una teoría unificada de campos electromagnéticos y gravitacionales se usa un método
quiral. Los fotones que satisfacen las ecuaciones de Maxwell, para una onda electromagnética se consideran como componentes
físicos básicos. El objetivo de esta teoría es unificar el fenómeno de la invarianza relativística, mecánica de onda y la creación
del par electrón positrón, con las ecuaciones de Maxwell, para obtener una teoría de la materia totalmente electromagnética.
Considerando esta teoría se discuten algunos aspectos de los sistemas GPS (Global Positioning Systems).
Palabras clave: Potencial quiral, teoría de la materia, onda-partícula.
ABSTRACT
In this paper, a chiral approach is used for developing a unified theory of electromagnetic and gravity fields. The photons which
satisfy Maxwell’s equations for an electromagnetic wave are taken as the basic physical components. The goal of the theory
is to unify the phenomena of relativistic invariance, wave mechanics and pair creation with Maxwell’s equation to obtain an
electromagnetic field theory of matter. With this theory some aspects of GPS (Global Positioning Systems) systems are discussed.
Keywords: Chiral potential, matter theory, wave-particle.
INTRODUCTION
A chiral approach is suggested for developing a unified
theory of electromagnetic and gravity fields. Photons which
satisfy Maxwell’s equations for an electromagnetic wave
are taken as the basic physical component. The extent of
the photon in its direction of travel permits a part of the
photon to modify the geodetic of another part.
A photon with a self disturbed orbit, for which a centroid
can be defined, has the key property by which matter
differs from light. Matter has a speed which is less than
that of light. The centroid of the orbit has a speed which
is less than the speed of the photon which travels with
the speed of light. We refer to this chiral approach as the
electromagnetic field theory of matter.
Chiral approach means that our Universe is observable area
of basic space-time where temporal coordinate is positive
and all particles bear positive masses (energies). The
mirror Universe is an area of the basic space-time, where
from viewpoint of regular observer temporal coordinate
is negative and all particles bear negative masses. Also,
from viewpoint of our-world observer the mirror Universe
is a world with reverse flow of time, where particles travel
from future into past in respect to us. The two worlds are
separated with the membrane - an area of space-time
inhabited by light-like particles that travel along light-like
right or left-handed (isotropic-chiral) spirals.
The goal of the theory is to unify the phenomena of relativistic
invariance, wave mechanics and pair creation with Maxwell’s
equation for electromagnetic waves. Section 2 enumerates
advantages of an electromagnetic field theory of matter. Section
3 considers how the de Broglie relation and the Schrodinger
equation might be derived from Maxwell’s wave equation.
Section 4 treats the relation between electromagnetic and
inertial energy. Section 5 comments on parity failure in
weak interactions. Appendix A derives an application on
GPS satellites using chiral potential.
1 Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: [email protected]
36
H. Torres-Silva: Chiral field ideas for a theory of matter
ADVANTAGES OF AN ELECTROMAGNETIC
FIELD THEORY OF MATTER
An electromagnetic field theory should not be confused
with an electric charge theory of matter which is not
relativistically invariant [1]. An electromagnetic field,
however, is relativistically invariant from the start.
One simplicity is that special relativity is not a separate
hypothesis. The Lorentz contraction of electromagnetic
fields was realised before special relativity. If matter is
composed only of electromagnetic fields then matter is
automatically Lorentz invariant. In particular, matter
cannot exceed the speed of light.
Another area of simplicity is pair creation where two
electromagnetic fields (photons) produce an electron and
a positron. If the particles are electromagnetic fields, then
pair creation is like the transformation of electromagnetic
field from one state of motion to another. We suggest
since masses are unique that this should be thought of
as the construction of a quantized state of the EM wave
(i.e. standing wave a de Broglie type phase relation) [2].
The distinction between matter and antimatter would then
be sought as a natural law of conservation of a property
inherent in the separate initial photon and divided between
the particles. Such a difference is inherent the photon
polarization. For example, the photons can have rightand left-handedness.
It should be further noted that a fast electron is like an
EM wave having a transverse EM field with equal electric
magnetic field energies. Also the momentum of fast
particles, like EM waves, is the energy divided by c.
The uncertainty principle of quantum mechanics would
also be more consistent with an electromagnetic theory
of matter. That is, particles which have an inherent
wave nature would be expected and not a surprise. Also,
quantized absorption of EM energy would not be viewed
as a charge accelerated by an electric field but a merging
of two EM waves. The merged EM fields would have the
required frequency and wavelength to be the quantized
wave of the electron in the final state.
The appearance of the fine structure constant, α = e2 /hc,
in the ratio of the masses of fundamental particles would
be expected and not a coincidence as in the ratio of the
mass of the π meson and the electron.
The development of an EM field theory of matter requires
the accomplishment of at least two objectives, namely
(i) predict the Coulomb force between electron and
positron (etc.), and (ii) derive the Schrodinger equation
from Maxwell’s equation for an electromagnetic wave.
We suggest approaches to these objectives in the next
sections.
DERIVATION OF THE PARTICLE
WAVELENGTH FROM CHIRAL
POTENTIAL WAVES
We start with the potential vector equation.
Assuming ejwot time dependence, Maxwell’s time-harmonic
equations [2] for isotropic, homogeneous, linear media
(without charges) are
∇ × E = − jω 0 B (1)
∇ × H = jω 0 D (2)
∇•B = 0 (3)
∇•D = 0 (4)
Chirality is introduced into the theory by defining the
following constitutive relations to describe the isotropic
chiral medium [3]
D = ε E + εT ∇ × E (5)
B = µ H + µ∇ × H (6)
Where the chirality factor indicates the degree of
chirality of the medium, and the ε y µ are permittivity
and permeability of the chiral medium, respectively. Since
D and E are polar vectors and B and H are axial vectors,
it follows that ε and µ are true scalars and T is a pseudo
scalar factor. This means that when the axes of a righthanded Cartesian coordinate system are reversed to form
a left-handed Cartesian coordinate system, T changes in
sign whereas ε and µ remain unchanged.
Since ∇•B = 0 always, this conditions will hold identically
if B is expressed as the curl of a vector potential A since
the divergence of the curl of a vector is identically zero.
Thus
B=∇× A
(7)
And A must be perpendicular to both ∇ and B and lie
in ∇ and ∇ x B plane. However, A is not unique since
only its components perpendicular to ∇ contribute to
the cross product. Therefore, ∇ • A, the component of
37
A parallel to ∇, must be specified. The curl equation for
E, as in Equation, and Equation give ∇ x (E + jwA) = 0
where the quantity in parentheses should be parallel to
∇ and the curl of the gradient of a scalar function φ is
identically zero; so the general integral of the above
equation is
E + jω 0 A = −∇φ Let

 xˆ
∇ × A =  ik x

 Ax

Ay

zˆ 
ik z 

Az 

= xˆ (−ik cos θ Ay ) −
(8)
− yˆ (ik sin θ Az − ik cos θ Ax ) + zˆ(ik sin θ Ay )
Substituting Equation into Equation we obtain
yˆ
0
(13)
so equation (12) is expressed as
∇ × A = µ H + µT ∇ × H = B (9)
Substituting Equation and Equation into Equation
gives
∇ × ∇ × A+ j
ω 0 µεT
1 − ko2T 2
∇×E = j
ω 0 µε
1 − ko2T 2
E−
ω 0 2 µεT
1 − ko2T 2
∇2 A +
+2
k
A+
1− k T 2

ω 02 µε T
ω 02 µε T φ 
j
A
A
∇
×
=
∇
∇
−
•
(
)


1 − k02T 2
1 − k02T 2 

B,
∇•A = j
ω 0 µεφ
1 − k02T 2
(10)
(11)
And eliminate the term in parentheses. Then Equation
will be simplified to
∇2 A +
k02
1 − k02T 2
A+ 2
ω 02 µε T
1 − ko2T 2
(∇ × A) = 0 (12)
− k (1 −
k02T 2 ) + k02
2ik02 kT sin θ
0
A 
x

Ay  = 0

Az 

−2ik02 kT cos θ  

− k 2 (1 − k02T 2 ) + k02  
k 2 (1 − k02T 2 ) = k02 = ω 02 / c 2 ⇒ k = ± k0 / 1 − k02T 2 (14)
{− k (1 − k T ) + k )}
2
2
0
2
2
0
⇒ k = k0 / (1 ± k0 T )
2
− 4 k02T 2 (sin 2 θ + cos2 θ ) = 0
(15)
for the transverse fields.
Our approach to deriving the Schrodinger equation from
Maxwell’s equation starts with the assumption that an
electron is an electromagnetic ware travelling in a circular
orbit in the observer’s rest frame. We suggest that the orbit
is a geodetic in a space-time curved by the photon’s own
electromagnetic energy.
We note that the model of the self- trapped wave must look
like an electron to observers in all inertial frames. The
observer at rest only sees the static Coulomb field. The
moving observer, with speed u, sees (i) some magnetic
field from the current associated with the charged particle,
and (ii ) (the wave-motion of the particle with a wavelength λm = h/mu. For the observer whose speed relative
to the circulating photon approaches the velocity of light,
the photon appears as a photon. This is appropriate since
the electric and magnetic fields of a fast electron become
transverse as its speed approaches c.
From equation (14), where k = ± k0 / 1 − k02T 2 , if we
The solution of the potential vector equation can be
solved as follows:
38
2
for the longitudinal field, and
Here ∇ • A is arbitrary, so in order to specify ∇ • A, for
unique A, we may define a chiral Lorentz gauge
−2ik02 kT cos θ
The solution of the determinant is
with k0T = w0T/c. Placing the value of ∇xE from equation
into the above equation, using the vector identity ∇ x ∇ x A
= ∇ (∇ • A) – ∇2 A and equation (8) enables us to write
the above equation as
2
0
2
0
 − k 2 (1 − k 2T 2 ) + k 2
0
0

2ik02 kT cos θ


0

make k = w/c, k0T = w0 T/c ≡ u/c then particle momentum
is consistent with the photon model. The observer with
speed u in the usual theory of special relativity notes that
the electron has energy [3].
E=
m0 c 2
1− β2
(16)
p=
m0 u
1− β2
(17)
From equations (16) and (17) as v approaches c and the
rest mass becomes a small part of the energy
p≈E /c
We will try to illustrate how the photon frequency can
lead to the appropriate particle wavelength for all inertial
observers. For the observer at rest with respect to the
circulating photon. The field appears static, the period
is infinite, the frequency is zero, and the wavelength
infinite. At high speed we will show that the photon’s
wavelength in the observer’s frame is consistent with the
quantum mechanical expression for the corresponding
particle. We relate the photon energy hv0 to the rest mass
of the electron m 0 using the special relativity relation
E=mc 2, by.
hν 0 = m0 c 2 = ω 0 (19)
and inverting equation (19)
h / m0 c = c / v = λ 0 (20)
In the limit of u → c the matter wavelength is the wavelength
associated with the photon reduced by the usual special
relativity Lorentz contraction factor
1− β2 .
Then using equations (20), (19) and (16) in order we
obtain
λm = λ0 1 − β 2 = c / ν 0 1 − β 2 =
h hc 2 hc
=
≈
p Eν E
p=
(22)
hc
mc
2
=
hc
(21)
E
hν u
c2
(23)
2
2
2
where hν = E = mc = m0 c / 1 − β . Def ining a
wavelength related to the particle
(18)
In special relativity the frequency transforms as the energy
and this is the correct expression for the momentum of a
photon. The momentum of a photon in the rest frame is
effectively zero because the geodetic closes on itself.
λm =
More generally, for any value of u from equations (19)
and (21)
where β = u/c = w0T/c, and momentum
This result is the same as obtained from the theory of the
electron for v - c using Equation (17)
λm =
h c2
c
c
1− β2 =
= λ = λ0
p uν
u
u
(24)
In terms of frequencies from equation (24)
νm = ν
u
c
(25)
We interpret this result as follows. The electron has
intrinsically the frequency of its parent photon. The
observer going by at the speed of light sees the circulating
wave stretched out to its limit and associates the full
frequency, v, with the particle frequency.
The observer moving more slowly passes the nodes in
the EM wave more slowly and interprets this as a lower
frequency, vm<v . The observer at rest sees no change and
concludes vm=0. These frequencies are consistent with
the de Broglie wavelength for matter.
The Schrodinger equation describes the wave motion of
the centroid of a photon which is a solution of Maxwell’s
wave equations in a distorted space time.
A word is in order about the problem of interference patterns
of scattered electrons such as produced by the diffraction by
two slits in a barrier. Margenau [4] considers the difficulties
of this problem. He concludes that there are no known
interactions that can explain how an electron can go through
one slit and be appropriately scattered by the other.
An electromagnetic field which acts like a particle, may
possible avoid this dilemma by either (i) the electric and
magnetic field goings partly through each slit, or (ii)
having a scattering which differs from known interactions
because of the fluctuating EM field properties of the
39
electron such as in our photon model. We think that the
second possibility is the correct explanation.
RELATION BETWEEN ELECTROMAGNETIC
AND INERTIAL ENERGY
In this section we relate the photon electromagnetic and
the elementary charge electric field energies with the
inertial and gravitational energies.
When an electron positron pair is created we distinguish
two changes in energy: (i) the electromagnetic field energy
of the photons is transformed into the electric field of the
pair, and (ii) the transformed photon which we recognise
as a particle has inertia with respect to the cosmology. By
special relativity the inertial energy per unit mass is c2.
The inertial mass relates to the electrical energy of the
charge by m2 = e2/r. Now a more detailed discussion of
the effect of the cosmology on inertia and the gravitational
red shift are required to clarify the distinctions between
the four types of energy.
is the photon energy when it is emitted by an atom
or produced as a result of pair annihilation. Now hv;
depends on the gravitational potential at the location
where it is emitted since for atomic radiation hviαe4 mi /
h2 and for pair radiation hvi = mic2. In this way we see
that hviαmi, and both depend in the same way on the
gravitational potential. This is consistent with general
relativity [7]. In particular it agrees with the gravitational
red shift. For example, a photon radiated from the sun
has energy hvsαms, and the corresponding atom on the
earth has the transition energy hveαme; vs is to the red
of ve as given by general relativity because the masses
are related by
me = ms 1 −
2 ∆φ
c2
(27)
where ∆φ. is the difference in the gravitational potential
energy per unit mass.
Thus the photon energy hvs has not changed energy during
its travel from sun to earth.
We assume, following E Mach, that the inertia depends
on the cosmology. That is, we take inertial energy equal
to the negative of the cosmological gravitational potential.
From general relativity, then, we assume
This is the justification for the assumption that the
photon energy does not change as it moves through the
cosmology.
m 2 MG
u2
mc −
= 0 ⇔ k 20T 2 = 2 R
c
We now return to the relation between the electrical
energy and inertial energy and compare both with the
photon energy. The subscripts relating to location in the
gravity field are retained
2
(26)
where M is the total mass of the universe (~1056 gm),
R is the radius of the universe (~1028 cm) and G is the
gravitational constant (0,67x10-8 dynecm2/gm2) [5-7].
Consequently, in pair creation, the inertial energy gained
is cancelled exactly by the loss of gravitational potential
energy. We note that equation (11) is independent of m.
We have assumed c 2 to be invariant and hence the
gravitational potential energy per unit mass must also
be independent of location in the cosmology. This is
consistent with the assumption that every location in the
cosmology senses the expansion of the cosmology in the
same way and that there is no distinguished location in
the cosmology.
The next important hypothesis is that the photon energy
does not change as it moves through the cosmology. This
concept must be distinguished from the dependence
of frequency on the local gravitational potential. We
use a subscript i to correspond to initial value, i.e. hvi
40
hν i = mi c 2 = e 2 / ri (28)
and we must subscript the particle radius for consistency.
Thus photons which are produced by annihilation
can only receate the correct amount of energy where
the gravitational potential is the same as at point of
origin. The problem can be solved by special relativity.
There are four energies with significant differences. A
photon can be transformed into an electron and both
photon and electron have equal and positive energy.
The photon energy does not change as it traverses the
cosmology. The particle energy depends on the local
cosmological metric. No net gravitational energy is
produced by the creation of the particle since the
gain of inertial energy is just cancelled by the loss in
gravitational energy.
As a particle traverses space the inertial energy is
always equal to the electrical energy of the particle
and the inertial energy is always cancelled by the
gravity energy.
potential (at distance r from the center of the Earth of
mass mE). In the case of a photon, we replace m by E/c2,
where E = hf is the photon’s energy.
PARITY
If the photon travels downward in Earth’s gravitational
field, it therefore loses potential energy of (hf/c2)∆V
and gains an equal amount of kinetic energy h∆f. We
thereby deduce that the falling photon is gravitationally
blue-shifted by
The photon model of the electron has a natural explanation
of the failure of reflection symmetry (parity) in weal
interactions. Wigner [8] has given a phenomenological
discussion of the implication of the C0β decay experiment
in which the spin of the β is determined to be opposite
to its momentum. He points that the mirror image of an
electron is a positron.
We invoke the polarization of the photons the electron
positron pair. We assume that one particle has righthanded polarization photon and the other a left- handed
one. Thus, the particles are mirror images as required. In
fact, it is the difference in handedness which distinguishes
two charge states.
APPENDIX A: RELATIVISTIC EFFECTS
ON CLOCKS ABOARD GPS SATELLITES
Consider a clock aboard a satellite orbiting the Earth,
such as a Global Positioning System (GPS) transmitter.
There are two major relativistic influences upon its rate
of timekeeping: a special relativistic correction for its
orbital speed and a general relativistic correction for its
orbital altitude. Both of these effects can be treated at an
introductory level, making for an appealing application
of relativity to everyday life.
First, as observed by an earthbound receiver, the
transmitting clock is subject to time dilation due to its
orbital speed. From our results of chiral potential, (14), a
clock aboard a spaceship traveling at speed u runs slow
(compared to a stationary clock) by a factor of [9]
u2
u2
1
= 1− 2 = 1− β2 ≈ 1− 2 γ
c
2c
(A1)
provided u << c, as would be the case for a satellite. Thus
when one second of proper time elapses, the moving
clock loses u2/2c2 = K/E 0 seconds, where K and E 0 are
the kinetic and rest energies of the clock, respectively.
Second, a clock at the higher gravitational potential of
orbit runs faster than a surface clock. The gravitational
potential energy of a body of mass m in Earth’s gravity
is U = mV, where V = –GmE/r is Earth’s gravitational
∆f = f
∆f
c2
(A2)
(This expression can also be straightforwardly deduced
[10] using the equivalence principle to treat Earth’s
downward gravitational field as an upward accelerating
frame, and then calculating the Doppler shift in the light
between emission high up and observation low down in
this moving frame.) If the clock’s ticking is synchronized
to a light wave, the orbiting clock will be observed at
Earth’s surface to be ticking faster due to this gravitational
frequency shift. Therefore, when one second of Earth
time elapses, the clock at high altitude gains ∆V/c2 =
∆U/E 0 seconds, where U is the gravitational potential
energy of the clock.
The sum of the two relativistic effects can be compactly
expressed as
∆t K − U
=
τ
E0
(A3)
where ∆t is the time lost by the orbiting clock when a time
interval τ elapses on the surface-bound clock.
Here K–U is the Lagrangian of the orbiting clock where
the reference level for the gravitational potential energy
is chosen to lie at Earth’s surface.
As a concrete example, let’s calculate the size of these
two effects for a GPS satellite, located at an altitude of
r = 26,580 km, about four times Earth’s radius of rE =
6380 km. From Newton’s second law, we have
a=
gr 2
F
v 2 GmE
⇒
= 2 ⇒ v2 = E m
r
r
r
(A4)
where Earth’s surface gravitational fieldis g = GmE / rE2
= 9.8 m/s2. Hence the fractional time loss due to the
satellite’s orbital speed is −grE2 / 2c 2 per second, or –7.2
µs/day. Meanwhile, the general relativistic fractional time
gain due to the satellite’s altitude is
41
∆V
c2
=
rE
1  GmE GmE  grE
−
+
 = 2 (1 − ) (A5)
2 
r
rE  c
r
c 
which works out to be +45.6 µs/day. Notice that the
gravitational effect is more than six times larger than
the speed effect: the dominant GPS correction is general,
not special relativistic! If we instead consider satellites
in progressively lower altitude orbits, their speeds will
increase according to Eq. (4), while the gravitational
potential difference in Eq. (5) will decrease. Eventually we
will reach an altitude at which the two corrections exactly
cancel, so that the satellite’s clock will run synchronously
with an earthbound clock [10, 11].
Section 5 pointed out that the mirror image property of
the positron and electron required by the failure of parity
conservation in weal interactions can be attributed to the
handedness of the photon which are transformed into the
electron positron pair.
REFERENCES
[1]
H.A. Lorentz. “The Theory of Electrons”. B.G.
Teubner, Leipzig. 1909.
[2]
H. Torres-Silva. “New interpretation of the atomic
spectra of the hydrogen atom: a mixed mechanism
of classical LC circuits and quantum wave-particle
duality”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 24-30. 2008.
[3]
Nº 1, pp. 6-23. 2008.
[4]
H Margenau. Open Vistas. Yale Univ. Press, New
Haven. 1961.
[5]
N. Salingaros. “Invariants of the electromagnetic
field and electromagnetic waves”. Am. J. Phys.
Vol. 53, pp. 361. 1985.
[6]
C.W. Allen. “Astrophysical Quantities”. 3rd
ed. Athlone, London. 1973.
[7]
A. Einstein. “The Meaning of Relativity”. Princeton
Univ. Press. Princeton, New Jersey. 1950.
[8]
M. Gogberashvili, Octonionic version of Dirac
equations, International Journal of Modern Physics
A. Vol. 21 Nº 17, pp. 3513-3523. 2006.
[9]
D.C. Giancoli. “Physics for Scientists and Engineers”.
3rd ed. Chap. 37. Prentice Hall, Upper Saddle
River, NJ. 2000.
[10]
N. Ashby. “Relativity and the Global Positioning
System”. Phys. Today. Vol. 55, pp. 41-47. May,
2002.
[11]
S.P. Drake. “The equivalence principle as a
stepping stone from special to general relativity”.
Am. J. Phys. Vol. 74, pp. 22-25. January 2006.
This occurs when
u2
2c 2
=
∆V
c2
⇒
grE2
r
= grE (1 − E ) ⇒ r = 1.5rE (A6)
r
2r
i.e., at an altitude of half an Earth radius.
SUMMARY
An outline has been presented of an electromagnetic field
theory for matter. The advantages of the theory are given
in section 2. Their seemingly distinct areas of physics see
unified with Maxwell’s equation for EM waves. They are
relativistic invariance, pair creation, and wave mechanics.
Light is relativistically invariant, hence, particles made
out of photon are relativistically invariant. If matter is
a form of electromagnetic energy, then pair creation is
a transformation from energy. If particles are made out
of photons they have an intrinsic wave nature and their
wave motion is expected.
In section 3 we elaborated on how the wave nature of
photon, which forms an electron, leads to wave mechanics
of the particle. The full frequency is approached as the
velocity of the particle, relative to the observer, approaches
the velocity of light.
In section 4, we gave the relation between the
electromagnetic and inertial energy. No energy is
added in pair creation: the electromagnetic energy
is transformed between photon and particle states.
The inertial energy is a property of the particle in the
cosmology. The cosmological gravitational potential is
the negative of the inertial energy so that these mutually
cancel. Therefore, no net inertial plus gravitational
energy is required for pair creation.
42
Ingeniare.
H. Torres-Silva:
Revista chilena
The de
close
ingeniería,
relation vol. 16
between
Nº 1,
the 2008,
Maxwell
pp. 43-47
system and the dirac equation when the electric field is parallel to the magnetic field
THE CLOSE RELATION BETWEEN THE MAXWELL SYSTEM AND THE DIRAC
EQUATION WHEN THE ELECTRIC FIELD IS PARALLEL TO THE MAGNETIC FIELD
LA ESTRECHA RELACIÓN ENTRE EL SISTEMA DE MAXWELL Y LA ECUACIÓN
DE DIRAC, CUANDO EL CAMPO ELÉCTRICO ES PARALELO AL CAMPO MAGNÉTICO
H. Torres-Silva1
RESUMEN
En el presente artículo se propone una simple igualdad que considera el operador de Dirac y los operadores de Maxwell
bajo un enfoque quiral. Esta igualdad establece una conexión directa entre las soluciones de los dos sistemas. Además
se muestra que es válida cuando una relación muy natural
 se cumple entre la frecuencia de la onda electromagnética y
la energía de la partícula Dirac, si el campo eléctrico E es paralelo al campo magnético H . Este análisis se basa en la
forma cuaterniónica de la ecuación de Dirac y la forma cuaterniónica de las ecuaciones de Maxwell. En ambos casos
las reformulaciones con cuaterniones son completamente equivalentes a la forma tradicional de los sistemas de Dirac y
Maxwell. Esta teoría es una nueva interpretación de la mecánica cuántica. Este trabajo prueba que la mecánica cuántica
representa la electrodinámica de ondas quirales curvilíneas cerradas. Esto está enteramente de acuerdo con la moderna
interpretación y resultados de la teoría cuántica de campo.
Palabras clave: Cuaternión, ecuación de Dirac, Sistema de Maxwell.
ABSTRACT
In the present article we propose a simple equality involving the Dirac operator and the Maxwell operators from a chiral
approach. This equality establishes a direct connection between solutions of the two systems. Moreover, we show that
the connection is valid when a fairly natural relationship between
the frequency of the electromagnetic
wave and the


energy of the Dirac particle is fulfilled, if the electric field E is parallel to the magnetic field H . Our analysis is based
on the quaternionic form of the Dirac equation and on the quaternionic form of the Maxwell equations. In both cases
the quaternionic reformulations are completely equivalent to the traditional form of the Dirac and Maxwell systems.
This theory is a new quantum mechanics (QM) interpretation. The research below shows that the QM represents the
electrodynamics of the curvilinear closed chiral waves. This concords entirely with the modern interpretation and results
of the quantum field theory.
Keywords: Quaternion, Dirac equation, Maxwell system.
INTRODUCTION
The relation between the two most important in
mathematical physics first order systems of partial
differential equations is among those topics which attract
attention because of their general, even philosophical
significance but at the same time do not offer much for
1
the solution of particular problems concerning physical
models. The Maxwell equations can be represented in a
Dirac like form in different ways (e.g., [3, 5, 9]). Solutions
of Maxwell’s system can be related to solutions of the
Dirac equation through some nonlinear equations (e.g.,
[11]). Nevertheless, in spite of these significant efforts
there remain some important conceptual questions.
43
For example, what is the meaning of this close relation
between the Maxwell system and the Dirac equation
and how this relation is connected with the waveparticle dualism. In the present article we propose a
simple equality involving the Dirac operator and the
Maxwell operators under chiral approach. This equality
establishes a direct connection between solutions of
the two systems and moreover, we show that it is valid
when a quite natural relation between the frequency
of the electromagnetic wave and the energy of the

Dirac particle is fulfilled when E is parallel to H .
Our analysis is based on the quaternionic form of the
Dirac equation obtained in [7] and on the quaternionic
form of the Maxwell equations proposed in [6] (see also
[8]). In both cases the quaternionic reformulations are
completely equivalent to the traditional form of the Dirac
and Maxwell systems. Chiral approach means that our
Universe is observable area of basic space-time where
temporal coordinate is positive and all particles bear
positive masses (energies). The mirror Universe is an
area of the basic space-time, where from viewpoint of
regular observer temporal coordinate is negative and all
particles bear negative masses. Also, from viewpoint of
our-world observer the mirror Universe is a world with
reverse flow of time, where particles travel from future
into past in respect to us. The two worlds are separated
with the membrane — an area of space-time inhabited
by light-like particles that travel along light-like right
or left-handed (isotropic-chiral) spirals. On the scales
of elementary particles such space can be attributed
to particles that possess spirality (e. g. photons). The
membrane prevents mixing of positive and negative-mass
particles and thus their total annihilation. Exchange
interactions between the two worlds can be effected
through particles with zero relativistic masses (zeroparticles) under physical conditions that exist on surfaces
of collapsers in degenerated spacetime (zero-space).
The complex imaginary unit i commutes with ik , k = 0, 3 .
We will use the vector representation of complex
quaternions: q=Sc(q)+Vec(q), where Sc(q) = q 0 and
3

Vec(q) = q = ∑ k =1 qk ik . That is each complex quaternion
is a sum of its scalar part and its vector part. Complex
vectors we identify with complex quaternions whose scalar
part is equal to zero. In vector terms, the multiplication
of two arbitrary complex quaternions q and b can be
written as follows:

 
 

q ⋅ b = q0 b0 − < q , b > +  q × b  + q0 b + b0 q ,
where
3
 
< q , b >:= ∑ qk bk ∈C
k =1
and
i1
 
 q × b  := q1
b1
We shall consider continuously differentiable H(C)
-valued functions depending on three real variables
x = (x1, x 2, x 3). On this set the well known (see, e.g.,
[1, 4, 7 and 8]) Moisil-Theodoresco operator is defined
by the expression
The algebra of complex quaternions is denoted by H(C).
Each complex quaternion q is of the form q = ∑ qk i
k =0
3
{
the quaternionic imaginary units:
i02 = i0 = −ik2 ; i0 ik = ik i0 = ik , k = 1, 2, 3;
i1i2 = −i2i1 = i3 , i2i3 = − i3i2 = i1;
i3i1 = − i1i3 = i2
44
3
D := ∑ ik ∂ k , where ∂ k =
k =1
{ }
i3
q3 ∈C 3 .
b3
∂
.
∂x k
The action of the operator D on an H(C) -valued function
f can be written in a vector form:
PRELIMINARIES
where qk ⊂ C , i0 is the unit and ik
i2
q2
b2
}
k
k = 1, 2, 3 are


Df = − div f + grad f0 + rot f .
(1)


That is, Sc( Df ) = − div f and Vec( Df ) = grad f0 + rot f .
In a good number of physical applications (see [4 and 8])
the operators D α = D+Mα and D – α = D–Mα are needed,
where α is a complex quaternion and Mα denotes the
operator of multiplication by α from the right-hand side:
Mα f = f⋅α. Here we will be interested in two special
cases when α is a scalar, that is α = α0 or when α is a

vector α = α . The first case corresponds to the Maxwell
equations and the second to the Dirac equation.
H. Torres-Silva: The close relation between the Maxwell system and the dirac equation when the electric field is parallel to the magnetic field
THE DIRAC EQUATION
and
Following [7], we consider the Dirac equation in its
covariant form
3
 γ


0

∂
+
γ k ∂ k  + imc Φ ( t , x = 0 .
 
∑
t

  c

k =1
)
 0 −i −1 0   F0 
 −1 0 0 −i  F 
  1 .
Φ =  −1  F  = 
 1 0 0 −i  F2 
 

 0 i −1 0   F3  We have the following important equality
For a wave function with a given energy we have
)
Φ (t , x = a ( x
)
ε
i t
e
, where α satisfies the equation
3
 iε
imc 
 c γ 0 + ∑ γ k ∂ k +   a ( x = 0. 

k =1
)
(2)
D :=
( (
3
iε
imc .
γ 0 + ∑ γ k ∂k +
c

k =1
)
) (
)
) )
1
 −Φ
 i +i Φ
 −Φ
 i − Φ
 +Φ
 i +i Φ
 +Φ
 i
F =   Φ  = − Φ
1
2 0
0
3 1
0
3 2
1
2 3
2
(
The inverse transformation
A-1
(
is defined as follows
(
Φ =  −1  F  = −iF1 − F2 , − F0 − iF3 , F0 − iF3 , iF1 − F2
)
Let us present the introduced transformations in a more
explicit matrix form which relates the components of a
C4-valued function Φ with the components of an H(C)
-valued function F:
 0 −1
0
1 i
F =   Φ  = 
2  −1 0

0 i
(3)


1 ε
where α := −  i i1 + mci2  . This equality shows us that
 c

instead of equation (2) we can consider the equation
Dα f = 0 (4)
and the relation between solutions of (2) and (4) is
established by means of the invertible transformation
 : f = q .
Let us introduce an auxiliary notation f := f ( t , x1 , x 2 , − x3 ).
The transformation which allows us to rewrite the Dirac
equation in a quaternionic form we denote as A and
define in the following way [7]. A function Φ : R3 → C 4
is transformed into a function F : R3 → H (C ) by the
rule
Dα = −γ 1γ 2γ 3 D −1 , Denote
 
1 0  Φ
0

 
0 −i   Φ
1

 
0 −1  Φ
2
  
i 0  Φ
3
THE CHIRAL MAXWELL EQUATIONS
We will consider the time-harmonic Maxwell equations
for a sourceless isotropic chiral homogeneous medium

*

 *



 
with: E = ε −1  D + T ∇ × D and B = µ  H + T ∇ × H 




 *
*
[12], so H = H + T ∇ × H . Then we have


rot H = −iωε E ,
(5)


rot E = iωµ H ,
(6)

div E = 0 ,
(7)

div H = 0 .
(8)
Here T is the chiral scalar parameter, ω is the frequency,
ε and µ are the absolute permittivity and permeability
respectively. ε = ε0εr and µ = µ 0 µ r, where ε0 and µ 0
are the corresponding parameters of a vacuum and
εr, µ r are the relative permittivity and permeability
of a medium.
Taking into account (1) we can rewrite this system as
follows


DE = iωµ H ,
(9)
45


DH = −iωε E .
(10)
This pair of equations can be diagonalized in the following
way [6] (see also [8]). Denote



ϕ := −iωε E + κ H 


ψ := iωε E + κ H ,
(12)
ω
ε µ is the wave number.
c r r


Applying the operator D to the functions ϕ and ψ one

can see that ϕ satisfies the equation
Where κ := ω εµ =
( D − κ )ϕ = 0 ,
( D + κ )ψ = 0 .
(14)
Solutions of (13) and (14) are called the Beltrami fields
(see, e.g., [10]).
In the preceding sections it was shown that the Dirac
equation (2) is equivalent to the equation Dα f = 0 with


1 ε
α := −  i i1 + mci2  and the Maxwell equations (5)-(8)
 c

are equivalent to the pair of quaternionic equations


D−κ ϕ = 0 and Dκ ψ = 0 . Now we will show a simple
relation between these objects. Suppose that
 2 ω 2 .
2
=
=
= 2
κ
α
T2
c
(15)
L et us i nt roduce t he fol lowi ng operators of
multiplication
P ± :=




f = P + iωε E + κ H + P − −iωε E + κ H


= iωε P + − P − E + κ P + + P − H
(
)
(

iωε  
=
E ⋅α + κ H
κ
)
(
(
)
)


is a solution of (4) if E and H are solutions of (5)-(8).
It should be noticed that (16) works in both directions.
We have
Dκ = P + Dα + P − D−α
and
The fact that the Maxwell system reduces to equations


(13) and (14), where the functions ϕ and ψ are purely
vectorial provokes the natural question whether
it had

any sense to consider full quaternions ϕ and ψ and
hence four-component vectors E and H or the nature
definitely eliminated their scalar parts. Some arguments
supporting the idea of nonzero scalar parts can be found,
for example, in [2].
As we have seen equality (16) is valid under the condition
(15). Let us analyze this condition. Note that


 
1  ε2
α 2 = − < α , α >= 2  2 − m 2c 2  .
 c



Thus, when E is parallel to H , (15) has the form
1 κ ±α
M
.
2κ
It is easy to verify that they are mutually complementary
and orthogonal projection operators, and the following
equality is valid [8]
46
Where ϕ and ψ are solutions of (13) and (14) respectively
but in general can be full quaternions not necessarily
purely vectorial. In particular, we have that
D−κ = P + Dα + P − D−α .

THE RELATION ω (κ) WHEN
E

IS PARALLEL TO H
1
f = P +ψ + P −ϕ ,
(13)

and ψ satisfies the equation
(16)
Moreover, as P± commutes with D±κ, we obtain that any
solution of (4) is uniquely represented as follows
(11)
and
Dα = P + Dκ + P − D−κ .
1
T
2
=κ2 =

1  ε2
− m 2c 2  2  2
 c

(17)
H. Torres-Silva: The close relation between the Maxwell system and the dirac equation when the electric field is parallel to the magnetic field
REFERENCES
or equivalently
( ω )2 εr µr = ε 2 − m 2c4 .
[1]
F. Brackx, R. Delanghe and F. Sommen. Clifford
analysis. Pitman Res. Notes in Math. 1982.
[2]
K. Carmody. Circular and hyperbolic quaternions,
octonions, and sedenions-further results. Applied
Mathematics and Computation. Vol. 84 Nº 1,
pp. 27-47. 1997.
(18)
[3]
In general, if in (17) we formally use the de Broglie
equality p = κ =  / T , we again obtain the fundamental
relation (18).
W. Greiner. “Relativistic quantum mechanics”.
Springer-Verlag. 1990.
[4]
K. Gürlebeck and W. Sprößig. “Quaternionic
analysis and elliptic boundary value problems”.
Akademie-Verlag. 1989.
[5]
K. Imaeda. “A new formulation of classical
electrodynamics”. Nuovo Cimento. Vol. 32 B Nº 1,
pp. 138-162. 1976.
[6]
M. Gogberashvili. “Octonionic version of Dirac
equations”. International Journal of Modern Physics
A. Vol. 21 Nº 17, pp. 3513-3523. 2006.
From this equation in the case ε r = µr = 1 , that is for a
vacuum, using the well known in quantum mechanics
relation between the frequency and the impulse: ω = pc
we obtain the equality
ε 2 = p2c 2 + m 2c 4 .
Thus relation (15) between the Dirac operator and the
Maxwell operators is valid if the condition (17) is fulfilled

which quite is in agreement with (18), if and only if E
is parallel to H .
CONCLUSIONS
The main result of this paper is that the Dirac equation
can be derived from the Maxwell’s equation under a chiral
approach (equation 15). The suggested theory is the new
quantum mechanics (QM) interpretation.
The below research proves that the QM represents the
electrodynamics of the curvilinear closed (non-linear)
waves. It is entirely according to the modern interpretation
and explains the particularities and the results of the
quantum field theory.
Chiral approach means that our Universe is observable
area of basic space-time where temporal coordinate is
positive and all particles bear positive masses (electrons).
The mirror Universe is an area of the basic space-time,
where from viewpoint of regular observer temporal
coordinate is negative and all particles bear negative
masses (positrons). Also, from viewpoint of our-world
observer the mirror Universe is a world with reverse
flow of time, where particles travel from future into past
in respect to us. The two worlds are separated with the
membrane - an area of space-time inhabited by light-like
particles that travel along light-like right or left-handed
(isotropic) spirals (chiral photons).
[7]
V.V. Kravchenko. “On a biquaternionic bag model”.
Zeitschrift für Analysis und ihre Anwendungen.
Vol. 14 Nº 1, pp. 3-14. 1995.
[8]
V.V. Kravchenko and M.V. Shapiro. “Integral
representations for spatial models of mathematical
physics”. Addison Wesley Longman Ltd., Pitman
Res. Notes in Math. Series. Vol. 351. 1996.
[9]
I. Yu. Krivsky, V.M. Simulik. “Unitary connection
in Maxwell-Dirac isomorphism and the Clifford
algebra”. Advances in Applied Clifford Algebras.
Vol. 6 Nº 2, pp. 249-259. 1996.
[10]
A. Lakhtakia. Beltrami fields in chiral media.
World Scientific. 1994.
[11]
J. Vaz, Jr., W. Rodrigues, Jr. “Equivalence of Dirac
and Maxwell equations and quantum mechanics”.
International Journal of Theoretical Physics. Vol. 32
Nº 6, pp. 945-959. 1993.
[12]
H. Torres-Silva and M. Zamorano Lucero. Chiral
Electrodynamic. URLs: http://www.chiral.cl
47
vol. 16 Nº 1,
Nº 1, 2008,
2008 pp. 48-52
DIRAC MATRICES IN CHIRAL REPRESENTATION AND THE CONNECTION
WITH THE ELECTRIC FIELD PARALLEL TO THE MAGNETIC FIELD
MATRICES DE DIRAC EN REPRESENTACIÓN QUIRAL Y LA CONEXIÓN
CON EL CAMPO ELÉCTRICO PARALELO AL CAMPO MAGNÉTICO
H. Torres-Silva1
RESUMEN
En este trabajo se presenta una expresión de la transformación general de Foldy-Wouthuysen a la representación quiral de
las matrices de Dirac interactuando con un campo de fermión. La
 hipótesis
 es que a través de la multiplicación de la matriz
de Pauli por las ecuaciones quirales de Maxwell en el caso de E = iη H , se obtiene la ecuación quiral de Dirac. Esta es la
prueba del teorema de que la mecánica de ondas de partícula cuántica representa una electrodinámica especializada.
Palabras clave: Transformación de Foldy-Wouthuysen, ecuación quiral de Dirac, electrodinámica.
ABSTRACT
In this paper we offer an expression of the general Foldy-Wouthuysen transformation in the chiral representation of Dirac
matrices interacting with fermion
 field.Our hypothesis is that through the multiplication of the Pauli matrix and Maxwell’s
chiral equations in the case of E = iη H , one obtains the Dirac’s chiral equation. This is the proof of the theorem that the
wave mechanics of quantum particles represent a specialized electrodynamic.
Keywords: Foldy-Wouthuysen transformation, chiral Dirac equation, electrodynamics.
CHIRAL DIRAC MATRICES
The paper offers an expression of the general FoldyWouthuysen transformation in the chiral representation of

Dirac matrices interacting with fermion field ψ ( x , t .
The paper [1, 2] discuss the theory of interacting
quantum fields in the Foldy-Wouthuysen representation
[3]. These papers offer, in particular, the relativistic
nonlocal Hamiltonian HFW in the form of a series in
terms of powers of charge e. Quantum electrodynamics
in the Foldy-Wouthuysen (FW) representation has been
formulated using Halmitonian HFW and some quantum
electrodynamics processes have been calculated within
the lowest-order perturbation theory. As a result, the
conclusion has been made that the FW representation
describes some quasi-classic states in the quantum field
theories. Both particles and antiparticles are available in
these states. Particles, as well as antiparticles, interact
)
with each other. However, there is no interaction of
real particles with antiparticles – such interaction is
possible only in intermediate (virtual) states. The FW
representation modification is required to take into a
account real particle/antiparticle interactions. In the
papers [1, 2] such modification has been made using the
symmetry identical to the isotropic spin symmetry owing
to invariance of final physical results under change of
sings in the mass terms of Dirac Hamiltonian HD and
Hamiltonian HFW. In the modified Foldy-Wouthuysen
representation, real fermions and antifermions can be
in two states characterized by the values of the third
1
component of the isotropic spin S 3f = ± ; real fermions
2
and antifermions interacting with each other must have
3
opposite signs of S f . Quantum electrodynamics in
the modified FW representation is invariant under P–,
C–, T– transformations. Violations of the introduced
1 Instituto de Alta Investigación. Universidad de Tarapacá. Antofagasta Nº 1520. Arica, Chile. E-mail: [email protected]
48
H. Torres-Silva: Dirac matrices in chiral representation and the connection with the electric…
symmetry of the isotropic spin lead to the corresponding
violation of CP– invariance. The standard model in the
modified FW representation was formulated in the papers
[1, 4]. It has been shown that formulation of the theory
in the modified FW representation doesn’t require that
Higgs bosons should obligatory interact with fermions
to preserve the SU (2)– invariance, whereas all the
rest theoretical and experimental implications of the
Standard model obtained in the Dirac representation are
preserved. In such a case, Higgs bosons are responsible
only for the gauge invariance of the boson sector of the
±
theory and interact only with gauge bosons Wµ , Z µ ,
gluons and photons.
)
0
αi = 
σ
i
0 I i
I 0 
0
,γ 5 = 
, γ = γ 0α i (1)
 ,β = γ = 

0
−
I


 I 0
0
σ
Here we propose to change the Foldy-Wouthuysen
transformation form by using the chiral representation
of Dirac matrices.
σ i
i
αc = 
0
0 I
I 0  i
0 
0
,γ =
, γ = γ c0α ci (2)
 , βc = γ c = 
 I 0 5  0 − I  c
−σ i 
The chiral representation (2) is commonly used in the
modern gauge field theories and in the Standard Model,
in particular.
First consider the structure of equations describing
the components of the wave functions ψD (x) for the
two representation of Dirac matrices considered in the
paper.
In relations (1), (2) and below the system of units with  = c = 1
is used; x, p, are 4-vectors; the inner product is taken as
∂
; σk
xy = xµyµ = x0y0 – xkyk µ = 0,1,2,3, k = 1,2,3; p µ = i
∂x µ

1, µ = 0
are Pauli matrices; α µ =  i
; ψD (x) is the
α , µ = k , k = 1, 2, 3
)
)
)
four-component wave function, ϕ ( x , χ ( x ,ψ R ( x ,ψ L ( x
)
)
)
)
)
)
)
)
) ;
)
(3)
With representation (2), relation (3) looks like
)
)
ψ ( x 
 
p0ψ D ( x = (α ⋅ p + β m ;ψ D ( x =  R  ;
ψ L ( x 
 
 p0ψ R ( x = σ ⋅ pψ R ( x + mψ L ( x 
;

 
 p0ψ L ( x = −σ ⋅ pψ L ( x + mψ R ( x 
(4)
 
p0 − σ ⋅ p
  −1
ψ L (x =
ψ R ( x = ( p0 + σ ⋅ p mψ R ( x ;
m
 
p +σ ⋅ p
  −1
ψ R (x = 0
ψ L ( x = ( p0 − σ ⋅ p mψ L ( x ;
m
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Relations (4) use the operador equality:

p02 = E 2 = p2 + m 2 Comparison between relations (3) and (4) shows that with
the substitution below,
 
m ↔ σ ⋅ p, β ↔ γ 5 (5)
These relations transform into each other.
The Foldy-Wouthuysen transformations for the energy and
chiral representations of Dirac matrices also transform
into each other if the substitution (5) is made.
Thus, the general Foldy-Wouthuysen transformation
with Dirac matrices in the chiral representation
( ) (1 + δ
chir
0
U FW
= U FW
chir
chir
1
)
+ δ 2chir + +δ 3chir + .... , as well
as the fermion Hamiltonian in the Foldy-Wouthuysen
representation
are the two-component wave functions.
The following operator relations are valid for the free
Dirac equation with representation (1):
)
)
)
In the papers mentioned above, the energy representation
of Dirac matrices derived by Dirac himself is used:
i
ϕ (x
 
p0ψ D ( x = (α ⋅ p + β m ;ψ D ( x = 
 χ (x
 
 p0ϕ ( x = σ ⋅ pχ ( x + mϕ ( x 
;

 
 p0 χ ( x = σ ⋅ pϕ ( x − m χ ( x 
−1  
χ = ( p0 + m σ ⋅ pϕ ;
−1  
ϕ = ( p0 − m σ ⋅ pχ;
chir
H FW
= γ 5 E + qK1chir + q 2 K 2chir + + q3 K 3chir + ... can be obtained. From the corresponding expressions for
en
en
with Dirac matrices in the energy representation
U FW
, H FW
 
(see [1, 2]) with substitution m ↔ σ ⋅ p, β ↔ γ 5 , we
have
49
the relations
 
 p0ψ R ( x = σ ⋅ pψ R ( x + mψ L ( x

 
 p0ψ L ( x = −σ ⋅ pψ L ( x + mψ R ( x
)
)
)
)
)  )
(6)
Also the relations (6) can be obtained under the chiral
approach of Maxwell’s Equations where the electric field




E is parallel to the magnetic field H [5], that is E = iη H ,
where η = µ0 / ε 0 .
If one wanted to describe the hydrogen gas by means
of electrodynamics one should start from the firmly
established experience that the hydrogen gas may
absorb and reemit electromagnetic energy, and that
without external intervention there is no indication
that the gas to contain electric charges [6-11]. Thus
we consider the hypothesis witch visualizes the gas
as charge free electromagnetic field as the starting
point with the lest number assumptions; and so we
try characterize the field by the covariant chiral
Maxwell system [5]


∂ 
rotE + µ (1 + Tm ∇×)
H = 0, divε (1 + Te ∇×) E = 0 (7)
c∂t


∂ 
rotH − ε (1 + Te ∇×)
E = 0, div µ (1 + Tm ∇×) H = 0 (8)
c∂t
 
Here, T is the chiral scalar factor with divTe ,m ∇ × E ( H ) =
 
Te ,m div∇ × E ( H ) = 0, and the condition of charge-free
( )
 
1   mc  
rotE ( H ) =  E ( H ) ±
E(H ) T

(10)
Our hypothesis is that through the multiplication of
the
 matrix
 Pauli for chiral Maxwell’s equations with
E = iη H , one obtains the chiral Dirac equation (6).
Using the algebraic relation [12]
50
(11)

ω
mc
 σˆ ⋅ ∇ ± i c  ψ E (ψ H ) = −  ψ H (ψ E ) (12)
Below the system of units with  = c = 1 equation (12)
is exactly equal to the chiral Dirac equation (6), if
ψ E (ψ H ) = ψ R (ψ L ) .
To probe this close connection we can obtain the well known
normal Dirac equation, we get for (7, 8) the equations


∂ 
rotE + µ0 (1 + Tm ∇×)
H = 0, divE = 0 c∂t
(13)


∂ 
rotH − ε 0 (1 + Te ∇×)
E = 0, divH = 0 c∂t
(14)


with E ⊥ grad ε , H ⊥ grad µ . Equations (13) and (14)
can be transformed as:
µ0 (1 + Tm ∇×) → µ (ω , mc 2 ),
ε 0 (1 + Tm ∇×) → ε (ω , mc 2 ).
So, scalar multiplication of the rot equations in (10, 11)
by the Pauli-vector, and using the algebraic relation [12]



(σˆ ⋅ ∇ σˆ ⋅ A = divA + iσˆ ⋅ rotA we have
)(
(9)
 
Solv i ng t h e wave e q u a t io n fo r E H w it h
Te = Tm = T =  / 2mc , and by considering we have



where ψ E (ψ H ) = σˆ ⋅ E (σˆ ⋅ η H ) .
by means of


divE = 0 besides divH = 0 
in equation (10) together with the two div equations (9),
transform that system (10) in to
CHIRAL APPROACH OF
MAXWELL’S EQUATIONS

(σˆ ⋅ ∇ ) (σˆ ⋅ A) = divA + iσˆ ⋅ rotA )
 ε ∂



(σ ⋅ ∇ σ ⋅ H − c ∂t iσ ⋅ E = 0 



 µ ∂


σ ⋅H = 0 
(σ ⋅ ∇ σ ⋅ E +
∂
c
t




 E ⊥ grad ε , H ⊥ grad µ




)(
)
(
)
)(
)
(
)
(15)
Equation (15) can be expressed in terms of in matriz
notation this reads

 0 σ 
 ε1 0  1 ∂   i σ ⋅ E

 ⋅ ∇ −  0 µ1 c ∂t   σ ⋅ H
 σ 0 
 


E ⊥ grad ε , H ⊥ grad µ
(
(
) = 0 
)   (16)


H. Torres-Silva: Dirac matrices in chiral representation and the connection with the electric…
Denoting the quantity on witch the differential operators
act by ψD, that is

i σ ⋅ E


 σ ⋅ H
)
)
(
(
 iE3
  iE
= +
  H3

 H+
'
iH −   Ψ 1

−iE3   Ψ '2
=
H−   Ψ '
  3
− H3   Ψ '
 4
Ψ ''1 

Ψ ''2 

Ψ ''3 
Ψ ''4 
 0 σ
 σ 0  = γ ≡ α (17)
(18)
between the Pauli and Dirac matrices, we get for (13)
system


 ε1 0  1 ∂ 
 γ ⋅ ∇ − 
 Ψ D = 0

 0 µ1 c ∂t 

 




 E ⊥ grad ε , H ⊥ grad µ
(19)
Ψ D = ψ D e − iω t (21)
ω  ε1 0  

Ψ = 0 γ ⋅ ∇ − i 
c  0 µ1  D

γ ⋅ ∇Ψ +
i  ω + mc

c 
0
2

 Ψ D = 0 (23)
ω − mc 2 
0
(23’)
And the transformation to a chiral Dirac equation is trivial
by using relation (5).
The equations (20) as well as (21) or (22), show in addition
that the electrodinamical and the wave mechanical field
component are connected by simple linear relation, the
same holding true for the refraction (ε, µ) in relation to
the scalar T.
This isomorphism can be checked easily and directly
because the eight Eq. (10, 11) may be combined into two
systems of four equations each, in the following way:



±i rotH  ic −1ε E + divH = 0
3
3




±i rotH  ic −1ε E − rotH + c −1ε E = 0 
1
2


2
1
 (24)


−1 
±idivE − rotE 3 − c µ H3 = 0





−1
−1
±i rotE 2  ic µ H 2 − rotE 1 + c µ H 1 = 0 
(
(
)
)
(
)
)
)
(
(
)
Inserting here the first or second wave function of (21) into
the first system (upper signs) or the second one (lower signs),
respectively, the wave functions of (20) ends up immediately,
in both cases and we are back to Dirac again
Using a chiral representation of the Foldy-Wouthuysen
transformation for the Dirac equation we show that the
same result can be obtained with a chiral electrodynamics
using the matrix Pauli.
With this we proof the theorem that waves mechanic of
quantum particle represents a specialized electrodynamics.
The result seems unambiguous and incompatible with the
current doctrine which rest on a particle interpretation.
(22)
If we use equation (13) in (22), it’s agreement with the
Dirac amplitude equation
)
CONCLUSION
finally yields the amplitude equation
)
(20)
Independently represent a system of functions solving
(16). From this, a separation of the time dependence
according to
)
(
Here one has to bear in mind that each of both columns
matrix (14) that is
 iE1 + E2 
 iE3 
 −iE 
 iE − E 
3  2
ΨD =  1
and Ψ D = 
 H1 − iH 2 
 H3 




 H1 + iH 2 
 − H3 
 
p0ψ D ( x = (α ⋅ p + β m ;ψ D ( x with X ± = X1 ± iX 2 and considering the well-known
connection
is complete. Now nor ma l izing eq. (23) wit h
 = 1, c = 1, γ ⋅ ∇ = α ⋅ p , we can write as
REFERENCES
[1]
V.P. Neznamov. Physics of Elementary Particles
and Atomic Nuclei (EPAN). Vol. 37 Nº 1. 2006.
[2]
V.P. Neznamov. Voprosy Atomnoi Nauki I Tekhniki.
Ser: Teoreticheskaya I Prikladnaya Fizika. Issues
1-2, p. 41, hep-th/0411050. 2004.
51
[3]
L.L. Foldy and S. A. Wouthuysen, Phys. Tev 78,
29. 1950.
[4]
V.P. Neznamov. Hep-th/0412047. 2005.
[5]
H. Torres-Silva and M. Zamorano Lucero. Chiral
Electrodynamic. URLs: http://www.chiral.cl
[6]
J.R. Oppenheimer. Phys. Rev. Vol. 38, p. 725. 1931.
[7]
H.E. Moses. Sup. Nuovo Cimento. Serie X. Vol. 7.
Nº 1. 1958.
[8]
T. Ohmura Prog. Theor. Phys. Vol. 16, p. 684. 1956.
52
[9]
S.N. Gupta. Theory of longitudinal photons in
quantum electrodynamics. Proc. Phys. Soc. Vol. 63,
pp. 681-691. 1950.
[10]
F. Reines and W. H. Sobel, Test of the Pauli
Exclusion Principle for Atomic Electrons, Phys.
Rev. Lett. Vol. 32, pp. 954. 1974.
[11]
W. Heitler Quantum Theory of Radiation, 2nd Ed.,
Oxford University Press, Oxford, p. 1. 1944.
[12]
H. Sallhofer. “Maxwell Dirac isomorphism”.
Z. Naturforsch. Vol. 41 a, p 1067. 1986.
Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008, pp. 53-59 H. Torres-Silva: Maxwell equations for a generalised lagrangian functional
MAXWELL EQUATIONS FOR A GENERALISED LAGRANGIAN FUNCTIONAL
ECUACIONES DE MAXWELL PARA UNA FUNCIONAL DE LAGRANGE GENERALIZADA
H. Torres-Silva1
RESUMEN
En este trabajo se aborda el problema de la construcción de la funcional de Lagrange de un campo electromagnético. Se
introducen las ecuaciones generalizadas de Maxwell de un campo electromagnético en el espacio libre. La idea principal
se basa en el cambio de función de Lagrange en virtud de la acción integral. Por lo general, la funcional de Lagrange, que
describe el campo electromagnético, se construye con el cuadrado del tensor de campo electromagnético. Ese término
cuadrático es la razón, desde un punto de vista matemático, de la forma lineal de las ecuaciones de Maxwell en el espacio
libre. Se obtienen las ecuaciones no lineales de Maxwell sin considerar esta suposición. Las ecuaciones obtenidas son
bastante similares a las conocidas ecuaciones de Maxwell. Se analiza el tensor de energía del campo electromagnético en
un enfoque quiral de la Lagrangiana de Born Infeld en relación con la constante cosmológica.
Palabras clave: Lagrange, acción, ecuaciones de Maxwell, Born Infeld.
ABSTRACT
This work deals with the problem of the construction of the Lagrange functional for an electromagnetic field. The generalised
Maxwell equations for an electromagnetic field in free space are introduced. The main idea relies on the change of
Lagrange function under the integral action. Usually, the Lagrange functional which describes the electromagnetic field
is built with the quadrate of the electromagnetic field tensor Fik . Such a quadrate term is the reason, from a mathematical
point of view, for the linear form of the Maxwell equations in free space. The author does not make this assumption
and nonlinear Maxwell equations are obtained. New material parameters of free space are established. The equations
obtained are quite similar to the well-known Maxwell equations. The energy tensor of the electromagnetic field from a
chiral approach to the Born Infeld Lagrangian is discussed in connection with the cosmological constant.
Keywords: Lagrange, action, Maxwell equations, Born Infeld.
INTRODUCTION
The action integral (built to formulate the least-square
principle [1]) for a process in an electromagnetic field
has the following form:
I=

∫ ∫  ( S ) − ( ρV − j ⋅ A) − 2 ( µ
1
v
T V
−1
0

B2 − ε 0 E 2  dVdt (1)

)
All physical phenomena in the electromagnetic field take
place so the action integral has the minimal value δΙ = 0.
The theory of the electromagnetic field [1] leads to the
1
Lagrange motion equations for an electric charge in the
electromagnetic field and defines the electromagnetic
field tensor:
Fik =
∂Ak
∂x
i
∂Ai
−
∂x k
(2)
Eqns. (2) are equivalent to the first pair of Maxwell
equations:
∂Fik
∂x
l
+
∂Fkl
∂x
i
+
∂Fli
∂x k
=0
for
i≠k ≠l ≠i
(3)
53
Or, in three-dimensional notation:
curlE = − B
speed in free space for these new Maxwell equations will
also be equal to “c”.
divB = 0 and
(4)
Making calculations of variation for the functional I with
respect to the four-dimensional potential Ai one obtain
the second pair of Maxwell equations:
∂F
ik
∂x k
= − µ0 j i (5)
Or in equivalent vector notation:
)
div ( ε 0 E = ρ and curl
(
I=
The second pair of Maxwell equations in the case of
the generalised action integral eqn. (8) is obtained after
evaluating the variation of the action with respect to the
four-dimensional potential Ai. We can denote
µ0−1 B
)
= j + ε 0 E (6)
From the mathematical point of view, the demand for a
linear form of Maxwell equations for free space compels
one to assume that the field term (the third in integral
eqn. (1)) must be built with the electromagnetic field tensor
Fik second power (the exponent of any power function
under the action integral is one less after calculating the
variation). So, the equations obtained are linear with
respect to the electromagnetic field tensor Fik. The action
integral I could be rewritten in the following form:
MAXWELL EQUATIONS FOR THE
GENERALISED FUNCTIONAL
∫ ∫ ( Sv − ( ρV − j ⋅ A) − ∆eV ) dVdt
(7)
TV
1
1
1
∆ = ∆eV = µ0−1 B2 − ε 0 E 2 = µ0−1F ik Fik
2
2
4
The linear form of the Maxwell equations, from the
mathematical point of view, is arbitrarily assumed by
eqn. (7). In addition the linear character of Maxwell
equations for free space (as well as for air) has been
confirmed by many experiments. There is no doubt that
linear Maxwell equations, within experimental precision, are
satisfied, however, we could not reject other mathematical
forms of the electromagnetic field equations.
Is we assume, more generally, that the action integral is built
with the help of a function f (·), it could be written:

1

1
I = ∫ ∫  Sv − ( ρV − jA − f  µ0−1 B 2 − ε 0 E 2   dVdt (8)
2
2


TV
)
(the codomain for the function f (·) is the real set).
Under this assumption new Maxwell equations having
the same mathematical structure are obtained. The wave
54
)
(
)
1

= ∫ ∫ − j iδ Ai − f ′ ( ∆  2 µ0−1F ik δ Fik  dVdt
4

TV
)
According to eqn. (2) we could write
0 = δI
1
 ∂δ Ak ∂δ Ai  
= − ∫ ∫ j iδ Ai + f ′ ( ∆  µ0−1F ik 
− k   dVdt
 ∂x i
∂x  
2
TV
))
(
After interchanging the indices ‘i’ and ‘k’ in the final
term we obtain:
Where
(
1

δ I = ∫ ∫ − j iδ Ai − δ f  µ0−1F ik Fik  dVdt
4

TV
(
))
(
))
1
∂δ A
∂δ Ai 
0 = ∫ ∫ j iδ Ai + f ′ ( ∆  µ0−1F ik ( i k − F ik
) dVdt
2
∂x
∂x k 
TV

∂δ Ak 
= ∫ ∫ j k δ Ak + f ′ ( ∆  µ0−1F ik
 dVdt

∂x i 
TV
Hence:
)
)


∂f ′ ( ∆ µ0−1F ik δ Ak ∂f ′ ( ∆ µ0−1F ik
−
0 = ∫ ∫  j k δ Ak +
δ Ak  dVdt
i
i


∂x
∂x
TV
Using the Gauss theorem and taking into account the fact
of the disappearance of the four-dimensional potential at
the boundary of the four-dimensional space we obtain
)
∂f ′ ( ∆ F ik
∂x
k
= − µ0 j i (9)
Eqns. (9) include the second pair of Maxwell equations
in the form eqns. (5) and (6). Whereas, on using the
function f (·) as the identify function, its derivative will
be equal to one; i.e.
1

f ′  µ0−1F ik Fik  = f ′ ( ∆ = 1
4

)
H. Torres-Silva: Maxwell equations for a generalised lagrangian functional
This means that the Maxwell equations in the form
eqn. (9) include the classical Maxwell equations in the
form of eqn. (5) and (6).
INTERPRETATION OF THE ESTABLISHED
EQUATIONS
In three-dimensional notation, eqn. (9) has the following
form:

1
 
1
div  ε 0 f ′  µ0−1 B2 − ε 0 E 2  E  = ρ
2
2
 


1
 
1
curl  v0 f ′  µ0−1 B2 − ε 0 E 2  B
2
2
 

1
 
1
∂
ε 0 f ′  µ0−1 B2 − ε 0 E 2  E 

2
∂t 
2
 
The second pair of Maxwell equations could be rewritten
in the same form as the well-known Maxwell equations
(eqns. (5) and (6)):
)
)
div ( ε E = ρ
curl ( vB = j +
∂
(ε E ∂t
)
(10)
Where it was denoted:
1

1
ε = ε 0 f ′  µ0−1 B2 − ε 0 E 2 
2
2

1

1
µ 0−1 = µ0−1 f ′  µ0−1 B2 − ε 0 E 2 
2
2

(11)
) )
)
f (⋅ = (⋅ + (− k 0 2 T 2 ) (⋅
)
)
⇒ f ′ (⋅ = 1 + κ (⋅ Thus:
(
ε = ε 0 1 + κµ0−1 B2 − κε 0 E 2
µ 0−1
=
µ0−1
(
1 + κµ0−1 B2
(
)
curl µ 0−1 B = j +
curlE = − B
divB = 0
∂
(ε E (13)
∂t
)
With modern levels of measurement accuracy, we are
able to use laboratory devices that enable determination
of the value of magnetic flux density (or electric field
strength) with very high accuracy (0.01%), and the material
parameters with the same relative error. In a magnetic
field B=2T the variation of this material parameter will
not be observed according to eqn. if:
κ < π × 10 −11  J −1   
(14)
The constant ‘κ’ is so small only in the case of strong
magnetic or electric fields may the linear Maxwell equations
deformation be observed and detected.
GENERALIZATIONS OF MAXWELL THEORY
FROM BORN-INFELD THEORY
There are nonlinear electromagnetic field theories, e.g.
Born-Infeld theory of the charged particle [2, 3]. In this
Born-Infeld theory the nonlinear Maxwell equations are
obtained from the following action integral:


c2
c2
I = ∫ ∫ b 2  1 − 1 + 2 I1 − 4 I 22 ,  dVdt = ∫ ∫ b 2 (1 − R dVdt 

b
b

TV
TV
)
)
− κε 0 E
)
div ( ε E = ρ
The derivative f’(·) which appears in the Maxwell equations
is the reason for the nonlinear character of the generalised
Maxwell equations with respect to electric field strength
and magnetic flux density. The level of ‘deformation’ of the
Maxwell equations in comparison with the linear Maxwell
equations is determined by the constant ‘κ’. The less is
the value of constant ‘κ’, the less is the influence of the
nonlinear term in eqn. (12). We may evaluate (roughly)
the value of this unknown constant.
Lets us assume that the function f (·) can be almost linear
or linear. Many experiments confirm that if the nonlinear
character of the electromagnetic field equations exists, it
cannot be strong; it must be weak. It seems to be reasonable
to consider only the first nonlinear term of the Taylor
series. Here we propose that:
  = ε 0 µ0
εµ
The Maxwell equations (in spite of their nonlinear
character) still have the same form:
And
= j+
According to eqn. (11) and (12) the permittivity and
reluctivity (or permeability) for free space have been
changed with respect to the strong electric or magnetic
field. The strong electromagnetic field causes a change
of the free space coefficients. The multiplication ‘ε’ by
‘µ’ is independent of the function f (·) and equal to ε0µ0:
2
)
(12)
(15)
On this basis, the equations obtained are supposed to
be valid inside the electric particle. For fields that are
weak compared to the critical strength b, the Born-
55
Infeld Lagrangian becomes the Lagrangian of classical
Maxwell theory.
So the energy, the momentum, and the Poynting vector,
are now given, respectively, by
The well-known Born-Infeld Lagrangian is usually
written as


E 2 + ( E ⋅ B / a) 2


2
2
2

  1 + B − E − ( E ⋅ B)
 (21)
3 
2
4
= ∫d x
a
a


2
2
2 
−
(
⋅
)
+a2 ( 1 + B E − E B ) 


a2
a4
L BI =
b2
µ0 c
(1 − R ) ,
2
R = 1+
c2
b
2
I1 −
c2
b
I 22 , (15’)
4
ε field
Where
I1 = B2 −
1
c2
E2 =
c
1
F F ik , I 2 = B ⋅ E = Fik F ik , b i s a
2 ik
4
maximum electric field strength (in the absence of magnetic
field). If b2 is very much larger than E2 and c2 B2, then
LBI ≈ − (1 / 2 µ0 I1 and we recover linear Maxwell theory.
We remark here that in the limit as c → ∞, LBI tend to zero,
while cLBI approach a well-defined, non-zero limit.

Pfield = ∫ d 3 x
E×B
1+
)
F=
1
F F ik = B2 − E 2 2 ik
1
G 2 = ( Fik* F ik )2 = ( B ⋅ E )2 4
LBI = a 2 (1 − R ) ,
R = 1+
F
a
−
2
G ,
a4
 F ik - * F ik G/a 2 
∂ν 
 =0,
R


∂ j Fik + ∂ k Fji + ∂i Fkj = 0 T ik =
F µ j Fjk + G 2ηik / a 2
R
+ ηik a 2 R 1+
B − E 2 ( E ⋅ B)2 −
a2
a4
2
(23)
The volume density of the Lagrangian function in a region
outside the electrical charges and currents is equal to
(15)
( )
f ∆ (24)
Where
∆=−
v0
4
Fαβ F αβ = −
v0
F F gαγ g βδ
4 αβ γδ
= a0 Fαβ Fλγδ gαγ g βδ
(25)
And gik means the second-order metric tensor of space
[1, 3, 4].
(18)
Let us evaluate the energy tensor T ik by the definition
[1, 5] in the following form:
(19)
(20)
In deriving this result, use has been made of the identity
*
Fµν
Fνρ = −Gδ ρν .
56
a4
(17)
Here, we find that the symmetric energy-momentum
tensor for that theory is given by
(22)
ENERGY TENSOR OF THE
ELECTROMAGNETIC FIELD
The field equations for Born-Infeld theory are
( E ⋅ B)2 (16)
1 ik ρσ
*
Fρσ is the dual field strength tensor,
Where Fik = ε
2
Making c=1 we have that equation (15) can be expressed as
2
a2
−
E×B
S=
Since the Lagrangian density must be a Lorentz scalar,
the electromagnetic field has only two gauge invariant
Lorentz scalars, namely
B2 − E 2
1
− gTik =
2
(
∂ f (∆
)
−g
∂gik
)


∂ ∂ f (∆ −g
− l 
 gik 
∂x 
∂ l 

 ∂x 
(
)
)


 (26)



The tensor satisfies the energy conservation law in the
tour-dimensional form:
Ti k, k = 0 (27)
Substituting eqn. (20) into eqn. (26) one obtains:
(
2 ∂ f (∆
Tik =
)
−g
∂gik
−g
)
Because function f (∆) is independent of the derivatives
of the metric tensor.
Thus one obtains:
Tik = 4 a0
∂f δ
F F − gik f ( ∆ ∂∆ i kδ
)
(28)
For the function f (·) given by the first two Taylor series
terms we could write:
)
f ( ∆ = ∆ + κ∆
Thus the energy tensor is equal to:
(
Tki = Tki |κ = 0 +2κ∆ Tki |κ = 0 +δ ki ∆
T = Tαα = Tαα
κ =0
(
+ κ Tαα
κ =0
c2
κ∆ (32)
The sign of constant k thus, take into account an electrostatic
field forced by one charged particle. Such a field has
spherical symmetry. The Riemannian curvature scalar
for two-dimensional space, where only one external
charge is situated, must be non-negative, R ≥ 0. Is not,
the Riemannian curvature scalar tensor is negative, the
space would have two radii of curvature, one positive and
the other negative. This is impossible with respect to the
assumed spherical symmetry of the electric field (forced
by one electric charge), therefore:
κ ≤ 0
(33)
We can obtain an especial result. Und er a chiral approach,
using equations (21, 22) with E = iB ,we obtain S = 0 and an
electromagnetic term which correspond to a cosmological
constant given by 8π Gε 0 / c 4 = 1.8382 ⋅ 10 −54 Volt −2 . This
allows the close connection between the electromagnetism
and the gravitation (see annex).
)
The trace of this energy tensor is equal to:
32π G
In the case of a nonlinear electromagnetic field theories, e.g.
Born-Infeld theory of the electromagnetic particle [5].
)
f ′(∆ = 1+κ
⇒
R=−
)
+ 4 ∆ = 4κ∆ (29)
because, in the case κ = 0, the trace of the covariantcontravariant energy tensor disappears:
CONCLUSIONS
According to the main Einstein equations, for energy
field for which one can introduce the energy tensor [1, 3,
5] it could be written:
Generalised Maxwell equations include the classical
Maxwell equations of the electromagnetic field for weak
fields. The reluctivity and permittivity of free space
are changed. If the constant ‘κ’ cannot be omitted, the
Riemannian-Christoffel curvature tensor is not equal to
zero. The constant ‘κ’ is not positive: κ ≤ 0. In the case
of a nonlinear electromagnetic field theory, e.g. BornInfeld theory of the electromagnetic particle [5], we can
obtain an especial result. Under a chiral approach, with
E = iB, we obtain S = 0 and an electromagnetic term
which correspond to a cosmological constant.
1
8π G
Rik − gik R = − 2 Tik 2
c
ANNEX
Tαα |κ = 0 = 4 a0 F ij Fij − δαα ∆ = 0
The trace of the energy tensor is not negative. The trace
is equal to zero if and only if the constant ‘κ’ vanishes.
(30)
Contraction with respect to the indices ‘i’ and ‘k’ gives
the Riemannian curvatura scalar of the electromagnetic
field:
R=−
8π G
c2
T
(31)
Substituting eqn. (29) into eqn. (31) one finally obtains
This work discovers the space-time curvature carried by
the electromagnetic field and provides a new unification
of geometry and classical electromagnetism. The new
unification contains the Einstein equations to handle the
mechanics and permits the derivation of the Maxwell
equations from the full second Bianchi identities. This
is a purely classical work and quantum considerations
are merely mentioned.
57
Central to this work are the requirements that the
electromagnetic field be expressed as a two form F
and fit into general relativity under the demand that
the total stress-energy tensor used in the Einstein
equations contain the Maxwell stress-energy tensor
T Max. In the notation with the conventions of [1] and
in S.I. units T Max is
i
TMax
k =
ε 0 ji
( F Fjk + *F ji * Fjk )
2
−12
where ε 0 = 8.85418782 ⋅ 10 farad/meter is the electric
vacuum permittivity.
Originally [2] general relativity was conceived as a
unification of mechanics and geometry that explained
gravitation. It was just a bonus [3] that electromagnetism
also entered the unification via equation. If the Maxwell
stress-energy tensor carried all the properties of the
electromagnetic field, showing electromagnetism to be
entirely reducible to mechanics, that would have been
the end of the story.
However, the electromagnetic field has polarization or
phase information that is not contained in the Maxwell
stress-energy tensor [4]. Since Weyl’s conformal tensor, the
totally traceless piece of Riemann curvature, is supposed
to contain the phase or polarization information carried
by gravitational radiation, one should expect it to do the
same for electromagnetic radiation.
This is born out by the discovery of a piece of the
Weyl conformal tensor that depends explicitly on the
electromagnetic field and contains this polarization or
phase information. It is denoted by TMax CF, called “the
local gravitational field of the electromagnetic field’’,
and given by:
C ikjl = 8π
Gε 0  3 ik
1
( F Fjl − *F ik * Fjl ) − δ ikjl F ik Fik +
4 2
4
c 

1
+ ηikjl F ik Fik 
4

where G = 6.6726 ⋅ 10 −11 Newton-meter2/kilogram 2 is
Newton’s gravitational constant, c = 2.99792458 ⋅ 108
meter/second is the speed of light, is a fully antisymmetric
tensor. The traces in the expression for CF are the Lorentz
invariants of the electromagnetic field FikFik = –2 (E2–
c2B2) and * FikFik = 4c (E∙B), where E is the electric
field strength in Volt/meter and B is the magnetic field
strength in Tesla.
58
The major discovery of this work is the expression for
CF. The arguments that led to that expression are quite
general and should defeat the criticism that CF was
built on algebraically special black holes and will fail
elsewhere. It would be useful to have a physical solution
to the Einstein-Maxwell equations with non-zero currents
that were not overwhelmed by symmetry. Then one could
extend this analysis into the currents and see how the full
second Bianchi identity works there. Further successful
examples will give knowledge and comfort; but will not
prove the generality for CF that is claimed here. However,
a single credible counterexample or the observation of a
magnetic monopole will vitiate this work.
The small coupling constant required by the Einstein
Gε
equations, 8π 40 = 1.8382 ⋅ 10 −54 Volt −2 , permits the
c
superposition of electromagnetic fields. It has also led
many to believe that the gravitational consequences of
electromagnetism are insignificant.
Nothing could be further from the truth. It is a matter of
principle to unify classical electromagnetism and gravitation
and the curvature-based unification presented here allows
the electromagnetic field to appear as an algebraically
special piece of curvature. This fulfills the nineteenth
century speculation that gravity and electromagnetism
are both aspects of Riemann curvature.
This theory is not experimentally vacuous. The smallness
of the coupling constant merely means that it could be along
time before curvature detectors are sufficiently sensitive
while withstanding an intense electromagnetic field; or
sufficiently sensitive over very long distances having less
intense fields. One wonders about the consequences of
CF in the environment around very strongly magnetised
neutron stars [7]. Further, what are its consequences in the
Jacobi equation for geodesic separation that might apply to
trans galactic travel? When two electromagnetic fields are
superposed could the interaction terms in the curvature have
any bearing on the problem of emission or absorption?
The physical geometry of space-time is determined by
specifying the metric tensor or the full curvature tensor
[6]. The Einstein equations, which link classical mechanics
to physical geometry, may be written as
M1αβ
γδ =
8π G  1 αβγ  ρ 1 ρ 
 − δγδρ Tλ − 4 δ λ T 
c4  2


and
M 2αβγδ =
8π G  1 αβ 
 − δγδ T 
c 4  12
where T is the total stress-energy tensor and T its trace.
There is no mention of Weyl’s conformal tensor that would
complete the specification of the physical geometry.
Placing constraints on Weyl’s conformal tensor is
the novel feature of this work. Such constraints are
meant to limit the solutions to those with a physical
gravitational field. If the constraints are too limiting
and they forbid physical solutions, then they will have
to be altered. Similar constraints might deal with the
embarrassing number of Ricci flat universes, which
may or may not describe gravitational radiation. It is
an open question whether the Einstein equations will
have to be extended to the full curvature to handle
gravitational radiation.
[2]
A. Einstein. “The Principle of Relativity”. Dover
Publications, New York. 1952.
[3]
D. Jackson. Classical Electrodynamics. 3rd ed, pp. 273280, John Wiley & Sons, New York. 1998
[4]
T.T. Wu and C.N. Yang. Concept of Non integrable
Phase Factors and Global Formulation of Gauge
Fields. Phys. Rev. D. Vol. 12, pp. 3845-3857. 1975.
[5]
H. Torres-Silva. “A new relativistic field theory
Nº 1, pp. 111-118. 2008.
[6]
Nº 1, pp. 6-23. 2008.
[7]
W. E. Thirring. “An alternative approach to the
theory of gravitation”. Ann. Phys. USA. Vol. 16,
pp. 96-117. 1961.
REFERENCES
[1]
C. W. Misner, K. S. Thorne, and J.A. Wheeler.
Gravitation. W. H. Freeman, San Francisco.
1973.
59
vol. 16 Nº 1,
Nº 1, 2008,
2008 pp. 60-64
ASYMMETRICAL CHIRAL GAUGING TO INCREASE THE COEFFICIENT
OF PERFORMANCE OF MAGNETIC MOTORS
CALIBRE QUIRAL PARA AUMENTAR EL COEFICIENTE
DE RENDIMIENTO DE MOTORES MAGNÉTICOS
H. Torres-Silva1
RESUMEN
Este trabajo introduce un recalibre físico quiral asimétrico usado para aumentar el coeficiente de rendimiento de un motor eléctrico. Se
presenta una revisión de la teoría de calibres y se examina el descarte de la condición de Lorentz para obtener el recalibrado quiral. Se
introduce el coeficiente de rendimiento y se analiza un motor magnético bajo el enfoque quiral que permite un proceso Beltrami.
Palabras clave: Calibre quiral, motor magnético, Lorentz.
ABSTRACT
This paper introduces a physical chiral asymmetrical regauging to increase the coefficient of performance of an electric motor. A
review of gauge theory and a consideration of the disposal of the Lorentz condition to achieve the chiral regauging are presented.
The coefficient of performance terminology is introduced. A magnetic motor is discussed under a chiral approach which gives a
Beltrami process.
Keywords: Chiral gauge, magnetic motor, Lorentz.
INTRODUCTION
In this paper we investigate a process referenced in recent
permanent magnet (PM) motor patents [1]. These specially
designed PM motors claim to capture and use environmental
energy as an additional energy input. The technique that
allows this energy transfer to occur is called asymmetrical
regauging (ASR). The physics behind the ASR process will
be examined by reviewing gauge theory, the Lorentz gauge,
and the effect of discarding the Lorentz gauge to include
the vacuum chiral current density. The term coefficient of
performance (COP) is introduced to adequately describe
the energy transfer of these motors.
REVIEW OF THE LORENTZ GAUGE
To understand how environmental energy may be utilized
in a motor, to theoretically gain a COP >1, a review of
the Lorentz gauge is first presented. The equations used
in standard practice to design motors are derived from
1
Maxwell’s equations. It has been accepted practice, to
apply the Lorentz gauge to these equations to make them
simpler. In abbreviated steps, we start with Maxwell’s
equations [14].
All the information in Maxwell’s four equations can be
reduced to the following equation:
 2

∂V 
∂2 A 
 ∇ A − µ0 ε 0 2  − ∇  ∇ • A + µ0 ε 0 ∂t  = − µ0 ε 0 J (1)


∂t 

The Lorentz gauge is then applied to reduce the complexity
of these two equations. Mathematically, applying any
gauge, is represented by (2,3) where gamma is an
arbitrary, differentiable scalar function called the gauge
function [5].
)
V (t , x )  V ′(t , x ) = V ( t , x −
(2)
)
(3)
A(t , x )  A'(t , x ) = A(t , x ) + ∇Γ ( t , x
60
∂Γ (t , x )
∂t H. Torres-Silva: Asymmetrical chiral gauging to increase the coefficient of performance of magnetic motors
To specifically apply the Lorentz gauge, the “Lorentz
Condition” is imposed by choosing a set of potentials
(A, V) such that
∇ • A = µ0 ε 0
∂V
∂t
(4)
Equations (4) are the ones on which all the equations for
motor design are currently based. Since the magnetic
vector field and the voltage scalar field are both changed
at the same time, this can be referred to as symmetrical
gauging, so
2
∇ A − µ0 ε 0
∇ 2V − µ0 ε 0
∂2 A
= − µ0 J
∂t 2
∂2V
∂t
2
=−
1
ρ
ε0
(5)
)
−∇ ( ∇ • A + µ0 ε 0
(6)
∂V
∂t
(7)
ASYMMETRICAL REGAUGING
∂V
−∇ ( ∇ • A + µ0 ε 0
= µ0 j ∂t
(8)
and JA = σEA
(9)
Asymmetrical regauging is the equivalent of discarding
the Lorentz condition. Further ASR is any process that
changes the potential energy of a system and also produces
a net force in the process [6].
Understanding the vacuum and its polarization are
essential steps to utilizing energy from the environment.
According to T.D. Lee, he define the vacuum state as
the lowest energy state of the system [7]. Hence, the
vacuum is considered to be the worst case model of the
environment. Maxwell’s equations must be modified, in
the vacuum, since ρ and J vanish. Classically, this causes
the Ampere-Maxwell law to be revised.
∇ × B = µ0 ε 0
∂E
∂t ∂D
,
∂t (11)
where D = ε0E + PA and B = µ0H + µ0M .
(12)
This leads to the result that
∂PA
∂(ε T ∇ × E)
= jA =
.
∂t
∂t
(13)
(10)

D = D = ε 0 E + PA = ε 0 E + ε T ∇ × E
(14)
B = µ0 H + µ0 M = µ0 H + µ0 T ∇ × H.
(15)
and
Hence, T is the chiral factor which allows to extract
vacuum energy, Thus discarding the Lorentz condition
in classical electrodynamics leads to new equations that
include the effect of the vacuum polarization.
Invoking the Lorentz condition in classical electromagnetics
discards the vacuum polarization component that exists
in quantum electrodynamics [6] since
)
∇ × H = jA +
Here we are considered for D, B a chiral term so, [14]
Notice that (5) is (1) with the middle term, (7),
eliminated.
In [6, 8], the authors show that if the vacuum current density
factor is included, the above equation changes to
COEFFICIENT OF PERFORMANCE
The energy transfer of electrical machinery is generally
described using the term “efficiency”. Efficiency is defined
as the power output divided by the total power input from
all sources. The underlying assumption when defining the
energy of any system is that all the energy input is from
an identifiable and measurable energy sources(s). In an
ideal system the efficiency would be one. The equation
for efficiency (η) is normally stated [2] as
η=
POut
[ Watts].
PIn
(16)
Coefficient of performance is a broader energy transfer
term that defines the measure of energy output divided
by the operator’s energy input. COP is used to describe
any machinery that has additional energy input from the
environment. For example, COP is commonly used to
describe the energy exchange of heat pumps [3] or solar
collectors. Unlike the term “efficiency”, the COP can be
greater than one. See figure 1 for the energy flow diagram.
The following equation defines COP mathematically.
61
COP ≡
POut
PIn (Operator )
[ Watts]
(17)
B= ∇× A
Energy Input
from the
Environment
Energy
Input
from
Operator
Heaviside theory the magnetic field B is connected with
its generating vector potential A by the relation
(18)
In TTS theory this law has to be replaced in the simplest
case by [14-16].
Dissipative
Elements
Energy
Output
Figure 1. Energy flow for machines described by COP.
CHIRAL MAGNETIC RESONANCE EFFECTS
Besides electrical spacetime devices, self-running
magnetic motors have been constructed in a repeatable
and reproducible way ([4], see also figure 2, [1, 2]). The
functioning of these devices cannot be explained by
Maxwell-Heaviside electrodynamics but it can be explained
with the chiral electrodynamics.
B = ∇ × (1 + T ∇×) A (19)
where ω is the spin connection vector again. In the
following we discern between the magnetic field of
the assembly and the magnetic field of the surrounding
spacetime itself, denoted by Bs. The torque T acting on
the magnetic dipolo moment m of the assembly due to
the external field Bs is
T = m × Bs
(20)
Under normal conditions there is no resulting torque
because of Bs = 0. Spacetime is force free and does not
bear a magnetic field. So there is no rotation of stationary
magnets. The situation becomes different if it were possible
to create a magnetic field from spacetime. In order to
understand how this can be achieved we have first to look
closer on the fields of the surrounding spacetime. In case
of Bs = 0 it follows from Eq. (19) that
∇× A=−
1
A
T (21)
where A is the vector potential of the spacetime itself. In
contrast at Maxwell-Heaviside theory, this is no gaugable
quantity but is uniquely defined and has a physical meaning.
In case of A consisting of plane waves, ω takes a special
form and Eq. (21) can be expressed as ∇ × A = kBA with
a wavelength kB = 1/T, [14].
Figure 2. Johnson magnetic motor [1] Schematic
representation of spacetime vector potential
for a magnetic assembly: magnet stator
including rotor magnets, flow with vortices
(force field).
As was described in the preceding section of this edition,
the Cartan torsion of spacetime introduces the spin
connection as an additional quantity occurring in the laws
of nature so that they take a generally covariant form. In
particular this holds for the magnetic field. In Maxwell
62
This equation is known as Beltrami equation in the literature
[7]. It describes a flow with longitudinal vortices where
streamlines have a helical form. In the case of chiral
potential this means that there is no force field present,
in accordance with our prior assumption.
Taking the curl at both sides of Eq. (21) gives
∇ × (∇ × A) = ∇ × (−
1
A) ⇒ (∇ 2 + k B2 ) A = 0 (22)
T
This is a Helmholtz equation for the spacetime surrounding
the magnetic assembly. Because of the assumption of
H. Torres-Silva: Asymmetrical chiral gauging to increase the coefficient of performance of magnetic motors
Bs = 0 there is no torque on the magnets, they remain at
rest. Torque can be created by disturbing the Beltrami
flow. For the Helmholtz equation this means that the
balance to zero is no more fulfilled. Assuming a periodic
imbalance leads to
(∇ 2 +
1
T2
) A = Rcos( k ⋅ r )
(23)
with a vector R having units of inverse square meters,
therefore it can be interpreted as a curvature. κ is a wave
vector and can be interpreted as the frequency of a driving
force which the right hand side of the equation constitutes.
If restricted to one coordinate (x) the equation reads
(
∂2
∂x
2
+
1
T2
) Ax = Rx cos( kr )
(24)
It can be seen that this is a differential equation for a
resonance without damping (α = 0). The resonant oscillation
occurs in case k = 1/T with A x going to infinity.
Because of violating the Beltrami condition, A creates a
force field according to Eq. (24), which creates a torque
being big enough to spin the magnetic assembly and to
maintain the rotation. This is the mechanism how spacetime
is able to do work via a resonance mechanism.
In total we have shown qualitatively how energy can be
obtained from spacetime via magnetic assemblies. This
could be the basis for development of an engineering for
such devices.
It has already been shown in quantum electrodynamics
that the vacuum behaves like a dielectric [9]. The vacuum
sprouts positron-electron pairs as shown in the Feynman
diagram.
It has been shown that by discarding the Lorentz gauge,
the Ampere-Maxwell law equation evolves to include the
current density of the vacuum. Also, the task remains to
develop the equation and determine the process to apply
it to magnetic motors. Future work is planned to study
the magnetic motor to ascertain the exact mechanism
involved that allows this motor to exchange energy with
the vacuum.
CONCLUSION
It has been suggested in at least one recent patent that it is
possible to make use of energy from the environment as
an extra source in permanent magnet motors. This paper
presents a new term “coefficient of performance’ which
may be used to more adequately describe the energy
transfer of such an electromechanical system. This paper
also shows the physics behind one possible explanation
for this phenomenon.
The physics is explained by first considering how the
Lorentz gauge is used to give us the design equations
used today. The Lorentz gauge is then discarded to show
how the current density of the vacuum may be included in
the Maxwell-Ampere equation. They term asymmetrical
regauging is introduced for this procedure. The particle
physics explaining the vacuum polarity is introduced. A
thorough investigation for practical application of this
new equation is encouraged by suggestion that further
study be applied to the “Wankel motor”. Future work is
planned to study the magnetic motor to ascertain the exact
mechanism involved that allows this motor to exchange
energy with the vacuum.
REFERENCES
[1]
J. Bedini. “Device and Method of a Back EMF
Permanent Electromagnetic Motor Generator”.
US: Bedini Technology, Inc. 2002.
[2]
A. Trzynadlowski. “Introduction to Modern Power
Electronics”. New York: John Wiley & Sons. Inc.
1998.
[3] K. Annamalai and I. Puri, “Advanced Thermodynamics
Engineering”. New York. CRC Press. 2002.
[4] D. Griffiths. “Introduction to Electrodynamics”.
New York. Prentice-Hall. 1999.
[5]
B. Thide. “Electromagnetic Field Theory”. Uppsala:
Upsilon Books. 2004.
[6] P.K. Anastasovski. “Classical Electrodynamics
without the Lorentz Condition: Extracting Energy
from the Vacuum”, Physica Scripta. Vol. 61, p. 513.
1999.
[7]
T.D. Lee. “Particle Physics and Introduction to user
Field Theory”. New York: Harwood Academic
Publishers. 1981.
63
[8]
B. Lehnert and S. Roy. “Extended Electromagnetic
Theory”. Singapore: World Scientific. 1998.
[9]
D. Griffiths. “Introduction to Elementary Particles”.
New York. John Wiley & Sons, Inc. 1987.
[10]
[11]
[12]
64
M.W. Evans and H. Eckardt. “Spin connection
resonance in magnetic motors”. Documento 74
en la serie sobre ECE. URLs: www.aias.us
D. Reed. “Beltrami vector fields in electrodynamics
– a reason for reexamining the structural foundations
of classical field physics?” Modern Nonlinear Optics,
Part 3. Second Edition. Advances in Chemical
Physics. Vol. 119. Recopilado por Myron W. Evans.
John Wiley & Sons. 2001.
G. Kasyanov. Phenomenon of electrical current
rotation in nonlinear electric systems, Comment
on the Violation of the law of charge conservation
in the system, New Energy Technologies. Vol. 2
Nº 21, pp. 28-30. 2005.
[13]
“Motor de Johnson de imanes permanentes”. US
Patent 4151431. 1979.
[14]
Nº 1, pp. 6-23. 2008.
[15]
H. Torres-Silva. “Chiral field ideas for a theory
of matter”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 36-42. 2008.
[16]
H. Torres-Silva. “A metric for a chiral potential
pp. 91-98. 2008.
H. Torres-Silva: Podolsky’s electrodynamics under a chiral approach
PODOLSKY’S ELECTRODYNAMICS UNDER A CHIRAL APPROACH
ELECTRODINÁMICA DE PODOLSKY BAJO UN ENFOQUE QUIRAL
H. Torres-Silva1
RESUMEN
En este trabajo se muestra que un nuevo esquema conduce a la electrodinámica de Maxwell y a la electrodinámica de
Podolsky, partiendo con relaciones constitutivas quirales en lugar de la usual ley de Coulomb.
Palabras clave: Electrodinámica de Podolsky, ecuaciones de Maxwell.
ABSTRACT
In this paper we show that a new approach leads to Maxwell’s and Podolsky’s electrodynamics, provided we start from
chiral constitutive relations instead of the usual Coulomb’s law.
Keywords: Podolsky’s electrodynamics, Maxwell’s equations.
INTRODUCTION
“On the Question of Obtaining the Magnetic Field,
Magnetic Force, and the Maxwell Equations from
Coulomb’s Law and Special Relativity”, where it can
be shown that any attempt to derive Maxwell equations
from Coulomb’s law of electrostatics and the laws of
special relativity ends in failure unless one makes use
of additional assumptions. Kobe [1] gave the answer: all
one needs to arrive at Maxwell equations is
(i) Coulomb’s law;
(ii) the principle of superposition;
(iii) the assumption that electric charge is a conserved
scalar (which amounts to assuming the independence of
the observed charge of a particle on its speed [2];
(iv) the requirement of form invariance of the electrostatic
field equations under Lorentz transformations, i.e. the
electrostatic field equations are thought as covariant
space-space components of covariant field equations.
Neuenschwander and Turner [3] obtained Maxwell
equations by generalizing the laws of magnetostatics,
which follow from the Biot-Savart law and magnetostatics,
to be consistent with special relativity.
1
The preceding considerations leads us to the interesting
question: what would happen if we followed the same
route as Kobe did, using an electrostatic force law other
than the usual Coulomb’s one? We shall show that if we
start from the force law proposed by Podolsky [4], i.e.,
F( r) =
QQ '
4πε 0
 1 − er / a e − r / a  r
−

ra  r
 r2
(1)
where a is a positive parameter with dimension of length,
Q and Q’ are the charges at r and r = 0, respectively, and
F(r) is the force on the particle with charge Q due to the
particle with charge Q’ and if we follow the steps previously
outlined, we arrive at the outstanding electrodynamics
derived by Podolsky in the early 40 s. In other words,
we shall show that the same route that leads to Maxwell
equations leads also to Podolsky equations. A notable
feature of Podolsky’s generalized electrodynamics is that
it is free of those infinities which are usually associated
with a point source. For instance, (1) approaches a finite
value QQ’/8πε0 a2 as r approaches zero. Thus, unlike
Coulomb’s law, Podolsky’s electrostatic force law is finite
in the whole space.
65
In Sec. II we derive the equations that make up Podolsky’s
electrodynamic under the chiral approach [5, 7, 11].
In Sec. III we arrive at Podolsky’s field equations by
generalizing the equations of Sec. II, so that they are form
invariant under Lorentz transformations. For consistency,
we show in Sec. IV that (1) is indeed the electrostatic
force law related to Podolsky’s theory. The conclusions
are presented in Sec. V. Natural units  = c = 1,are used
throughout. As far as the electromagnetic theories are
concerned, we will use the Heaviside-Lorentz units
with c = 1.
To begin with let us establish some conventions and
notations to be used from now on. We use the metric
tensor
1 0 0 0 
 0 −1 0 0 

=
 0 0 −1 0 


 0 0 0 −1
with Greek indices running over 0, 1, 2, 3. Roman indices
i, j etc, - denote only the three spatial components. Repeated
indices are summed in all cases. The space-time four

vectors (contravariant vectors) are x µ = (t , x ) , and the

covariant vectors, as a consequences are x µ = (t , − x ) .
The four-velocities are found, according to
dx µ

= γ (1, v )
dτ

uµ = γ (1, − v )
uµ =
where τ is the proper time (dτ2 = dt 2 – dx2), and γ denotes
dt / dτ = (1–v2) –1/2. Let us then generalize (6) so that it
satisfies the requirement of form invariance under Lorentz
transformations. To do that, we write the mentioned
equation in terms of the Levi-Civita density εnml, which
equals +1 (–1) if n, m, l is an even(odd) permutation of
1, 2, 3, and vanish if two indices are equal. The curl
equation becomes
εjkl∂El = 0
(2)
It we define the quantities
66
F0i = -F0i = Ei = –Ei
εjkl∂kF0l = 0
We imagine now the curl law to be the space-space
components of a manifestly covariant field equation
(invariance under Lorentz transformations). As a result,
we get
εµαvαβ ∂vFαβ = 0 (4)
where Fαβ is a completely antisymmetric tensor of rank
four with ε0123 = + 1.
CHIRAL FIELD EQUATIONS
η µν = ηµν
Equation (10) can be rewritten as
(3)
Of course, this generalization introduces the components
F00, F01, and Flk, for which at this point we lack a physical
interpretation. Note that the F0i are not necessarily static
anymore.
On the other hand, as is well-known, the charge density
ρ is defined as the charge per unit of volume, which has as
a consequence that the charge dq in an element of volume
d 3 x is dq = ρd 3x. Since dq is a Lorentz scalar [3],
ρ transforms as the time-component of a four-vector,
namely, the time-component of the charge-current

four-vector u = ( ρ, j ) . The electric charge, in turn,
is conserved locally [3], which implies that it obeys a
continuity equation
∂µ j µ = 0
(5)
Assuming e jω t time dependence, Maxwell’s time-harmonic
equations [1] for isotropic, homogeneous, linear media
are
∇ × E = –jωB
∇ • B = 0
(6)
∇ × H = –jωD + J
∇ • D = ρ
(7)
Chirality is introduced into the theory by defining the
following constitutive relations to describe the isotropic
chiral medium [5, 7]
D = ε E + εT ∇ × E (8)
B = µ H + µT ∇ × H (9)
Where the chirality admittance T indicates the degree of
chirality of the medium, and the ε y µ are permittivity
and permeability of the chiral medium, respectively. In
natural units ε = 1, µ = 1 (the factor 1/4 is absorved in
the current value). Since D and E are polar vectors and
B and H are axial vectors, it follows that ε and µ are
true scalars and T is a pseudoscalar. This means that
when the axes of a right-handed Cartesian coordinate
system are reversed to form a left-handed Cartesian
coordinate system, T changes in sign whereas ε and µ
remain unchanged.
Since ∇ • B = 0 always, this conditions will hold identically
if B is expressed as the curl of a vector potential A since
the divergence of the curl of a vector is identically zero.
Thus by rearranging equation (7) we have
(1 − ko2T 2 )∇ × B = jω E + ( J + T ∇ × J ) (1 + T 2 )∂ν F µν = j µ 0
0
0
+ jordinary
≈ jordinary
Now if j 0 = jchiral
, we can imagine now
a particle of mass m and charge Q at rest in a lab frame
where there is an electrostatic field E. Newton’s second
law allows us to write
dp
= QE dt
dp
= Qγ E = Qu 0 E
dτ
where u 0 is the time part of the velocity four-vector u µ.
For the component along de xi direction, we have
(1 + T 2 ∂t 2 )∇ × B = jω E + ( J + T ∇ × J )
dpi
= Qu 0 F 0 i
dτ
or
(1 + T 2 ∂t 2 )(∇ × B −
∂E
∂
∂2
) = (−T 2 2 E + 2T ∇ × E ) + ( J + T ∇ × J)
∂t
∂t
∂t
(11)
In order that the right-hand side of this equation transforms
like a space-component of a four-vector, it must be
rewritten as
dpi
= Quν F ν i
dτ
In relativistic form we have
(1 + T 2 ∂i ∂i )∂ j E j = j 0 (12)
0
0
where j 0 = jchiral
is given by a chiral current
+ jordinary
plus a ordinary current of electrons and protons, the chiral
current is given in ref [1], ∂i = ∂ / ∂x i and ∂i = ∂ / ∂xi . Note
2
i
0j
0
that ∂i = −∂i Using (11), yields (1 + T ∂i ∂ )∂ j F = j
In order that the left-hand side of the preceding equation
transforms as the time-component of a four-vector, we
must write it as
(1 + T 2 )∂ j F 0 j = j 0
whose covariant generalization is
dp µ
(16)
= Quν F νµ dτ
If (16) is multiplied by p µ = muµ, where m is the rest
mass, the result is
1 d
( p p µ ) = Qmuµ u µ F νµ
2 dτ µ
However,
2
pµ p µ = m 2γ 2 (1 − v ) = m 2γ 2γ −2 = m 2
where
(15)
In terms of the proper time this becomes
(10)
In terms of (3) can now be rewritten as
(14)
Therefore, we come to the conclusion that
i
µν
2
2
2
 = ∂i ∂ = η ∂ µ ∂ν = ∂ / ∂t − ∇ (13)
The requirement of form invariance of this equation
under Lorentz transformations leads then to the following
result
uµ u µ F νµ = 0
Using this result Kobe [1] and Neuenschwander and
Turner [3] showed that Fvµ is an antisymmetric tensor
67
(Fvµ = –Fµv). Since Fvµ is an antisymmetric tensor of
second rank, it has only six independent components,
three of which have already been specified. We name
therefore the remaining components
containing the Lorentz force. For v = 0, (15) assumes
the form
dU
= Qv ⋅ E
dt
1
B = ε ilm Flm
2
(23)
i
(17)
Note that F kl = −klj B j . Writing out the components of
(17) explicitly,
B1 = F23 = F 23 = − B1
B 2 = − F13 = − F 13 = − B2
B 3 = F12 = F 12 = − B3
Hence, a clever physicist who were only familiar with
Podolsky’s electrostatics and special relativity could

predict the existence of the magnetic field B , which
naturally still lacks physical interpretation.
The content of (12) and (14) can now be seen. For µ = 0,
(12) gives
∇ ⋅ B = 0
(18)
where U = p 0 is the particle’s energy. Accordingly, our
smart physicist, who was able to predict the B field only
from its knowledge of electrostatics and special relativity,
can now-by making judicious use of (22) and (23) - observe,
measure and distinguish the B field from the E field of
(15). The new field couples to moving electric charge,
does not act on a static charged particle, and, unlike the
electrostatic field, is capable only of changing the particle’s
momentum direction.
Equations (18-21) make up Podolsky’s higher-order field
equations. Of course, in the limit T = 0, all the preceding
arguments apply equally well to Maxwell’s theory.
Two comments fit in here:
(1) Equation (14) is consistent with the continuity
equation (13). In fact, if the divergence of (14) is taken,
we obtain
(1 + T 2 )∂ µ ∂ν F µν = ∂ µ j µ
showing that there are no magnetic monopoles in Podolsky’s
electrodynamics, while for µ = i we obtain
∂B
∇×E=−
∂t (19)
which says that time-varying magnetic fields can be
produced be B fields with circulation.
The components µ = 0 and µ = i of (14) give,
respectively,
(1 + T 2 )∇ ⋅ E = ρ (1 + T 2 )(∇ × B −
∂E
)= j
dt
(20)
(21)
For v = i, (16) becomes
68
(2) As was recently shown [8], it is not necessary to
introduce a formula for the force density fµ representing
the action of the field on a text particle. We have only
to assume that (– fµ) is the simplest contravariant vector
constructed with the current jµ and a suitable derivative
of the field Fµv. Applying this simplicity criterion to
Podolsky’s electrodynamics, we promptly obtain
f µ = − F µν jν
which are nothing but a generalization of Gauss and
Ampère- Maxwell laws in this order.
Since Fµv is an antisymmetric tensor ∂ µ ∂ν F µν is identically
µ
zero. On the other hand, according to (14) ∂ µ j . Thus,
the equation in hand is identically zero;

where, as we have already mentioned, j µ = ( ρ, j ) .
Therefore,
f 0 = − F 0 i ji = E ⋅ j
and
dp
= Q( E + v × B)
dt
(22)
f k = − F k β jβ = F 0 k j0 + F ik ji = ( ρ E + j × B) k
Thus, the force density for Podolsky’s electrodynamics is
the same as that for Maxwell’s electrodynamics, namely,
the well-known Lorentz force density.
THE FORCE LAW FOR PODOLSKY’S
ELECTROSTATICS

V ( k ) =
(2π )3 / 2 T 2 k 2 ( k 2 +
So,

V (r ) =
We show now that (1) is indeed the force law for Podolsky’s
electrostatics. It follows
That
F = QE
E = −∇V 
V (r ) =


(1 − T 2 ∇ 2 )∇ 2V (r ) = − ρ(r )
For a charge Q at the origin of the radius vector this
equation reduces to


(1 − T 2 ∇ 2 )∇ 2V (r ) = −Qδ 3 (r ) 
V ( k ) =
 − i k ⋅r  
∫ d ke V (k )
3
(2π )3 / 2
1
(2π )3 / 2
 

3  ik ⋅r
d
re
V (r )
∫
(26)
(27)


where d 3 k and d 3r , respectively, stands for volumes in
the three-dimensional k-space and the coordinate space. If
we substitute (26) into (25) and take into account that

δ 3 (r ) =
we obtain
1
(2π )3 / 2
)
 
e − i k ⋅r
k 2 (k 2 +
1
T2
)
1 − e−r / T
)
r
(2π )2 r
Q
(
1 − e−r / T e−r / T r

E = −∇V (r ) = Q / 4π (
−
)
rT r (28)
r2
It follows then the force law for Podolsky’s electrostatics
is
QQ ' 1 − er / a e − r / a r
(
−
)
4πε 0
ra r
r2
F( r) =
(25)
We solve this equation using the Fourier transform method.

First we define V ( k ) as follows:
1
T2
Accordingly, the electric field due to a charge Q at the
origin is given by
Eq. (20) can then be rewritten as

V (r ) =

∫d k
3
(2π )3 T 2
(24)


 ρ(r ')(1 − e − R / T )
V (r ) = ∫ d 3r '
4π R
Q
1
Integral (28) may be found in any textbook on the theory
of functions of a complex variable [8]. As a result,
where
Q
which is nothing but the force law for which we were
looking (see Eq. (1)).
Recently an algorithm was devised which allows one
to obtain the energy and momentum related to a given
field in a simple way [8]. Using this prescription we can
show that in the framework of Podolsky’s electrostatics
the energy is given by
ε field =
1 3  2
d x E + T 2 (∇ ⋅ E ) 2 


2∫
Making use of the expression for the electrostatic field
we have just found, we promptly obtain
 
3 − i k ⋅r
d
ke
∫
ε field =
Q2
2T
which tells us that the energy for the field of a point
charge has a finite value in the whole space. This is
indeed a important feature of Podolsky’s generalized
electrodynamics.
69
Equations (28) and ∇ × E = 0 are the fundamental laws
of Podolsky’s electrostatics. We will slightly analyze an
interesting feature of Podolsky’s electrostatics by computing
the flux of the electrostatic field across a spherical surface
of radius R with a charge Q at its center. Using (28) we
arrive at the result
∫ E ⋅ dS = Q(1 − (1 + R / T )e
R/T
)
Rs = 2mGN / c 2 = T
which tells us that
∫ E ⋅ dS = 0,
∫ E ⋅ dS = Q,
R << T
R >> T
Therefore, a sphere of radius R << T, unlike what happens
in Maxwell’s theory, shields its exterior from the field
due to a charge placed at its center. We remark that in
Maxwell’s electrostatics a closed hollow conductor shields
its interior from fields due to charges outside, but does
not shield its interior from the field due to charges placed
inside it [6]. Note, however, that in order not to conflict
with well established results of quantum electrodynamics,
the parameter a must be small. Incidentally, it was shown
recently that this parameter is of the order of magnitude
of the Compton wavelength of the neutral vector boson
z, λ ≈ 2.15 × 1016 cm, which mediates the unified and
electromagnetic interactions [7].
ABOUT ELECTRON SIZE
In actuality, we don’t know how big the electron is. All of
our measurements point to the electron having no size, but
we haven’t measured down far enough. The electron, if it
were a black hole, would have to be smaller than 1x10 -57
meters, quite a bit smaller than we’ve ever measured!
But, another reason that the electron is not considered
a black hole, even assuming that its radius is infinitely
small, is that it obeys the laws of quantum field theory.
Normally, when one speaks about black holes, one is
talking about them in terms of Einstein’s theory of general
relativity. No one is sure how nature merges Einstein’s
theory with quantum field theory. So we aren’t really
sure if the idea of a black hole makes sense on distance
scales as small as the (possible) radius of the electron..
Our best idea to unify general relativity with quantum
field theory is an idea called string theory, but string
theory still appears to be a long way from being put to
any experimental tests.
70
According to general relativity all massive objects possess
an event horizon known as the Schwarzschild radius. This
is a surface in three-dimensional space surrounding the
object. Any light rays emitted from within this radius are
unable to escape. If an object exists entirely within its
Schwarzschild radius then it is referred to as a black hole.
This radius grows with the mass of the object according
to the formula:
Notice that for our sun we obtain a radius of 2.95 kilometers.
This is well within the interior of the sun so it is not a
black hole. For an electron we would obtain 1.35 x10 -51
m. If the electron were a point particle, it seems it would
be within even this fantastically small radius and would
indeed be a black hole!
However, as a subatomic particle the electron is also a
quantum-mechanical object. Recall the wave-particle
duality hypothesis of de Broglie. All objects have a wave
function which represents the probability of locating that
object at a particular point in space. During a collision this
wave function momentarily collapses and the particle is
truly at one ‘point’ in space, but it immediately starts to
spread out again after the instant of collision. The typical
spread of the wave-function of a point particle is given by
the Compton wavelength: λ = h / mc = 2π  / mc = 4π T ,
in according with our chiral theory and this can be
considered the true quantum-mechanical “size” of the
object [7]. Notice that this size gets smaller as the mass
gets larger. For you or I or the sun this quantum-mechanical
size is essentially zero (there’s not much uncertainty as
to where the sun is!) but for an electron the size is 2.42
x10 -12 m. Though still small, this is much, much larger
than the Schwarzschild radius. So quantum-mechanically
most of the electron is ‘outside’ its event horizon. That’s
why it and other subatomic particles are not black.
FINAL REMARKS
Despite the simplicity of its fundamental assumptions,
Podolsky’s model has been little noticed. Currently some
of its aspects have been further studied in the literature
[7, 8, 12, 13]. In particular, the classical self-force acting
on a point charge in Podolsky’s model was evaluated and
it was shown that in this model, unlike what happens in
Maxwell’s electrodynamics, the electromagnetic mass
is finite and enters the particle’s equation of motion in a
form consistent with special relativity.
To conclude we call attention to the fact the same
assumptions that lead to Maxwell’s equations lead also to
Podolsky’s equations and our chiral equations, provided
we start from a generalization of the Coulomb’s law
instead of the usual Coulomb’s law. Yet, in spite of the
great similarity between the three theories, Podolsky’s
generalized electrodynamics and chiral electrodynamics
lead to results that are free of those infinities which are
usually associated with a point source.
[7]
Nº 1, pp. 6-23. 2008.
[8]
Antonio Accioly, Am. J. Phys. Vol. 65, p. 882.
1997.
[9]
F.W. Byron, Jr. and R.W. Fuller. “Mathematics of
Classical and Quantum Physics”. Addison-Wesley
Publishing Company. New York. Vol. 2, pp. 366367. 1970.
[10]
Jon Mathews and R.L. Walker. “Mathematical
Methods of Physics. W.A. Benjamin, Inc. New
York, p. 58. 1965.
[11]
Nº 1, pp. 111-118. 2008.
REFERENCES
[1]
D. H. Kobe. Am. J. Phys. Vol. 54, p. 631. 1986.
[2] A. Accioly, Brazilian Journal of Physics. Vol. 28,
p. 35, 1998.
[3]
D. E. Neuenschwander and B. N. Turner, Am. J.
Phys. Vol. 60, p. 35. 1992.
[4]
B. Podolsky, Phys. Rev. Vol. 62, p. 66. 1942.
[12]
L.V. Belvedere, C.P. Natividade, C.A.P. Galvão,
Z. Phys. C56, p. 609. 1992.
[5]
pp. 91-98. 2008.
[13]
A.J. Accioly and H. Mukai, Z. Phys. C 75, p. 187.
1997.
[6]
B. Podolsky and P. Schwed, Rev. Mod. Phys.
Vol. 20, p. 40. 1948.
71
vol. 16 Nº 1,
Nº 1, 2008,
2008 pp. 72-77
SPIN AND RELATIVITY: A SEMICLASSICAL MODEL FOR ELECTRON SPIN
ESPÍN Y RELATIVIDAD: UN MODELO SEMICLÁSICO
PARA EL ESPÍN DEL ELECTRÓN
H. Torres-Silva1
RESUMEN
La relación cuántica m0 c 2 =  ω 0 puede ser considerada como la equivalencia entre dos expresiones para la energía en
reposo de la partícula, si ω0 se considera la velocidad angular de giro de partículas en su marco en reposo. La invariancia del
intervalo relativista espacio- tiempo ds 2 = c 2 dt 2 − dr 2 para tal movimiento de espín (isotropía espacial) conduce al impulso
de espín Sz =  / 2 para todas las partículas sin estructura, independientemente de sus valores de masa. La inercia es una
propiedad intrínseca debido al movimiento de spin de las partículas. Los signos de los valores de masa que se producen
en las soluciones de la ecuación de Dirac podrían estar relacionados con la orientación del espín, según lo sugerido por
la relación fundamental ± m0 c 2 = ±  ω 0 . Además se refiere al electrón, y más concretamente con dos de las principales
propiedades: su función de onda compleja, y su giro intrínseco. En su interpretación estándar no hay una clara imagen
del espacio real de lo que es oscilante en la onda, o lo que está girando en el espín. De hecho, es la creencia generalizada
de que ningún modelo sencillo puede dar cuenta de la rotación de espín de los electrones. Por el contrario, en el presente
trabajo se muestra que un crudo modelo mecánico de rotación de vórtices coherentes explica cuantitativamente no sólo el
espín, sino también la propia función de onda. Las consecuencias de esto son examinadas en este trabajo.
Palabras clave: Espín, relatividad, ecuación de Dirac, función de onda, modelo semiclásico.
ABSTRACT
The quantum relationship m0 c 2 =  ω 0 may be regarded as the equivalence between two expressions for the rest energy
of the particle, if ω0 is considered as the spin angular velocity of the particle in its rest frame. The invariance of the
relativistic space-time interval ds 2 = c 2 dt 2 − dr 2 to such a spin motion (space isotropy) leads to the spin momentum
Sz =  / 2 for all structureless particles irrespective of their mass values. The inertia is an intrinsic property due to the
spin motion of the particles. The signs of the mass values occurring in the solutions of the Dirac equation might be related
to the orientation of the spin motion, as suggested by the fundamental relationship ± m0 c 2 = ±  ω 0 . In addition, it deals
with the electron, and more specifically with two key properties: its complex wavefunction and its intrinsic spin. In the
standard interpretation, there is no clear real-space picture of what is oscillating in the wave, or what is rotating in the
spin. Indeed, it is generally believed that no simple model of rotation can account for the spin of the electron. On the
contrary, the present paper shows that a crude mechanical model of coherently rotating vortices can account quantitatively
not only for spin, but also for the wavefunction itself. The implications of this are discussed in this paper.
Keywords: Spin, relativity, Dirac equation, wavefunction, semiclassical model.
SPIN AND RELATIVITY
As we know, the spin cannot be motivated in the frame of
classical mechanics. Even in the nonrelativistic quantum
1
theory, the nature of spin remains unclear. The spin results
solely from Dirac’s equation [1]. Although the Pauli and
Dirac matrices undoubtedly show the spin existence,
there is some mystery as to the physical origins of and
72
H. Torres-Silva: Spin and relativity: a semiclassical model for electron spin
in the visualization of the spin [2, 3]. One may says that
spin is an intrinsic property of the matter. It must have
to do with relativity even though this connection is not
entirely understood [4].
In this paper, I shall try to sketch a simple motivation
for the existence of spin starting from the fundamental
relationship, and namely:
m0 c 2 =  ω 0 (1)
where m 0 is the rest mass and m 0 c2 the rest energy of
the particle.
If the particle is considered as being a physical torus
spinning with angular velocity ω0 (Figure 1a, and 1b),
the right-hand side of equation (1) should be regarded as
another relativistic expression for the rest energy.
ds 2 = c 2 dt 2 − dr 2 (2)
Every physical process, such as translation, rotation, etc.,
must be related to expression (2). The invariance of ds2
to uniform translation (space homogeneity) leads to the
Lorentz corrections [6].
D
T
P
x
The particle as a moving object must also obey another
fundamental relationship, namely the relativistic elementary
“space-time interval” between the physical events of the
particle:
Let us now consider the uniform rotation (spin) of the
reference frame, with angular velocity ω0 in x,y-plane
around the z-axis, as shown in Figure 1. In this case,
we have
z
ω0
Therefore, it is reasonable to assume that the rest energy
of a particle is related to its spin motion, which is only
allowed in that system [3]. This reasoning allows us to
regard  ω 0 as an equivalent expression for the rest
energy of the particle.
P’
y
x → x '; y → y '; z = z ';
dϕ
t → t ' = t + dt; ω 0 =
= const;
dt
and ds2 (equation 2) becomes
r⊥2 = x2 + y2
(a)
rp
(b)
Figure 1. a) A spinning reference frame. b) A torus
electron model
We may argue the existence of the spin motion in the
space-time frame of the particle as follows: unlike
space coordinates, time is not directly measurable
(observable). The simplest way to estimate time is
to consider uniform motion. One can obtain time by
comparing covered distances ([5]: time is the number
of motion). This leads to the necessity of introducing
motion in the space-time reference frame. The only
allowed motion in the rest frame of the particle should
be that of rotation (spin).
2
ds '2 = (c 2 − r⊥2 ω 0 ) − 2ω 0 ( ydx − xdy)dt − dr 2 (3)
where r⊥2 = x 2 + y 2 represents the radius perpendicular
to the rotation z-axis, i.e. the distance from origin to the
points P, P’, etc. (Figure 1 ).
Expression (3) can easily be derived.[7] Note that the linear
velocity u = r⊥ ω 0 must obey the restriction imposed by
2
the special relativity, i.e. c 2 > r⊥2 ω 0 . For the limit case
c = r⊥ ω 0 , the rotating space becomes ‘closed’ with the
lateral radius r⊥ . This is all what Fock mentioned [7].
But such a rotating empty space is physically meaningless.
We must therefore actually ascribe this rotation to the
particle situated in the origin of this space. For that
particle, the set of points P, P’, etc. for which c = r⊥ ω 0 ,
should be considered the closure (frontier) of the particle.
We do not know too much about the shape of a spinning
particle, considered as being structureless, but we can at
least define for it a radial extension equal to:
73
r⊥ = c / ω 0 (4)
From equations (1) and (4), we have:
r⊥ =  / m0 c =  c (Compton radius).
(5)
This result shows that all structureless particles with rest
mass cannot be pointlike. For the limit case, c = r⊥ ω 0 ,
expression (3) becomes:
ds '2 = 2ω 0 ( xdy / dt − ydx / dt )dt 2 − dr 2 2 ω 0 ( xdy / dt − ydx / dt ) = c 2 .
(7)
If we use now the fundamental relationship from equation
(1), we have
Sz = m0 ( xdy / dt − ydx / dt ) = m0 r × u =  / 2 , (8)
z
where Sz is the classical expression for the z-component
of angular momentum. The result is interesting. This
mainly shows that the  / 2 value of the angular
(spin) momentum preserves the space isotropy. It must
be universal and characteristic for all structureless
particles with finite rest mass, independent of their
mass values.
If the two possible rotations around z-axis are considered
{
}
(ω 0 ∈ + ω 0 ; − ω 0 ) corresponding to x → y and y →x
rotations, both the ± / 2 values conserve the space
isotropy. The time reversal t→-t in (8) leads to − / 2
value. Note that for ± ω 0 values equation (1) becomes
±  ω 0 = ± mo c 2 . The mass values ±m 0 occurring in
74
sense of the spin angular velocity ω 0 .
Moreover, according to (1) the rest mass m 0 is tightly
connected with the spin motion represented by ω 0.
Therefore, a structureless elementary particle with a
finite rest mass and radial extension behaves as a small
mechanical top, its inertial properties not necessarily
being conditioned by the gravitational interaction with
the matter in universe (Mach’s principle).
(6)
We have already mentioned the invariance of to the uniform
translation (space homogeneity). Let us now consider the
invariance of ds2 to the uniform rotation (space isotropy).
In other words, for a noninteracting spinning particle,
space must remain unaffected by the uniform rotation.
From the invariance condition ds’2 (equation 6) ≡ ds2
(equation 2), we obtain:
Dirac’s equation might actually be related to the rotation
A SEMICLASSICAL MODEL
FOR ELECTRON SPIN
First, consider an electron with its center of mass at
rest, but spinning. The simplest possible model is a
spinning solid torus (figure 1b). Based on the goal of
having this describe the electron wavefunction, one
expects that the angular velocity is given by the PlanckEinstein relation E = ω . Since this is a real physical
rotation, the zero of energy is not arbitrary as in standard
nonrelativistic quantum mechanics, but must be given
by the relativistic rest energy E = mc 2 . (This also has
the property of being relativistically covariant when
we transform later to a moving reference frame.) For
rotation of a solid torus of radius R, the linear velocity
on the equator is u = Rω = Rmc 2 /  . But clearly, u can
be no greater than the speed of light c. This is a natural
cutoff, and provides an estimate of the maximum size
of this spinning ball:
Rmax = c / ω =  / mc = Rc (9)
This is the Compton wavelength Rc of the electron
˜0.4pm, which is much smaller than the typical å scale
that characterizes atomic orbitals (1å=100pm). If we
want to model an extended electron state, then clearly
Rc is too small.
Consider instead an extended state consisting of a parallel
array of torus vortices (see figure 1), each a solid body
of radius Rc rotating around its axis at ω = mc 2 /  . For
simplicity here, assume that there are N identical vortices,
each of mass mv = m/N. The angular momentum of each
vortex is then given by
Lv = Iω =
1
m R 2ω =  / 2 N ,
2 v c
(10)
1
mR 2
2
for a cylinder of uniform mass density. This is a crude
semi-relativistic model, but it does in fact give the proper
value for the total angular momentum for the electron,
S =/2.
where we have taken the moment of inertia I =
One can also estimate the magnetic moment of the
electron from this model. Treating the rotating charge
per vortex qv = e / N as a current iv = qvω / 2π , one
obtains simply
(
)
µ = Niv Av = ( eω / 2π ) π Rc 2 = e / 2m = µ B (11)
where µB is the Bohr magneton and Av is the cylindrical
cross sectional area per vortex. Again, this is the correct
result, perhaps fortuitously, but it does suggest that this
crude model may incorporate much of the essential
physics.
These calculations require only that all of the torus are
rotating at the same frequency around parallel axes.
But in addition, it is reasonable to assume a coherent
state where all of them are rotating in-phase as well,
as suggested in figure 1. This requires a rotating vector
field A(r,t). Furthermore, it is not necessary to assume
that the vortices have identical masses. More generally,
one could have a density function ρ(r), which would go
as the square of the field amplitude A(r), analogously to
the energy density in electromagnetic waves.
Now the phase angle θ (t ) = Et /  is constant across the
entire electron, but that can also be relaxed. Consider what
happens when we Lorentz-transform to a reference frame
moving with velocity v. Locations that are in phase in the
rest frame will not in general be in phase in the moving
frame. The proper way to deal with this is to make the
phase angle relativistically invariant, so that
Et ⇒ E ′t ′ − p ′ ⋅ r ′ (12)
1
where in the usual way E ′ = γ mc 2 ≈ mc 2 + mv 2 ,
2
(
2
p ′ = γ mv ≈ mv is the momentum, γ = 1 − v / c
2
)
−1/ 2
,
and the approximate forms are for v << c. This is invariant
because (E/c, p) and (ct, r) are relativistic 4-vectors, and
the phase angle goes as their inner product. So now the
rotating phase angle takes the form
θ (r , t ) = ( Et − p ⋅ r ) /  (13)
This corresponds to a plane wave with wavelength λ = h/p,
which is well known as the de Broglie wavelength. Note
that this follows directly from the earlier assumption that
the rotation frequency is given by mc 2 /  .
Once we have a wave satisfying the Einstein-deBroglie
relations, the rest of quantum mechanics follows
naturally. We have a rotating vector field given by a
spin axis (assumed to be uniform), an amplitude A(r,t),
and a rotating phase angle θ(r,t). If we compare to the
standard complex wavefunction in quantum mechanics,
Ψ(r,t) = •Ψ•exp(iφ), and map A and θ onto •Ψ• and φ,
we have a rotating wavefunction which satisfies the
time-dependent Schrödinger equation.
For example, consider a rotating vector field of the
form
A(r , t ) = A0 ux cos( kz − ω t ) ± u y sin( kz − ω t )  , (14)
(ux and uy are the unit vectors in the x- and y-directions),
which represents a plane wave traveling in the z-direction
with spin also in the z-direction (figure 2). This is a circularly
polarized transverse wave, with either positive or negative
helicity depending on whether the plus or minus sign is
chosen. For fixed t, the tip of the vector follows a helix;
for fixed z, circular rotation at an angular frequency ω
of a vector of length A0.
)
(
Now define θ = arctan Ay / Ax = kz − ω t , and
Ψ(r , t ) = A exp(iθ ) = A exp i( kz − ω t )  ,
(15)
and substitute this into the time-dependent Schrödinger
equation with the rest-energy explicitly added:
(
)
i∂Ψ / ∂t = H Ψ = −  2 / 2m ∇ 2 Ψ +  mc 2 + V (r )  Ψ (16)
The result is the simple, correct relation (for v<<c) that
ω =  2 k 2 / 2m + mc 2 . Note also that the complex
conjugate of Ψ might seem to yield negative energy, but
really just represents the spin of the opposite sign.
Thus far the model has been limited to a single plane
wave, but electrons are generally present in bound states,
with standing waves instead of travelling waves. Consider
for simplicity the one-dimensional particle-in-a-box,
75
with the electron confined between z = 0 and z = L. The
solution takes the form of discrete bound states given
by the complex wavefunctions Ψn and equivalent vector
fields An:
Ψ n = sin(nπ z / L ) exp(−iω t ) )(
(17)
)
An = sin ( nπ z / L ux cos ω t ± u y sin ω t x
(18)
Here n=1 corresponds to the ground state and n=2, 3,...
to the excited states, and the quantized energies En are
given as usual (but with the mc2 offset) by
En = ω n = mc 2 +  2 k 2 / 2m = mc 2 +  2 (nπ / L )2 / 2m (19)
and as before the ± corresponds to the two spin states.
Note that this vector wavefunction has separated into two
factors, the usual standing-wave envelope and the rotating
λ = 2π/κ = h/p
z
Ψ0
y
Figure 2. Picture of real-space helical wave representing electron with spin. Evolution of helix for wave propagating in
z-direction.
phase vector. The negative values of the sine (for n>1)
correspond to 180º shifts of the rotating phase.
It is likely that the spins of the constituent components
contribute their angular phase references to the composite
system, even if the total spin cancels out.
DISCUSSION AND CONCLUSIONS
The wave example given above is based on a helical
transverse wave, which is similar in form to a transverse
electromagnetic wave which is circularly polarized like
a chiral wave. Indeed, such a helical TEM wave carries
angular momentum, and forms the classical limit of a
photon [9, 10], with spin ± pointing along the direction
of motion. However, unlike the case of the photon, one can
transform to the rest frame of the electron, and from there
to any other direction. In general, the electron spin axis
would not be parallel to the momentum, and the rotating
spin field vector would follow a general cycloidal motion
rather than a simple helix. The spin and translational
motions are essentially decoupled in this model (no spinorbit interaction).
This model of coherently rotating vortices appears to
account for the complex wavefunction of the electron [6].
This suggests that the spin picture may be substantially
more general than simply a single electron, and that spin is
fundamental to all of quantum mechanics. In that regard,
it may not be a coincidence that all fundamental quantum
particles seem to have spin. Certain mesons have spin-0,
but they can be regarded as composites of spin-½ quarks.
And certainly atoms with spin-0 show quantum effects.
76
One may speculate as to the physical basis for such a
coherent vortex model. It seems to correspond to a very
rigid state of an intrinsically rotating fluid. Such a rigid
state may indicate a very strong cohesive energy associated
with long-range phase coherence among the vortices. Since
the lowest excitation of an electron involves creation of
an electron-positron pair, this cohesive energy might be
expected to be ~1MeV, larger than the rest energy of the
electron itself.
Speculating even further, the existence of such a highly
rigid state would have important implications for quantum
measurement. Any local interaction that would alter
the energy of part of an electron wavefunction would
jeopardize this cohesive energy. This, in turn, would create
an instability leading either to the rest of the electron
being pulled into the interaction region, or alternatively
to the expulsion of the electron from this region. This
suggests a real dynamical process which may provide a
physical basis for the “collapse of the wavefunction” in
quantum measurement.
Finally, if this rotating spin field is mathematically
equivalent to the usual Schrödinger equation, is it really
just a matter of preference which representation we
choose? Not entirely, because a real physical rotation,
with a definite frequency and spatial fine structure,
should be measurable. If one probes the behavior of
electrons at frequencies ~1020 Hz = mc 2/h, particularly
with a circularly polarized probe, one should expect to
see a sharp resonance in some sort of spectral response,
perhaps associated with spin-flip of the electron in a
large magnetic field. Furthermore, the fine structure of
the spin model identified a periodicity on the scale of
2 Rc = 2 / mc , which would correspond to a momentum
transfer k = π mc  1.5 MeV / c . It would be interesting
to see whether relevant measurements are consistent
with the model described in this paper.
possible to remove much of the abstraction and mystery
from quantum theory.
It is somewhat surprising that a simple mechanical
model for spin was not presented in the early days of
quantum mechanics. It seems that early researchers
were discouraged by apparent rotation velocities greater
than c [8]. It may be that the distributed coherent vortex
model provides a way around these difficulties. More
recently, Ohanian [9] showed that the relativistic Dirac
equation is consistent with a distributed circular energy
flow on a scale larger than Rc, which provides the basis
for the electron spin and magnetic moment. The present
semiclassical model is certainly cruder than the Dirac
equation, but also reproduces these results within a more
intuitive physical picture.
In conclusion, a new semiclassical picture for electron
spin is presented, in which a spinning vector field,
rotating at mc 2/h, is organized into a coherent array
of rigidly rotating vortices on the scale of Rc =  / mc .
The vector field F maps onto the quantum wavefunction
Ψ, providing for a unification of spin and quantum
mechanics. It is further suggested that the coherent
nature of this spin field may be associated with a cohesive
energy, which in turn may play a key role in quantum
measurement. While the specific details of this model
remain crude, its clear intuitive physical picture may
help to stimulate further research along similar lines.
By dealing with specific real-space models, it may be
REFERENCES
[1]
P.A.M. Dirac. “Principles of Quantum Mechanics”.
Clarendon Press. Oxford. 1958.
[2]
H. Torres-Silva. “The close relation between the
Maxwell system and the Dirac equation when the
electric field is parallel to the magnetic field”. Ingeniare.
Rev. chil. ing. Vol. 16 Nº 1, pp. 43-47. 2008.
[3]
Nº 1, pp. 111-118. 2008.
[4]
M. Gogberashvili. “Octonionic electrodynamics”.
Journal of Physics A. Vol. 39 Nº 22, pp. 7099-7104.
2006.
[5]
J.M. Lévy-Leblond. “Non-relativistic particles and
wave equations”. Communications in Mathematical
Physics. Vol. 6, pp. 286-311. 1967.
[6]
C. Moller. “The Theory of Relativity”. Clarendon
Press. Oxford. 1964.
[7]
V. Fock. “Theory of Space, Time and Gravitation”.
Pergamon Press. Oxford. 1964.
[8]
Eugen Merzbacher. “Quantum Mechanics”. 3rd
ed. John Wiley. New York. 1997.
[9]
Hans C. Ohanian. “What is Spin?”. Am. J. Phys.
Vol. 54, pp. 500-505. 1986.
[10]
pp. 91-98. 2008.
77
vol. 16 Nº 1,
Nº 1, 2008,
2008 pp. 78-84
EXTENDED EINSTEIN’S THEORY OF WAVES
IN THE PRESENCE OF SPACE-TIME TENSIONS
TEORÍA EXTENDIDA DE ONDAS DE EINSTEIN
EN LA PRESENCIA DE TENSIONES EN EL ESPACIO-TIEMPO
H. Torres-Silva1
RESUMEN
Se propone una modificación a la dinámica de Einstein en presencia de ciertos tipos de tensión del espacio-tiempo. La
estructura de las ecuaciones de movimiento para las perturbaciones gravitacionales es muy similar a las ecuaciones de
Maxwell para cuerpos quirales micro y macroscópicos caracterizados por T, cuando los operadores de µ y ε son como
µ(ε)→ µ(ε) (1+T ∇×). Se discute el límite de unificación del electromagnetismo y la gravitación en el tiempo de Planck. Como
aplicación de esta teoría se menciona el efecto de la birrefringencia en sistemas GPS (Global Positioning Systems).
Palabras clave: Tensiones, espacio-tiempo, electrodinámica, quiralidad.
ABSTRACT
A modification of Einstein’s dynamics in the presence of certain states of space-time tension is proposed. The structure of
the equations of motion for gravitational disturbances is very similar to Maxwell’s equations for micro and macroscopic
chiral bodies characterized by T, when the operators ε and µ are like µ(ε)→ µ(ε) (1+T∇×). The unification limit between
the electromagnetism and gravity is discussed. As an application of this theory we mention the birefringence effect in
Global Positioning Systems (GPS).
Keywords: Tensions, space-time, electrodynamic, chirality.
INTRODUCTION
Electrodynamics is perhaps the most successful theory
physicists have constructed. Its theoretical and experimental
properties have been simulated and sought for in many
others theories, such as the analysis of gravitational
phenomena. Much work has been done in this direction
and many authors have discussed the resemblance
between electrodynamics and gravidynamics [1]. However,
it appears to us that it is not difficult to improve the
theoretical aspects of this similarity more that has been
done I the past. We intend to make a small contribution
to this problem here.
In this vein, we shall propose a modification of Einstein’s
theory of general relativity under certain special states of
space-time. Since the brilliant 1916 proposal of Einstein’s
geometrization of gravitational phenomena, many physicists
1
78
have discussed alternative models of gravitation. These
can be divided into two classes:
i. Geometrical models.
ii. Non-geometrical models.
The first group accepts Einstein’s idea of geometrization
of gravity but denies (under certain circumstances) the
validity of the equations of motion proposed by Einstein’s.
The second group contains all attempts to construct a model
in which gravity has no direct link with the structure of
space-time. It is not our intention to discuss these models
here. We merely state their existence.
The king of theory we shall advocate here may be
classified as being of type i. Indeed, we shall assume that
gravitational phenomena is described by the structure
of space-time. This will be given by means of its metric
H. Torres-Silva: Extended Einstein’s theory of waves in the presence of space-time tensions
properties – represented by a symmetrical metric tensor
gµv (x) and by two others functions, like operators ε (x)
and µ(x), which are independent of the metric [2] and
intimate characteristics of space-time.
We think it will be convenient, for pedagogical reasons,
to limit our considerations in the present paper to the
case in which both ε and µ are constants in time, but ε,
µ are function of 3-D space. The meaning we would like
to propose for these two constants is obtained by a direct
analogy with the dielectric and permeability constants of
a given medium in electrodynamics like a Born Fedorov
approach [4].
However, we shall simplify our model by merely
stating that ε and µ can be provisionally identified
with the characteristics of certain states of tensions,
is free space-time, due to an average procedure on
quantum properties of gravitation [3]. This is perhaps
not difficult to assume if we can say exactly how the
equations of motion of gravity phenomena must be
modified by them, as we shall go later. In sec. II we
shall describe gravitational interaction by means of a
fourth-rank tensor R αβµν. We shall set up its algebraic
properties and give its dynamics. It is possible to
separate this tensor, for an observer moving with
four-velocity u µ, into four second-order symmetric
trace-free tensors E αβ, B αβ, D αβ and Hαβ. Our principal
result is then obtained by showing that we can select
a class of observers with velocity u µ in such a way as
to have the equations of motion for B αβµν. That is, for
E αβ , B αβ , D αβ and H αβ separated into two groups: one
containing only E αβ and B αβ (and their derivatives)
and the others containing only D αβ and H αβ (and their
derivatives). These equations have the same formal
structure of Maxwell’s equations in a given general
medium. So, we arrived at the conclusion that in our
theory there is a class of privileged observers in which
gravitational field equations admit the above simple
separated form. Any others observers, which is in
motion with respect to u µ, mixes the terms E αβ , H αβ,
B αβ and H αβ into the equations. This situation could
be thought of as defining a new type of ether but it is
only a preferred frame of observation.
In the remainder of the paper we discuss in some detail
a very particular situation of the above tensors, that is,
the case in which they can be reduced to two tensors
plus two operators: the above ε and µ. The we show that
Einstein’s theory is obtained from ours for a particular
set of values of ε and µ, that is, the case ε = µ = 1. It is in
this sense that we have called our theory a generalization
of Einstein’s dynamics.
THE R-FIELD
Definitions
Let us define in a four-dimensional Riemannian manifold a
fourth-rank tensor Rαβµν given in term of four second-order
tensors E αβ, Bαβ, D αβ and Hαβ as viewed by an observer
with velocity tangent vector (time-like and normalized
uµuµ = + 1). We set
Rαβ µν = V[α Dβ ][µV ν ] + V[α Eβ ][µV ν ] + δ [µ E ν ]
[α
ρ
σ [µ
− ηαβρσ V B
V
ν]
−η
αβρσ
β]
(1)
Vρ H σ [α Vβ ]
In which the bracket means anti-symmetrization and
ηαβ µν = − gε αβ µν ; g is the determinant of g µν and
εαβµν is the totally anti-symmetric Levi-Civita symbol.
The tensors E αβ, Bαβ, D αβ and Hαβ satisfy the following
properties:
Dα α = 0 ,
Dαβ = Dβα , DαβVα = 0
(2)
E α α = 0, Eαβ = Eβα ,
E αβVα = 0
(3)
H α α = 0, Hαβ = H βα , H αβVα = 0
(4)
Bα α = 0, Bα α = 0, BαβVα = 0
(5)
We lower and raise the co-ordinate indices by means of
the metric tensors gµν (x). Greek indices run from 0 to 3,
in our units we set ≡ velocity of light = 1. We can write
D αβ, E αβ, etc, in terms of Rαβµν and projections on uµ, by
using properties that will be given below.
Algebraic properties
From definition of Rαβµν it is easy to prove the following
properties [5]:
Rαβ µν = − Rβα µν (6)
Rαβ µν = − Rαβ ν µ (7)
Rα βαν = Eβν − Dβν (8)
Rα α = 0 (9)
79
Dynamics
By analogy with Einstein’s equations in vacuum we shall
impose on Rαβµν the equations of motion [4]
αβµν
R
;ν
=0
(10)
(where the semicolon means the covariant derivative).
Now, we shall use the properties given in subsection 1
above for projecting the system of Eq. (10) parallel and
orthogonal to the rest frame of a selected observer uµ from
the whole class of ν µ. We impose that the congruence
generated by u µ satisfy the properties:
u µu µ = + 1
(a)
1 λ ε
h h u = 0 (b)
2 [α β ] λ ; ε
1
= h[ µ λ hν ]ε uλ ; ε = 0 (c)
2
uα = uα ; λ u λ = 0 (d)
wαβ =
θ µν
(11)
hµν = g µν − u µ uν (12)
So, the congruence generated by uµ is geodesic, irrotational,
non-expanding and shear-free. The reason for selecting
such a particular class of observers will appear clear later.
Then, Eq. (10) assumes the form:
Dα µ;ν h µν hαε = 0
D α µ h(ασ hεµ) + h(ασ ηε )νρτ uρ Hτα;ν = 0
µν α
Bα µ;ν h h
ε
=0
B µν hµ(σ hλ )ν − h(ασ ηλ )νρτ uρ Eτα;ν = 0
(13)
(14)
In which a round bracket means symmetrization.
This set of equations has a striking resemblance with
Maxwell’s macroscopic equations of electrodynamics. Indeed,
we can formally understand the above set as being [4]
80

∇D = 0


D − ∇ × H = 0
(14a)
(14b)
Where the symbol ↔ is put over D, E, etc. only to represent
its tensorial character; ∇• and ∇× son generalizations of
the usual well-known operators ∇• and ∇×.
So, we can understand the reason for selecting the above
privileged set of observers, given by the tangential vectoruµ.
Eqs. (13)-(14) takes the form-only for this class of frame.
Any others observer which is in motion with respect to
lµ will mix into the equations of motion the set of tensors
(Eαβ , Bαβ) with the set (Dαβ , Hαβ). So, it is in this sense that
there is a natural selection of all observers in the Universe,
with respect to the equation of motion satisfied by Rαβµν.
ε and µ states of tension
A particular class of states of space-time is that in which
there is a specific linear function relating the tensors Bαβ
with Hαβ and E αβ with D αβ by means of two operators,
ε and µ.
Where h µν is the projector in the plane orthogonal to
uµ, that is

∇•B = 0


B + ∇ × E = 0
(13a)
(13b) We set
Bαλ = µHαλ , Dαλ = ε Eαλ (15)
If we put expressions (15) into definition (1) of Rαβµν, a
straightforward calculation shows that it is possible to write
Rαβµν in terms of the Weyl tensor Cαβµν and its “electric”

and “magnetic” parts E αβ and Hαβ , if we identify the


tensor E αβ with Eαβ and Hαβ with Hαβ .

Eαµ;ν h µν hαε = 0
(a)


ε Eα µ h(ασ hεµ) + h(ασ ηε )νρτ uρ Hτα;ν = 0
(b)

Hα µ;ν h µν hαε = 0
(a)


µH α µ h(ασ hεµ) + h(ασ ηε )νρτ uρ Eτα;ν = 0
(b)
(16)
(17)
By the same argument that guided us to Eqs. (13)-(14)
we see from the above set that we can identify ε as being
the gravitational analogue of the dielectric constant of
electrodynamics, and µ as being the permeability of
space-time.
Now, we recognize in Eqs (16)-(17) Einstein’s equations
for the free gravitational field for the particular case in
which ε = µ = 1. [4]
This Tαβµν tensor has properties very similar indeed
to the Minkowski energy-momentum tensor of
electrodynamics.
So, we propose to interpret Eqs (16)-(17) for the general
case (ε, µ different from unity) as the equations for the
gravitational fields for states of space-time that are
characterized macroscopically (in the sense discussed
in the introduction) by the operators ε and µ.
The scalar constructed with Tαβµν and the tangent vector
uµ, for instance, takes the form
u(T ) = T αβ µν uα uβ u µ uν (20)
And gives the ‘energy’ of the field
GRAVITATIONAL ENERGY IN
AN ε-µ STATE OF TENSION
There have been many discussions, since Einstein’s 1916
paper, concerning the definition of the energy of a given
gravitational field. We do not intend to comment hare on
this subject but we shall limit ourselves to considering
one reasonably successful suggestion of Bel [3] for the
form of the energy-momentum tensor of gravitational
radiation.
The point of departure come from the supposed resemblance
of gravitational and electromagnetic effects. So, he defines
a fourth-rank tensor Tαβµν given in terms of quadratic
components of the field (identified with the Riemann
tensor) and written in terms of the Weyl tensor Cαβµν.
T αβ µν
(18)
C αβ µν
1
= ηαβρσ C ρσ µν 2
(19)
α β µν
*
αβ µ ν
= C αβ µν . This property does not
hold for Rαβµν. This is related to the lack of symmetry:
Rαβ µν ≠ Rµναβ . Indeed, we have
*

u β uν = C αβ µν u β uν = Hα µ
αβ µ ν

R * u β uσ = µHαε
R
*
α β µν
β ν 

*
*

1
θ αβ µν =  Rαρ µσ C β ρ ν σ + Rα ρ µσ C ρ σ  (23)
2



1 
U(ε , µ) = θ αβ µν uα uβ u µ uν = (ε E 2 + µH 2 ) 2
(24)
We would like to make an additional remark by presenting
to special properties of θαβµν
*
=C
(b)
in complete analogy with the electrodynamical case in
a general medium.
Due to the symmetric properties of the Weyl tensor, we
have C *
(22)
(a)
Then, the energy U(ε, µ) as viewer by an observer uµ will
be given by
*
∗

 
E 2 = Eαβ E αβ

 
H 2 = Hαβ H αβ
In the context f our theory, for a space-time in the states
ε–µ of tension, we propose to modify Tαβµν into θαβµν
defined in an analogous manner by
Where the definitions of the dual C αβ µν is the usual:
(21)
Where
Bel’s super-energy tensor takes the form:
αρ µσ * β ν 

*

1
= C αρ µσ C β ρ ν σ + C
C ρ σ 
2



1 
u(T ) = ( E 2 + H 2 ) 2

1
θ α βα µ = (1 − ε ) E ρσ Cβρ µσ
2
θ =θ α µα µ
(a)
(25)
(b)
Property (25a) states that not all traces of θαβµν are null
for a general states of tension of space-time that the nonnull parts of the contracted tensor are independent of the
‘permeability’ µ. The second property (25b) states that the
scalar obtained by taking the trace of θαβµν twice is null,
independent of the states of tension of the space-time.
81
`THE VELOCITY OF PROPAGATION OF
GRAVITATIONAL WAVES IN ε–µ
STATES OF TENSION
Subtituting Eq. (30) in (29) and using (28b) we finally
find
In order to know the velocity of gravitational waves in
ε–µ states of space-time let us perturb the set of equations
(15) and (16). The perturbation will be represented by
the map:



E µν → E µν + δ E µν



H µν → H µν + δ H µν
(a)
(b)
(26)


In which δ Eµν , δ H µν are null quantities. Then, Eqs. (15)
and (16) go into the perturbed set of equations:

δE
β
α ;β
≈0
(a)
 1

εδ E + h(α λ η µ) ρστ uρδ Hτλ ;ρ ≈ 0 (b)
2

δH
β
α ;β
(a)
≈0
 1 λ ρστ

µδ H − h(α η µ) uρδ Eτλ ;ρ ≈ 0 (b)
2
(27)
(28)
Now, let us specialize the background to be a Minkowaki
(flat) space-time with µ (ε ) → µ (ε )(1 + T ∇×) In this case
the covariant derivatives are the usual derivatives and we
can use the commutative property in order to write:


1
ε (1 + T ∇×)δ Eαβ + h(α λ ηβ ) ρστ uσ δ H τλ ; ρ ≈ 0 (29)
2
By taking the derivative of Eq. (27b) projected in the
privileged direction uµ.
Now, multiplying Eq. (27b) by the factor
1
∂ h(α ν ηβ )στγ uτ
2µ
∂x σ
We find


1 ν στγ
1
h η
uτ δ Hγν ;σ −
h ν η στγ uτ uρ hε (γ ην )ψρψ δ Eρε |ψ |σ ≈ 0
2 (α β )
4 µ (α β )
82
(30)
(31)
Where ∇2 is the Laplacion operator defined in the threedimensional space orthogonal to u µ.
In the same way an analogous wave equation can be obtained

for Hα µ . From Eq. (31) we obtain the expected result: the
velocity of propagation of gravitational waves in ε–µ states
of tension of space-time is equal to 1 / ε µ when T → 0.
Thus we are shown that the point of departure, from the
supposed resemblance of gravitational and electromagnetic
effects turns a truly unification when T →  / 2 M P c
well
(Planck limit) [6], where Rµν ≡ Tµmax
ν


E = iη H
= 0.
At the Planck scale both EM and gravity have the same
equations. Because of the ultra strong nature of EM
fields at the Planck scale, self-cancellation occurs and
the equations for both gravity and EM are the vacuum
equations:
Where the covariant derivative is taken in the background and
we limit ourselves to the linear terms of perturbation.


1
(1 + T ∇×)2 δ Eα µ − ∇ 2δ Eα µ = 0 εµ
1
G µν = Rµν − Rg µν = 0 2
(32)
But the zero is a result of cancelling terms,
G µν = 8π G / c 4 ( Aµν − T0 g µν ) (33)
where Aµν, is the first part of the Maxwell stress tensor,
which we will call the action stress, and where T0 is the
normalization stress scalar. The tensor T0gµν, will be
called the reaction stress. We can calculate the value of
T0 approximately:
T0 = M pc 2 / (2π 2rp3 ) = c 4 Λ / 8π G where we have defined the “cosmological constant”
Λ =  rp−2  Tp−2 , which was first proposed by Einstein.
Since at this scale we have gµν = Αg ην / T0 , we can
simplify the vacuum equation: G µν = Λg µν – Λg µν)
This equation is of the form first proposed by Einstein
[7] with the cancellation of terms first proposed by
Zeldovich [8]. At the Planck scale the first “action”
term and the second “reaction” term cancel exactly
to make a vacuum equation. It is here that the GEM
splitting occurs; the terms cease to cancel with the
parameters of the reaction term changing. We have
then a splitting into two equations, the first being the
action term equation
G µν = ΛTµν / T0 (34)
which becomes the standard non vacuum equation
of GR:
G µν = 8π G / c 4 Tµν
where Tµν, is now the stress tensor due to presence of
electrons and protons that have now appeared due to
splitting. The second equation, the reaction portion with
its negative sign will become the EM equation. It splits
again to form two equations of the form Gµν = –Λ' Tµν /T'0.
The new parameters Λ' and T'0 are no longer quantities
associated with the Planck scale but a new scale associated
with particles such that T'0 = q2/8πr 04 and Λ' = r 0 –2 where
q is a particle charge and r 0 has changed from the Planck
length to a particle classical radius.
In the chiral approach we have ∂ / ∂t → ∂ / ∂t (1 + T ∇×)
so we have
±
∇ 2 + ω 2 (1 + T ∇×)2 ε 2 − 2iω (1 + T ∇×)G ⋅ ∇  E = 0 (32)

 αµ
the solution of the wave equation can give relative retardation
of right- and left-handed circularly polarized waves
like was observed in the experiments with “Pioneer-6”,
whereas in the case of linearly polarized waves the effect
was practically zero. If such birefringent effects like
polarisation dependent bending of light by the Sun, the
Earth or time delay of pulsar signals are observed with
other measurements, they will signal new physics beyond
Einstein’s gravity [5-7].
As application of this theory in the future, will be the
potential designs to improve the Global Positioning System
(GPS). The variety of GPS applications is astonishing.
In addition to the more obvious civilian and military
applications, the system’s uses include synchronizing
of power-line nodes to detect faults, very-large-baseline
interferometry, monitoring of plate tectonics, navigation
in deep space, time tamping of financial transactions, and
tests of fundamental physics. Two years ago, the value of
the GPS to the general community had already become
so great that USA turned off “selective availability”-the
system by which the highest GPS precision was available
only to the military. At the Arecibo radio telescope in the
1970s and 1980s, Joseph Taylor and colleagues verified
the general-relativistic prediction for the loss of energy
by a binary pulsar through gravitational radiation. Their
exquisitely precise long-term timing measurements
made use of the GPS to transfer time from the Naval
Observatory and NIST to the local reference clock at
Arecibo. The GPS constellation of highly stable clocks in
rapid motion will doubtless provide new opportunities for
tests of relativity. More than 50 manufacturers produce
more than 500 different GPS products for commercial,
private, and military use. More than 2 million receivers are
manufactured each year. New applications are continually
being invented.
Relativity issues are only a small –but essential– part of this
extremely complex system. Numerous other issues must
also be considered, including ionospheric and tropospheric
delay effects, cycle slips, noise, multipath transmission,
radiation pressure, orbit and attitude determination, and
the possibility of malevolent interference. Relativistic
coordinate time is deeply embedded in the GPS. Millions
of receivers have software that applies relativistic
corrections. Orbiting GPS clocks have been modified
to more closely realize coordinate time. Ordinary users
of the GPS, though they may not need to be aware of it,
have thus become dependent on Einstein’s conception
of space and time.
CONCLUSION
This theory deserves further investigation. In any case,
the model we are proposing on gravitational interaction
has many intriguing consequences that should be carefully
examined. Among these, we would like to point out the
possibility of avoiding collapse either locally (starts) or
globally (the Universe).
The gravitational optic approximation should be changed
accordingly and many qualitatively new gravitational
phenomena are to be expected to appear. We intend to
come black to these problems elsewhere.
REFERENCES
[1]
J.A. Wheeler. “Geometrodynamics”. Academic
Press Inc. New York. 1960.
[2]
H. Endo. “On Ricci curvatures of certain
submanifolds in contact metric”. Tensor. Vol. 49,
pp. 146-153. 1990.
83
[3]
M. Novello. “Generalization of Einstein’s theory of
gravity in the presence of tensions in space time”.
IC/75/61. International Centre For Theoretical
Physics. 1975.
[6]
Nº 1, pp. 6-23. 2008.
[4]
S.N. Gupta. “Einstein’s and other theories of
gravitation”. Rev. Mod. Phys. Vol. 29, pp. 337-350.
1957.
[7]
H. Torres-Silva. “Maxwell equations for generalised
lagrangian functional”. Ingeniare. Rev. chil. ing.
Vol. 16 Nº 1, pp. 53-59. 2008.
[5]
N. Rosen. “A bi-metric theory of gravitation”.
General Relativity Gravitation. Vol. 4 Nº 6,
pp. 435-447. 1973.
[8]
H. Torres-Silva. “Einstein equations for tetrad
fields”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 85-90. 2008.
84
H. Torres-Silva: Einstein equations for tetrad fields
EINSTEIN EQUATIONS FOR TETRAD FIELDS
ECUACIONES DE EINSTEIN PARA CAMPOS TETRADOS
H. Torres-Silva1
RESUMEN
Todo tensor métrico puede ser expresado por el producto interno de campos tetrados. Se prueba que las ecuaciones de
Einstein para esos campos tienen la misma forma que el tensor electromagnético de momento-energía si la corriente
externa total es igual a cero. Usando la teoría de campo unificado de Evans se muestra que la verdadera unificación de la
gravedad y el electromagnetismo es con las ecuaciones de Maxwell sin fuentes.
Palabras clave: Ecuaciones de Einstein, campos tetrados, tensor de momento-energía, geometría Riemann-Cartan, sistemas
Einstein-Maxwell.
ABSTRACT
Every metric tensor can be expressed by the inner product of tetrad fields. We prove that Einstein’s equations for these
fields have the same form as the stress-energy tensor of electromagnetism if the total external current jα = 0. Using the
Evans’ unified field theory, we show that the true unification of gravity and electromagnetism is with source-free Maxwell
equations
Keywords: Einstein equations, tetrad fields, metric tensor, energy tensor, electromagnetism, Riemann-Cartan geometry;
Einstein–Maxwell system.
INTRODUCTION
It is agreed that gravitation can be best described by general
relativity and that it cannot be explained by using fields as in
electromagnetism or as in the case of any other interaction.
Furthermore, it has been assumed that the metric tensor is the
best mathematical argument to use to study on gravitation.
Such opinions lead physicists to concentrate more on only
the metric tensor and, hence, to change it according to
circumstances. As a result, this method provides some
important results about gravitation. However, it is also
obvious that these results are not enough to understand
gravitation as well as, perhaps, other interactions.
In the present paper, instead of concentrating on the
metric tensor, we shall focus on tetrad fields. Our first
objective will be to find some reasonable mathematical
results with these fields. The complete interpretation of
the results will be out of the scope of this paper.
1
Gravitation curves the space-time and this effect is related
to the line element or invariant interval as
ds2 = gµν dxµ dxν
where gµν is the metric tensor and its elements are some
functions of the space-time.
The metric tensor with tetrad fields is given by [1, 2]
gµν = eµ • eν
(1)
where eµ are basis vectors or tetrad fields, and these are
some functions of the space-time also (µ, ν = 0, 1, 2, 3).
Similar to (1), the inverse metric tensor can be written as
gµν = eµ • eν
where eµ are basis vectors of the dual space or cotetrad
fields. However, we will refer to these fields as inverse
fields throughout this work.
85
There are some useful features of and equations for the
tetrad fields and inverse fields. First
g µα gαν = δ µν
e µ • eα gαν = δ µν (2)
e µ • eν = δ µν
Other equations and all detailed calculations are given
in the appendix section.
If the metric tensor is determined, it is well-known that
it is demanding work to find the Einstein equations. The
Christoffel symbols for the metric tensor (1) are
Γα µν =
1 α
1
f • e = fα • e
2 ν µ 2 µ ν
where f α ν = ∂α eν − ∂ν eα .
The Riemann tensor for the above Christoffel symbols
is
1 α
1
1
∂ fβν • e µ + f α ν • fβµ + f α β • fµν ,
2
4
4
the Ricci tensor is
1
Rµν = jν • e µ + f α ν •fαµ ,
4
and the Ricci scalar is
1
R = jβ •e + f αβ • fαβ
8
1 α
∂ fαν = ∂α ∂α eν is the non homogeneous
2
Maxwell equation.
where jν =
Finally the Einstein Tensor can be expressed as
Gµν =
 1 αβ
1 α
α
 f • f − gµν  f • fαβ + jα • e   . (3)
4  ν αµ
4

The expression in square brackets is the same as the
stress-energy tensor of electromagnetism except for the
inner products. Despite this difference, the equations
of motion of the tetrad fields have the same form as the
86
 1 αβ

1 α
 F ν • Fαµ − gµν  F • Fαβ   4
4

(4)
Several results can be obtained from (3). However, the
most significant of these is that the Einstein equations
for the tetrad fields certainly give the electromagnetic
stress-energy tensor. More precisely, the general relativity
reveals that there are some inherent constraints for tetrad
fields. This means there are also definite limits for the
metric tensor. Since every metric tensor can be written
in terms of tetrad fields, metric tensors cannot be chosen
or adjusted arbitrarily. Instead, metric tensors must be
found as inner products of tetrad fields after these fields
are determined to be consistent with
∂α ∂α eν = jν = 0.
Another formalism to obtain this result is with the unified
field theory of Evans [3, 4]. We take the notation and the
conventions from [1], where also more references to Evans’
work can be found. We assume that the reader is familiar
with the main content of tetrad formalism. Here we were
able to reduce Evans’ theory to just nine equations, which
we will list again for convenience. Spacetime obeys in
Evans’ theory a Riemann-Cartan geometry (RC-geometry)
that can be described by an orthonormal coframe eα , a
metric gαβ = diag (+1,–1,–1,–1), and a Lorentz connection
Γαβ = Γβα . In terms of these quantities, we can define
torsion and curvature, respectively:
T α := Deα , (5)
Rα β := d Γα β − Γα γ ∧ Γ γ β . (6)
β
Gµν =
EINSTEIN TETRAD EQUATIONS
Rα µβν =
Maxwell equations; that is ∂α∂α eν = jν, with jα = 0 and
is the Maxwell electromagnetic tensor
The Bianchi identities and their contractions follow
there from.
The extended homogeneous and inhomogeneous Maxwell
equations read in Lorentz covariant form
DF α = Rβ α ∧  β
D * F α = * Rβ α ∧  β ,
(7)
respectively. Alternatively, with Lorentz non-covariant
sources and with partial substitution of (7), they can be
rewritten as
(
)
d F α = κ 0 Rβ α ∧ e β − Γ β α ∧ T β , Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008
(8)
)
(
d *F α = κ 0 *Rβ α ∧ e β − Γ β α ∧ *T β . (9)
In the gravitational sector of Evans’ theory, the EinsteinCartan theory of gravity (EC-theory) was adopted by
Evans. Thus, the field equations are those of Sciama [5],
which were discovered in 1961:
)
(
1
η ∧ R βγ = κ Σα = κ Σαmat + Σαelmg , 2 αβγ
1
mat
elmg
η ∧ T γ = κ τ αβ = κ τ αβ
+ τ αβ
.
2 αβγ
(
(
)
(10)
(11)
)
Here ηαβγ = * eα ∧ eβ ∧ eγ . The total energy-momentum
of matter plus electromagnetic field is denoted by Σα , the
corresponding total spin by ταβ.
What we will do here is to set a new principle where
mat
elmg
τ αβ
+ τ αβ
= 0 , so that describes the truly unification
of electromagnetism and gravitation. The derivation of the
field equations and their properties are discussed in [7].
Now we have conditions to discuss the Unification of
Electromagnetism and Gravitation through “Generalized
Einstein tetrads” who H. Akbar-Zadeh has proposed [6] a
new geometric formulation of Einstein–Maxwell system
with source in terms of what are called “Generalized
Einstein manifolds”. We show that, contrary to the
claim, Maxwell equations have not been derived in
this formulation and that, the assumed equations can
be identified only as source free Maxwell equations in
the proposed geometric set up. A genuine derivation
of source-free Maxwell equations is presented within
the same framework. We draw a conclusion that the
proposed unification scheme can pertain only to sourcefree situations.
In a recent article [6], using the tangent bundle approach
to Finsler Geometry, H. Akbar-Zadeh has introduced a
class of Finslerian manifolds called “Generalized Einstein
manifolds’. These manifolds are obtained through some
constrained metric variations on an action functional
depending on the curvature tensors. The author has
then proposed a new scheme for the unification of
electromagnetism and gravitation, in which the spacetime
manifold, M, with its usual pseudo-Riemannian metric,
gµν (x), is endowed with a Finslerian connection containing
the Maxwell tensor, Fµν (x). Following this scheme, the
author arrives at a class of Generalized Einstein manifolds
containing the solutions of Einstein–Maxwell equations.
As for Maxwell equations, they are declared [1] to have
been obtained by means of Bianchi identities. We wish
to point out the following flaws in the treatment of
Einstein-Maxwell system.
First consider the treatment of Maxwell equations. Through
some constrained metric variations, and the use of Bianchi
identities, the author arrives at [1, eq (5.55)]:
∇ µ F µν = µ1uν , (12)
where µ1 and µν = jν are defined by [1, eqs (5.14) and
(2.7)]: using notations of [3]
µ1 = −ur ∇i Fri , ur =
v r .
F
(13)
Using notations of [1] throughout, νr are fiber coordinates of
the tangent bundle over M and ∇i is the usual Riemannian
covariant derivative defined through gij (x). Assuming
that µ1 is the proper charge density [1], the author then
identifies (1) as the Maxwell equations with source. The
author has, therefore, assumed that:
µ1 = µ1 ( x ) (14)
However, this assumption, together with definition (13),
already implies equation (12). To see this, differentiate
(13) with respect to ν j and then use (12) to obtain:
∇ i Fj j = u r
∂F
∂v
j
∇i Fr i
∂F
= u j , and using (13) again, we arrive at
∂v j
(12). Therefore, rather than being derived, (1) has in fact
been merely assumed.
noting that
More importantly, assumption (12) implies that µ1 = 0,
so that the assumed equations can be identified only as
source-free Maxwell equations. However, for a system
of charged particles, for which we can write Maxwell
equations, the velocity vector is a function of x. Therefore
(12) can not be identified as Maxwell equations with
source because µ j in this equation are independent of x
and (contrary to [6]) cannot be considered as a velocity
field. There is, in fact, a genuine derivation of sourcefree. Consequently the proposed geometric formulation
of Einstein–Maxwell system can pertain only to sourcefree situations. However if we include chiral currents
87
(appendix 1) the truly unification of electromagnetism
and gravitation is obtained [7].
We have shown that every metric tensor can be expressed
by the inner product of tetrad fields. We have proved that
Einstein equations for these fields have the same form
as the stress-energy tensor of electromagnetism if the
total external current jα = 0. Besides, using the unified
field theory of Evans we show that the truly unification
of gravity and electromagnetism is with the source
free Maxwell equations. However a truly unification of
electromagnetism and gravitation is obtained if chiral
currents are included.
APPENDIX 1
In his 1916 paper on The Foundation of the General
Theory of Relativity [8], Albert Einstein demonstrates
the conservation of energy by relating the total energy

1

tensor Tµv to the Bianchi identity  R µν − δ µν R = 0 ,
 ;µ

2
the Maxwell energy tensor Tµv Maxwell, the field strength
tensor F µv, and the energy tensor tµv of the gravitational
field are related according to:
)
µ
µ
µ
−κ T ν ; µ = −κ T ν Maxwell + t ν
;µ
 µ
1 µ 
=  R ν − δ ν R = 0
2
 ;µ

)
(

1
= κ  F uσ Fµν ;σ + Fνσ ; µ + Fσµ;ν 
(1.1)

2



1 µ
σµ
= κ  F uσ Fνσ − δ ν F τσ Fτσ  − Fνµ F ;σ 
4




;µ



 1 uσ
σµ
σ
u
= κ  F Fνσ + *F * Fνσ
− Fνµ F ;σ  = 0
;µ

2
)
(
The “dual” of the field strength tensor above is defined
1
as *F στ ≡ ε δγστ Fδγ using the Levi-Civita formalism,
2!
see, for example, [9, 10 and 12]. This also employs
ε
δγµσ
εαβνσ = −δ
δγµσ
αβνσ
, see [11-12]. Integral to the identity
of Tµν;µ with zero and thus to energy conservation is the
second of Maxwell’s equations:
88
(
)
1
+ Fσν ;τ + Fντ ;σ = 0 F
4 τσ ;ν
(1.2)
which in turn has its identity to zero ensured by the
Abelian relationship:
CONCLUSION
(
Fµν = Aν ; µ − Aµ;ν (1.3)
between the four-vector potential Au and Fuv. Absent (1.3)
above, or, if (1.3) above were to instead be replaced by
the non-Abelian (Yang-Mills) relationship of the general
form:
Fi µν = Aiν ; µ − Aiµ;ν − gfijk A j µ Ak ν ,
(1.4)
where i is an internal symmetry index, f ijk are group
structure constants, and g is an interaction charge, then
(1.2) would no longer be assured to vanish identically,
and so the total energy tensor as specified in (1.1) would
no longer be assured to be conserved, Tµv;µ ≠ 0. More
to the point, the total energy Tµv would no longer be
“total”, but would need to be exchanged with additional
energy terms not appearing in (1.1). It is to be observed
that non-linear A∙A interaction terms such as in (1.4)
are also central to modern particle physics, and so must
eventually be accommodated by an equation of the
form (1.1) if we are ever to understand weak and strong
quantum interactions in a gravitational, geometrodynamic
framework.
The set of connections in (1.1) do, of course, underlie
the successful identification of the Maxwell – Poynting
tensor for “matter” with the integrable terms in (1.1),
according to:


1
1
T µν Maxwell ≡ −  F uσ Fνσ − δ µν F τσ Fτσ  = −  F uσ Fνσ + *F uσ * Fνσ  1

4
2


(1.5)
as well as the identification of the non-integrable energy
tensor tµv of the “gravitational field”:
κ ν ≡ t µ ν ;µ = Fµν F σµ ;σ = Fµν J µ ,
(1.6)
which represents the density of energy-momentum
exchanged per unit of time, between the electric current
density Jµ and electromagnetic field Fµv (see [12], following
equation (65a)). In the above, we have employed Maxwell’s
remaining equation
J ν = F µν ; µ (1.7)
However, if we set:
1
−κ T ν Maxwell = R ν − δ µν R
2


1
= κ  F uσ Fνσ − δ µν F τσ Fτσ  (1.8)
4


µ
(
κ uσ
F Fνσ + *F uσ * Fνσ
2
)


1
κ TMaxwell = R = −κ  F µσ Fµσ − δ µ µ F τσ Fτσ 
4

 (1.9)
κ
= − F τσ Fτσ + *F τσ * Fτσ = 0
2
(
)
on account of the photon mediators of the electromagnetic
interaction being massless, and therefore traveling at
the speed of light. Thus, as stated by Einstein in 1919,
“we cannot arrive at a theory of the electron [and matter
generally] by restricting ourselves to the electromagnetic
components of the Maxwell-Lorentz theory, as has long
been known” [13].
In addition to the problem of matter, there are other
problems which arise from equation (1.1). Because (1.1)
relies upon the Abelian field (1.3), it is simply not valid
for non-Abelian fields. Thus, without a reconsideration of
(1.1), one cannot apply the General Theory of Relativity
to non-Abelian interactions. This immediately bars
understanding SU(2)W weak interactions, or SU(3)QCD
interactions, for example, in connection with Einstein’s
theory of gravitation.
Additionally, (1.1) excludes, a priori, the possibility that
magnetic and electric current of electromagnetic nature
might actually exist in nature. Here Einstein does not
considerer chiral electric and magnetic currents. Our
conjecture is that without particle current, Jµ = 0, we
can take into account chiral currents produced by the
µ
µ
electromagnetic field, so we have Jchielectric ≡ J(ce) ≠ 0 .
Besides we considerer no magnetic monopoles but we
µ
include chiral magnetic currents, Jchimagnetic
≡ J(µcm ) ≠ 0
[15].
(1.10)
and because the current four-vector for chiral magnetic
currents may be specified in terms of J (cm)τνσ and
*Fµv by
J(σcm ) = * J(σcm ) =
then, on account of (1.1), we find that κν = 0 in (1.6) and
so the current is thought to vanish, Jµ = 0. Additionally,
the trace equation vanishes:
)
(
J(cm )τσν ≡ Fτσ ;ν + Fσν ;τ + Fντ ;σ ,
µ
=
In particular, if we define the third-rank antisymmetric
tensor (following and extending the Yablon’ approach
[12]):
1 ατγσ
ε
J(cm )ατγ = *F µν ; µ (1.11)
3!
we see that (1.1), as it stands, expressly forecloses the
existence of magnetic monopoles and chiral magnetic
currents, because the vanishing of J(cm)τσν in (1.10)
σ
causes J(cm ) in (1.11) to vanish as well. Any theory which
allows chiral currents by using a non-Abelian field (1.4),
requires that (1.1) be suitably-modified for total energy
to be properly conserved, because Fµν ;o + Fνσ ; µ + Fσµ;ν
)
(
will no longer be identical to zero. For completeness, we
also define (see [5]):
)
(
J(ce)τσν ≡ − *Fτσ ;ν + *Fσν ;τ + *Fντ ;σ =
1
ε Jγ 3! γτσν ce
(1.12)
As we shall demonstrate, all of theses problems stem
from the fact that (1.1) relies upon the vanishing of
the antisymmetric combination of terms in (1.2) to
enforce the conservation of total energy. The term
Tµν;µ = 0 is solidly-grounded: it is the quintessential
statement that total energy must be conserved. The


Bianchi identity  R µν − 1 δ µν R = 0 is equally
2

 ;µ
solid: although one can also add a “cosmological” term
 µ
1 µ
µ 
 R ν − 2 δ ν R + Λδ ν  = 0 , one is assured by the very
;µ
nature of Riemannian geometry that either combination
of terms will always be zero. Not so, however, for
1 uσ
F
Fµν ;o + Fνσ ; µ + Fσµ;ν = 0 . T h is ter m relies
2
(
)
directly on the Abelian field (1.3) and on the supposition
that chiral magnetic currents (1.11) vanish. Absent this
supposition, Tµν is no longer conserved, and so can no
longer be regarded as the “total” energy tensor.
89
REFERENCES
[1]
[2]
[3]
1916. The Principle of Relativity. Dover, pp. 111164. 1952.
C.W. Misner, K.S. Thorne and J.A. Wheeler.
“Gravitation”. Freeman, pp. 310-378. 1973.
A. Waldyr, Jr. Rodrigues, Quintino A.G. de Souza.
“An Ambiguous Statement Called ‘Tetrad Postulate’
and the Correct Field Equations Satisfied by the
Tetrad Fields”. Int. J. Mod. Phys. D14, pp. 20952150. 2005.
M.W. Evans. “Spin connection resonance in
gravitational general relativity”. Acta Physica
Polonica B38, pp. 2211-2220. 2007.
[4]
M.W. Evans. “Generally Covariant Unified Field
Theory: The Geometrization of Physics”. Vols. 1
to 5. Abramis Academic. 2005.
[5]
D.W. Sciama. “The physical structure of general
relativity”. Rev. Mod. Phys. 36, pp. 463-469. 1964.
[6]
H. Akbar-Zadeh. “Generalized Einstein manifolds”.
J. Geom. Phys. 17, p. 342. 1995.
[7]
Nº 1, pp. 6-23. 2008.
[8]
A. Einstein. “The Foundation of the General
Theory of Relativity”. Annalen der Physik 49.
[9]
G.Y. Reinich. “Electrodynamics in the General
Relativity Theory”. Trans. Am. Math. Soc. Vol. 27,
pp. 106-136. 1925.
[10]
J.A. Wheeler. “Geometrodynamics”. Academic
Press, pp. 225-253. 1962.
[11]
C.W. Misner, K.S. Thorne and J.A. Wheeler.
“Gravitation”. W.H. Freeman & Co. 1973.
[12] J.R. Yablon. “Magnetic Monopoles a nd
Duality Symmetry Breaking in Maxwell’s
Electrodynamics”. arXiv:hep-ph/0508257v1.
August 24, 2005.
[13]
[14] J.R. Yablon. “Magnetic Monopole Interactions,
Chiral Symmetries, and the NuTeV Anomaly”.
a rXiv:hep-ph /0509223v1. September 21,
2005.
[15]
90
Essential Part in The Structure of the Elementary
Particles of Matter?”. Sitzungsberichte der
Preussischen Akad. d. Wissenschafter. 1919. The
Principle of Relativity. Dover, pp. 191-198. 1952.
H. Torres-Silva. “Maxwell’s theory with chiral
currents”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 31-35. 2008.
H. Torres-Silva: A metric for a chiral potential field
A METRIC FOR A CHIRAL POTENTIAL FIELD
UNA MÉTRICA PARA UN CAMPO POTENCIAL QUIRAL
H. Torres-Silva1
RESUMEN
En este trabajo se presenta un ejemplo de una métrica específica que geometriza explícitamente un potencial cuadrivector
tipo luz (campo quiral). La geometrización muestra que tal vector tiene la misma estructura geométrica que un campo
gravitacional Kerr. Se discute una proposición teórica que un cuerpo rotante genera, su gravitación y el calibre de campo
tipo magnético que puede ser identificado con un campo quiral geometrizado. Este campo quiral representa un tipo novedoso
de campo que no puede ser identificado con alguno de los campos electromagnéticos conocidos. Como aplicación de esta
teoría se discute la morfología de los planetas alrededor del sol.
Palabras clave: Potencial vector, campo de fuerza cero, campo quiral, geometrización espacio tiempo, morfología.
ABSTRACT
In this paper we present an example of a specific metric which geometrizes explicitly a light-like four-vector potential
(chiral field). The geometrization shows that such a vector has the same geometrical structure as a gravitational Kerr
field. We discuss a theoretical proposition that a rotating body generates, besides a special gravitational field, a magnetictype gauge field which might be identified with a chiral geometrized field. This chiral field represents a novel type of field
because we cannot identify it with any of the known electromagnetic fields. As an application of this theory we discuss
the morphology of the planets around the sun.
Keywords:Light-like vector potential, force-free field, complete geometrization spacetime, morphology.
INTRODUCTION
In this contribution, we construct a metric which appears
appropriate for a geometrization, within the framework of a
Riemannian spacetime, of a light-like 4-vector potential field
which can be assigned to an electromagnetic-type field. Such
field with a 4-vector potential Aα satisfies the relation
Aα Aα = 0, Aα Aα

= 0, Aα Aα
⊥
= 0 ⇒ Aα Aα = 0, (1)
and Aα is denoted by us as a chiral field. In accordance with our
information something emerged for the first time in the work
of M Evans in connection with the hypothesis of the existence
of a special kind of magnetic field (see, for instance [1]).
The starting point is the well known approach to the
geometrization of physical fields involving the construction
1
of spacetime geometries (the so called force-free geometries)
within which the geodesic equation proves to be identical
to the equation of motion of a particle when interacting
with such (nongravitational) fields. This method derives in
fact from the generalized Einstein’s equivalence principle
which asserts that “any trajectory is a geodesic of some
geometry” [2]. Furthermore, the laws of motion, in the
case of interacting particles, are given by the differential
equations of the geodesics for the metric in question at
the instantaneous position of each particle [3].
Pursuing this subject, we observe that for the formulation
of the geodesic equations also in the presence of
nongravitational forces, some efforts have been directed
towards applying changes to the metric and other efforts
to modifications of the connection [4], in a Riemann or a
Riemann-Cartan spacetime. There appeared also papers
91
which consider the possibility of applying a Finsler or
a Randers geometry or a fractal spacetime geometry
in order to establish unitary theories of gravitation and
electromagnetism in conjunction with a probabilistic
interpretation of the geometry of the background
spacetime [4].
However, all these alternative interpretations of force-free
geometries have not yet reached the same level of elaboration
and experimental verification as is the case for Einstein’s
general theory of relativity the formal structure of which
has continuously invited the development of gauge theories.
These are reasons that why we maintain in the present work
the framework of a Riemannian spacetime which helps us
to geometrize a vectorial field. We propose a geometrization
of a vectorial field in the sense that the associated physical
quantity (e.g., the four-vector potential Aα) enters directly
into the metric which may be interpreted, alternatively,
as an ‘interior’ (Tαβ = 0, to obtain the microscopic Dirac
equation) or ‘exterior’ (Tαβ = 0, to obtain the classical
solution of Mercury’s orbit) solution of Einstein’s equations.
However, from an Einsteinian point of view, the field
defined by Aα is completely (truly) geometrized (like the
gravitational field itself) if it leads to a determination of
the geometry of the (curved) vacuum spacetime in which
no other (non-geometrized) matter manifests its presence
in conjunction with a non-zero energy-momentum tensor.
We emphasize that the physical quantities (e.g., density,
pressure, electromagnetic field tensor, etc.) which generally
appear on the right hand side of Einstein’s equations
represent non-geometrized quantities, i.e., the source of
the (geometrized) gravitational field. Our conjecture is
that if Tαβ = 0, det Fαβ ≠ 0, then, Aα is completely (truly)
geometrized [5].
In the present paper we adhere to the Einstein’s general
relativity and thus the energy and momentum of the
geometrized chiral field are encapsuled solely in the
pseudotensor tαb on the same geometrical footing as any
gravitational field. We recollect that the general relativity
is a very special non-Abelian gauge theory and thus it is
possible that a truly spacetime geometrization can be applied
also to a non-Abelian analogue of the electromagnetic field.
The Yang-Mills field may serve as such a field.
general relativity, Schwarzschild and Kerr solutions, (which
in Eddington coordinates are described also by light-like
four vectors) have an electromagnetic analogue. Thus,
the Kerr metric, which represents the gravitational field
exterior to a spinning source which ‘drags’ space around
with it, has the same geometrical structure as a geometrized
chiral field like an Evans-Vigier field. On a microscopic
level, the Evans’ optical (light) magnet [6] produced by a
circulary polarised light beam appears as a natural and
physically possible hypothesis. A search for cyclically
symmetric equations, similar to spin angular momentum
relations but now refering to a magnetic-type field, seems
also tempting from a geometrical point of view. Of course,
as for gravitation or perhaps for the entire field of physics
we do not yet know the physical intrinsic mechanism of
such a magneto-rotation induction: ‘rotation generates
magnetic-type field and magnetic field generates rotation’,
and yet we attempt to model and describe it here.
A simple experimental proposal for the verification of
these hypotheses may be the detection of an AharonovBohm effect as arising, for example, in the usual two-slit
electron diffraction experiment in which the solenoid is
replaced by a rotating body. Indeed, the gravitational
field of a rotating astrophysical lens object plays the role
of both a double slit (by its electriclike and curvature
inducing effects by gravity) and an ‘external’ field
(with a magneticlike contribution of the gravitation). A
proposal for a laboratory experiment for an observation
of a gravitational Aharonov-Bohm effect in conjunction
with photons is described in [6].
In the final section we present a discussion on the possibility
of identifying a chiral field like an a modified Evans-Vigier
field within the set of known electromagnetic fields.
SPECIAL METRIC AND BASIC RELATIONS
Let us consider a null-like four-vector with components
( )
)
)
Aα x β ≡ ( A0 , A1 , A2 , A3 = ( A0 , A (2)
We denote by
A2 = ηαβ Aα Aβ = 0 Attempts have also been made to mix directly the
standard symmetric Riemannian metric tensor with an
antisymmetric (electromagnetic) field tensor, but the new
nonsymmetric metric cannot achieve a real geometrization
of the electromagnetic field [6].
A possible existence of a light-like 4-vector electromagnetic
field would be a proof that the most important metrics of
is the Minkowski (flat) diagonal metric [8]. We should
mention that A α (x β) is here a standard spacetime
92
(3)
its Minkowskian module in which
ηαβ = +1, −1, −1, −1 Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008
(4)
vector which may represent the vector potential of an
electromagnetic-type gauge field. For the moment, we
cannot foresee if Aα may be associated with a massive
or zero-mass field or if we must include the subject of a
gauge invariance. Consequently, all the calculations are
given in the tangent bundle of spacetime.
We propose to study under which conditions a metric gαβ
having the special form
and [βγ, σ] is the Christoffel symbol of the first kind.
Because, g = constants = –1, it follows that
gαβ = ηαβ + KAα Aβ (5)
Γαβγ = gασ  βγ , σ 
1
= Kgασ  Aσ Aβ
2

(
) +(A A ) −(A A )
,γ
σ
γ ,β
β γ ,σ
 (12)

Γαβα = 0 (13)
where k ∝ t is a constant still to be determined, can
define a chiral field like an Evans-Vigier field, A chiral
field is defined as
and thus following [8] the Ricci tensor is given by
A → (1 + T ∇×) A Rβγ = −ηασ  βγ ,σ ,α + K  Aα Aσ  βγ ,σ ,α + A,αα Aσ + Aα A,σα  βγ ,σ  


+ ηαµησν  βσ , µ  γα ,ν 
(6)
(
)
(
( )
det gαβ ≡g = − 1 + KA2 = −1 (7)
g
=η
α
β
− KA A FORCE-FREE CHIRAL FIELD
Introducing the parameter s defined by
and thus, the inverse (contravariant) metric is
αβ
(14)
≡ KR1 + K 2 R2 + K 3 R3 + K 4 R4
The determinant of the metric tensor gαβ is given by
αβ
)
(
− K ηαµ Aσ Aν + ησν Aα Aµ  βσ , µ  γα ,ν 
+ K 2 Aα Aµ Aσ Aν  βσ , µ  γα ,ν 
(T is the chiral factor).
)
(8)
ds 2 = gαβ dx α dx β (15)
the equations of geodesics,
The metric (5) is similar to the one which describes a
weak gravitational field, i.e.,
gµν = ηµν + hµν , g µν = η µν − h µν (9)
However, for the time being we do not impose yet any
condition on the value or the strength of the term KAα Aβ.
There follows that
α
αβ
αβ
αβ
duα
dx µ + Γαβγ u β uγ = 0, u µ =
ds
ds
(16)
duα
dC
+ KAα
− KCηασ Bσγ uγ ds
ds
(17)
C = Aβ u β . (18)
become
where
αβ
A = g Aβ = η Aβ , η Aα Aβ − g Aα Aβ = 0 (10)
Aα
and thus the indices of
may be raised and lowered
with either the metric gαβ or the Lorentz metric ηαβ. It
is easy to show that
Aα Aα ;β = Aα Aα ,β = 0 (11)
where the ordinary partial derivatives are denoted by
commas (or alternatively by ∂α and ∂/∂χα), and covariant
derivatives by semicolons. The Christoffel symbols are
At this point it is easy to see that an Evans-Vigier field
described by the metric (5) becomes a ‘force-free field’
with respect to the motion of a charged test particle
having the characteristic parameter e/m 0, and subject to
the constraint
KC = constant = T
e
e
=
.
c m c 2
0
(19)
93
With this constraint, the geodesic equation (4) reduces
formally to the Lorentz equation,
duα
e
= T η βα Fβν uν c
ds
(20)
We may identify Fβν with an electromagnetic field tensor
if the 4-vector potential Aµ is related to an electromagnetic
potential Aµ by a gauge transformation of the second
kind
Aµ = µ +
∂φ
∂x µ
(21)
Since it is possible to demonstrate that constraints such
as (6) and (8) are consistent and in fact do not contradict
each other along the trajectory of the test-particle (see,
for instance, [19]), we can assert that we have achieved
a local or a semilocal geometrization (i.e., one along a
curve) of the chiral field.
The final conclusion of this section is that any field
described by a metric of the form (5) may act on a
test particle with a Lorentz-type force (7). In such
geometrical terms, a Lorentz-type force was known
until now only for a weak gravitational field (see, for
instance, [6]).
In other words, in this case Einstein’s equations are used
merely for a definition of an energy-momentum tensor
which generates a given gravitational field.
In the following we will not use this identity aspect of
the Einstein’s equations since we intend to geometrize
the field Aα which may be considered as a gravitational
perturbation of a vacuum spacetime. Then the field equations
correspond to an ‘exterior case’ and are given by
Rβγ = 0 where R βγ is given by equation (14). In a way, the
constant K may be called a ‘coupling constant’ because
it characterizes the strength of the perturbation of the
vacuum spacetime generated by a chiral field like an
Evans-Vigier field. We assume that the form of the metric
(5) retains its independence from the value of K. In other
words, the metric gαβ given by (5) remains a solution
for any arbitrary value of K. Thus in the expression (14)
of Rβγ, each coeficient of K and of its powers must be
cancelled separately. In this way, following [6], we obtain
four equations:
94
Gαβ ≡ κ
Gαβ
κ
≡ κ Tαβ (22)
(24)
)
(25)
(
+η η
Bearing in mind that the metric tensor is given in our
account by equations (5) and (7), we need only derive the
Rαβ, R, and also the Einstein’s tensor from the gαβ and
establish in this way the components of the matter tensor
Tαβ. If this energy-momentum tensor coincides with one
which is known for a given (physical, phenomenological)
material scheme, we say that (5) represents a solution
of Einstein’s equations for such a scheme. If we do not
posses such a coincidence, we say that we face an exotic
matter which might determine the desired properties of
the spacetime (e.g., ‘traversable wormhole’ [21] or ‘warp
drive’ [22]). From this point of view the general theory
of relativity is not a closed theory, and sometimes the
Einstein’s equations seem to form a mathematical identity
if a suitable metric is chosen:
R1 = 0 = −ηασ  βγ , σ  ,α
R2 = 0 = C0 T  Aα Aσ  βγ , σ  + A,αα Aσ + Aα A,σα  βγ , σ  
,α


αµ σν
A COMPLETE GEOMETRIZATION
OF A CHIRAL FIELD
(23)
 βσ , µ  γα , ν 
)
(
R3 = 0 = −C0 T ηαµ Aσ Aν + ησν Aα A µ  βσ , µ  γα , ν  (26)
R4 = 0 = +C02T 2 Aα Aµ Aσ Aν  βσ , µ  γα , ν  (27)
We note that, in accordance with equation (13), the potential
Aα generates a new light-like vector aα which, by analogy
with the kinematics of a timelike congruence of curves,
may be called an ‘acceleration-potential vector’ and has
the following properties:
( )
aα = A;αβ Aβ = A,αβ Aβ = − b x γ Aα aα = gαβ aβ = ηαβ aβ (28)
ηαβ aα aβ = gαβ aα aβ = 0
aα a;αβ = aα a,αβ = aα A;αβ = aα A,αβ = 0 aα Aα = 0
(29)
We notice that equations (12) and (14) are satisfied
identically, and that equation (13) is reduced to the definition
of the acceleration potential (15). Thus the Einstein field
equations (11)-(14) become
2
(
 Aβ Aγ
)
∂ ∂
→ (1+ T ∇× )
∂t ∂t
=0
2 → −∇ 2 (1 + T ∇×)2 ,
(31)
(35)
I1 =
2
2
1
F F αβ = E − B ,
2 αβ
(36)
1
I 2 = − Fαβ F αβ = 2E ⋅ B ,
2
(37)
I 3 = −2 Aα T αβ Aβ ,
(38)
(30)
For the stationary case,
I 0 = Aα Aα , we have particular vectorial solutions
k
∇× A= 
A ,
1  kT
(
)
(32)
there arise two remarkable solutions of equation (30),
namely, the Schwarzschild-type solution, and the KerrSchild type metric.
Here kT is related to the angular velocity and, thus, to the
angular momentum of the source. We remind the reader
that the Kerr metric represents a vacuum field exterior
to a spinning source. Hence, a chiral field like an EvansVigier field and a typical gravitational field have the same
topological properties. It is important to stress that for
the Schwarzschild-type solution (31),
∇ × AS = 0, T → ∞ (no magnetic-type field) (33)
and for the Kerr-Schild type metric (19)
∇ × AKS ≠ 0, T ≠ 0 (magnetic-type field). (34)
An immediate consequence of these results is that rotating
bodies generate, besides a special kind of gravitational
field, also some magnetic-type gauge fields defined by lightlike vector potentials. For the time being all experimental
tests of general relativity (e.g., Advance of the perihelion
of Mercury, Bending of light, Gravitational red shift,
etc.) are expressed only as functions of the mass of the
central gravitating body. In order to evaluate the physical
implications of the chiral field we must evaluate all these
effects in terms of the light-like vector potential.
PHYSICAL CONTENT
OF CHIRAL CONDITION
Four Independent Electromagnetic Invariants
In Classical Electrodynamics there exist only four
independent electromagnetic (EM) field invariants [7],
namely (in units with c=1),
where Fαβ is the EM field tensor, Fαβ is the dual EM
field tensor and Tαβ is the Maxwell stress-energy tensor.
Salingaros [7] used these invariants to announce the
proposition: plane monochromatic EM (transverse) waves
are characterized by vanishing invariants I1 = I2 = I3 = 0 in
the Lorentz gauge. As we mentioned, a chiral field as an
Evans-Vigier field are defined by a vanishing invariant I0
= 0, but contrary to Evans-Vigier field, the conditions for a
chiral field are I0 = 0, I|| = 0, I⊥ = 0 and I3 = –2AαTαβ Aβ.
Rotation and Chiral Field
Following our preceding account, we may now state
that a geometrized chiral field like an Evans-Vigier field
represents a classical but exotic electromagnetic-type field
which possesses similar properties to gravitational fields
defined by Schwarzschild and Kerr metrics. The process of
geometrizing such field, through association of the vector
potential with part of the structure of spacetime, leads to
the supposition that, possibly, there exists a fundamental
relation between rotation and a magnetic-type field. It
should be emphasized that in a sense our results demostrate
a generalisation of and the reciprocity to a well known
physical phenomenon. Thus, considering a free particle
in an external electromagnetic field defined by the tensor
Fαβ, we observe the generation of a vorticity,
ωαβ = uα ;β − uβ ;α ,
(39)
which is related to the field tensor Fαβ via the (London)
equation of superconductivity [6]:
Fαβ =
∂Aβ
∂x
α
−
∂Aα
∂x
β
=
mc
ω .
e αβ
(40)
Equation (40) expresses that the four-vector potential
Aα is tangent to the particle trajectories at all points
95
and thus the particle velocity is proportional to the
vector potential as we have seen above. It is important
to stress that it is the external vectorial field Aα which
determines the motion of a test particle and not vice
versa. Moreover, generally, the four-velocity u α may be
defined as the vector-potential of an inertial-gravitational
field and may be assigned to each point of the spacetime
independently of the fact whether or not a test particle
resides at that point [30-31]. Hence, if the vacuum
spacetime is perturbed by the presence of the vectorial
field A α we can assert that the source of vorticity is
precisely this field.
Our generalisation arises from the fact that not only
does a normalized (Dirac) vector potential field [see eq.
(41)] generate a vorticity field, but yields also a relation
between the angular momentum of a rotating body and
a geometrized light-like vector potential. This result is
clearly illustrated by equations (32) and (36).
∇ × B = kB γ −1
B
v = ±

1/ 2 −
γ
2

 (µ ρ
0
The compressed gas forms a cylindrical volume of plasma
which is moving through the background plasma and
rotating with a finite angular velocity. As this mass of
plasma propagates through the surrounding gas, it loses
energy by accelerating the surrounding plasma. The cylinder
will lose energy and settle down to a minimum-energy
“relaxed’’ state, a force-free collinear cylindrical structure.
It is shown in detail elsewhere -- that the resulting “field
equations’’ for the flow are given by
96
(42)
)
where ρ: fluid density, B: magnetic induction field, ν:
velocity of the center of mass of a fluid element, and γ:
ratio of specific heats of the gas.
A pseudoplane solution to the force-free equation (1) is
given by
Br = − k 2 a
J1 ( kr
) sin θ
kr

 J ( kr
Bθ = k 2 a  1
− J0 ( kr
 kr
 1 ( kr cos θ
Bz = k 2 aJ
)
)
Morphology of the Solar System Set with ∇ ×Β = κΒ
It has been shown in detail elsewhere [8] that the Bode
numbers and measured velocity ratios of the planets are
accurately predicted by the eigenvalues of the Euler-Lagrange equations resulting from the variation of the
free energy of the generic plasma that formed the Sun
and planets. This theory is reviewed to show that the
equation (36) ∇ × A = κA can explain the velocity ratios
of planets, the Bode numbers correspond to the roots of
the first-order Bessel functions. The extrema of the roots
of the zeroth-order Bessel function predict the ratios of
the measured planetary velocities almost without error
for the outer planets. Both sets of roots correspond to
the same eigenvalue solution of the force-free equations.
Both the Titius--Bode series and Kepler’s harmonic law
are predicted by the “relaxed state solution’’ of the freeenergy equation for the generic plasma that formed the
Sun and planets. Newton’s law of gravitation is not used
in the calculations. Here we use the chiral approach where
Rµ = 0, and A || B.
(41)

) cos θ

where Br, Bθ, and Bz are the magnetic induction components
in the striated rings of the gas cylinder, and
a = kθθ + k z z , k 2 = k12 + k z2 where kθ, k1, and kz are constants supplied by the boundary
conditions given by the chiral approach. If we plot of J1
and J0 with the functions scaled to the geometry of the
solar system, we observe that for J1 = 0,
B 2 ∼ J02 , Bθ ∼ J0 . ,
This maximizes the magnetic and kinetic energy at the
origin. In the cylindrical structure formed by the supernova
explosion, the first root corresponds to the structure of
the star at the center of the hypothetical solar system, and
the second root corresponds to a ring of gas just outside
the star. The corresponding flow velocities in the rings
is given by eq. 42. The geometry of the configuration is
shown in figure 1.
The signs reverse for every other ring (corotational and
contrarotational) so that the azimuthal velocities are all
prograde. The azimuthal velocity of the gas in each ring
has a direct relationship to the velocities of the planets
Figure 1. Plots of Bessel functions versus distance (AU) of planets from the sun.
as they exist today. An examination of figure 2 shows
that the Bode numbers of the planets out to Jupiter are
predicted by the roots of the equations describing the
“relaxed state’’ of the primordial gas. Comparison of the
measured velocity ratio with the ratios of the extrema of
J0 (kr) show very close agreement.
For the outer planets, the Bode series fails completely for
Neptune and Pluto, but the plasma solutions, the Bessel
function roots, give exact predictions. We can observe
that if the asymptotic expansions J1 (kr) and J0 (kr) where
carried out, the theory could be checked all the way out
to and including Pluto.
The predicted ratios of the successive peak velocities of the
gas in the rings check the measured velocity ratios of the
inner planets within a few percent. The velocity ratios for
Uranus, Neptune, and Pluto are exact. the relaxed state of
the generic plasma predicts both the Bode number series
and Kepler’s harmonic law p2 = a3 where p: period of
the planet, a: average radius of the planet.
It is suggested that the rings of gas in the planet structure
“roll up” azimuthally to form balls of gas that eventually
evolve into the planet . The roll up of vortex rings to form
balls of gas is a well-known phenomenon which has been
observed in laboratory experiments.
A planet is predicted at 1.3 AU. No such planet exists
today. It is suggested that the missing planet suffered
a catastrophe either in the birthing process or at a later
time and that the residue is our moon.
Figure 2. Orbital speed as a function of the distance
from the sun.
REFERENCES
[1]
M.W. Evans. “General covariant unified field
theory”. Abramis Academic. Suffolk. Vol. 1.
2005.
[2]
A.J. Wheeler. “Geometrodynamics”. Academic
Press, pp. 225-253. 1962.
[3]
A. Einstein. “The Foundation of the General
Theory of Relativity”. Annalen der Physik
97
49. 1916. The Principle of Relativity. Dover,
pp. 111-164. 1952.
[4]
[5]
98
C. Möller. “Selected Problems in General
Relativity”. Brandeis University 1960 Summer
Institute in Theorical Physics. Lecture Notes.
Brandeis University. 1960.
Nº 1, pp. 6-23. 2008.
[6]
J. Argyris, C. Ciubotariu and I. Andreadis. “A
metric for an Evans Vigier field”. Foundation of
Physics Letters. Vol. 11, pp. 141. 1998.
[7]
N. Salingaros. “Invariants of the electromagnetic
field and electromagnetic waves”. Am. J. Phys.
Vol. 53, pp. 361. 1985.
[8]
D.R. Wells. “Quantization effects in the plasma
universe”. IEEE Trans. Plasma Sci. Vol. 17, pp. 270.
1989.
H. pp. 99-110
Torres-Silva: Chiral universes and quantum effects produced by electromagnetic fields
Ingeniare. Revista chilena de ingeniería, vol. 16 Nº 1, 2008,
CHIRAL UNIVERSES AND QUANTUM EFFECTS
PRODUCED BY ELECTROMAGNETIC FIELDS
UNIVERSOS QUIRALES Y EFECTOS CUÁNTICOS
PRODUCIDOS POR CAMPOS ELECTROMAGNÉTICOS
H. Torres-Silva1
RESUMEN
La estructura aceptada del espacio y el vacío se derivan de los resultados de la cosmología relativística y de la teoría cuántica
de campo. Se demuestra que una interfaz quiral entre regiones enantioméricas de un universo cerrado, o un universo derecho
y un universo izquierdo, relacionados por un elemento de simetría PCT a lo largo de la interfaz, representa un modelo con
todos los atributos requeridos por el vacío teórico. Se desprende que el comportamiento cuántico es entonces visto que es
inducido por la interfaz de vacío. La mecánica quántica emerge como un caso especial de la mecánica clásica, más bien
que siendo la última un subconjunto de la primera. Esto resuelve el problema observacional mecánico cuántico, explica
las coincidencias de los grandes números cosmológicos y toma en cuenta la antimateria en el cosmos.
Palabras clave: Vacío, interfaz quiral, campo cuántico, universo derecho (izquierdo).
ABSTRACT
The accepted structure of space and vacuum derives from the results of relativistic cosmology and quantum field theory.
It is demonstrated that a chiral interface between enantiomeric regions of a closed universe, or a (right) R-Universe and
(left) L-Universe, related by an element of PCT symmetry along the interface, represents a construct with all the attributes
required of the theoretical vacuum, in-so-far as quantum behaviour is then seen to be induced by the vacuum interface.
Quantum mechanics emerges as a special case of classical mechanics, rather than the latter being a subset of the former.
This removes the quantum-mechanical observational problem, explains the cosmological large-number coincidences,
and accounts for the anti-matter in the cosmos.
Keywords: Vacuum, chiral interface, quantum field, R(L)-Universe.
INTRODUCTION
The vacuum is surprisingly hard to fill, despite clear
pronouncements from both general relativity and quantum
theories. The theory of general relativity concerns the shape
of four-dimensional space and fields, whereas quantum field
theory details a structured vacuum state and particles.
The logical next step of advancing a model of the physical
vacuum, consistent with both theories, is the subject of
this paper. This is a basic assignment because of the
widely held belief that quantum theory and relativity
1
are essentially incompatible. The problem arises through
the presumed non-locality of quantum theory, in direct
conflict with the tenets of relativity. The difficulty,
first highlighted by Einstein, Podolsky and Rosen, now
commonly referred to as the EPR paradox. It will be
necessary to return to this dilemma as a crucial test
of any proposed vacuum. To establish the necessary
background, a brief summary of the implications of
relativistic cosmologies and of quantum field theory on
the nature of space and the vacuum is presented first. Our
conjecture is that quantum theory and general relativity
are essentially compatible when the matter is produced
99
by electromagnetic fields with R µν = 0 and Tµνem = 0 but
det Fνµ ≠ 0 . The electromagnetic field theory of matter
rises when we have two universes separated by a chiral
membrane. This defines the vacuum as an interface,
either between two universes or between two regions of
opposite chirality in the same universe.
Chiral approach means that our Universe is observable
area of basic space-time where temporal coordinate is
positive and all particles bear positive masses (energies).
The mirror Universe is an area of the basic space-time,
where from viewpoint of regular observer temporal
coordinate is negative and all particles bear negative
masses. Also, from viewpoint of our-world observer the
mirror Universe is a world with reverse flow of time,
where particles travel from future into past in respect to
us. The two worlds are separated with the membrane - an
area of space-time inhabited by light-like particles that
travel along light-like right or left-handed (isotropic-chiral)
spirals. In appendix 1 we show the difference in energy
between the interface and the two universes.
QUANTUM FIELD THEORY
The approach that reveals the nature of space and the
vacuum is Quantum Field Theory. Dirac produced the first
quantum field theory for massive spin half particles. The
energy spectrum was found to consist of both positive and
negative states, separated by a gap of energy ∆E=2mc 2 .
This Dirac sea is the vacuum which therefore consists of
an infinite number of negative energy electrons, protons,
neutrons and all other spin half particles, or fermions.
Any vacancy or hole in the Dirac sea, at level -E, can be
filled by an electron dropping down from the level at E.
An amount of energy 2E is radiated, while both hole and
electron disappear into the vacuum. The hole is therefore
equivalent to a particle (called a positron) of charge +e and
of positive energy E. The mass of the electron-positron
pair that disappears produces the radiated photon or
energy quantum hν = ∆E = 2(mc 2 ) .
This equation can be obtained from the chiral
electrodynamics developed in accompanying papers.
This prediction of anti-particles has been confirmed
experimentally for all fermions. The model implies that the
vacuum should also support an infinite negative sea of the
anti-particles to ensure electrical neutrality. Each electronpositron pair is linked by a photon as, seen in figure 1.
In this (Feynmann) diagram a positron differs from an
electron only through the direction of an arrow, and
100
positrons have been described as electrons moving
backwards in time, (chiral particle).
e+
e–
time
(↑)
γ = e+e-
Figure 1. (Feynmann) diagram of an electron and positron
produced by a light photon.
The lifetime of virtual particles is proscribed by the
uncertainty principle ΔΕ ⋅ Δt ∼ h. Real particles can be
created when the vacuum is polarized by a sufficiently
strong chiral field.
The vacuum is assumed to accommodate virtual particle/
anti-particle pairs which requires a symmetry between
matter and anti-matter worlds. Particles and anti-particles
have the same modulus of mass, equal but opposite
charges and magnetic moments, and if they are unstable,
the same lifetime. Collectively this is known as chargeparity-time (PCT) symmetry. In appendix 1 we explain
the PCT symmetry in our model.
The PCT theorem requires invariance for all fields under
this three-way combined operation. This is therefore also
the property of the vacuum with all its virtual particles
and intermediaries.
In view of the foregoing, the single most vexing, unresolved
problem is the imbalance between matter and anti-matter
in the observable universe. Speculations that link this
problem to the parity violation of weak interactions are
clearly at variance with the PCT theorem. Conventional
wisdom has it that anti-matter apparently disappeared
soon after the big bang, and outlandish suggestions of
its whereabouts in the universe abound. Contrary to this
we postulate regions of matter (R) and anti-matter (L)
separated by a radiation layer of chiral EM.
In Quantum theory we find quantum paradoxes dealing
with the quantum-mechanical observational problem,
non-locality and the EPR paradox. Quantum theory
demands that two systems, once in interaction, remain
correlated ever after until a measurement disturbs one of
them. This measurement then reveals not only the nature
H. Torres-Silva: Chiral universes and quantum effects produced by electromagnetic fields
of the system under study but also that of the remote
partner still linked to the first by a common wave function;
all this despite the absence of causality in the quantum
world. This instantaneous communication, through the
collapsing wave function, constitutes the non-locality of
quantum theory.
The information carried by a wave function is indeterminate
until a measurement selects a single result from an
infinite set, and changes the course of events irreversibly.
All information, however, does not perish when the
measurement selects one bit. The wave function persists
to allow alternative choices elsewhere. Each measurement,
therefore, splits the universe into two, each with independent
continued existence. Instead of a single quantum universe,
an infinite number of universes is therefore required by
the many-worlds quantum theory.
In view of the fact that quantum theory deals in noncommuting operators, illogical conclusions in the system
are not unexpected, and are actually provided for in
quantum logic, which is based on non-Boolean reasoning.
The only problem is that it destroys locality, causality,
reality, logicality and other coherent ideas on which a
consistent cosmology can possibly be constructed. When
postulating a vacuum structure, the real challenge is
therefore to account for the unpredictability of quantum
events. Massive objects behave classically. Even large
molecules behave classically. The behaviour of very
small particles, which can be considered as isolated in
the vacuum, however, is more erratic and more wave-like.
To preserve any notion of reality it is therefore necessary
to accept the macroscopic world as the norm. It contains
quantum world as a special case, and not the other way
around.
Noting that the time evolution of both classical and quantum
mechanics merely corresponds to a change of coordinates,
it is concluded that neither system can adequately describe
irreversible processes. Natural macroscopic processes such
as decay and lifetime are therefore outside the scope of
quantum mechanics, which appears as a simplified limiting
case, useful for the description of microscopic events
only. In view of this, the standard argument that the more
fundamental quantum theory contains classical theory
as a special limiting case cannot be sustained. Prigogine
finds that quantum theory is not complete, and suggests
irreversibility as another basic element in the description
of the physical world. However, when superimposing an
entropy operator on quantum mechanics the distinction
between classical and non-classical systems disappears.
The classical theory with irreversibility therefore contains
quantum theory as a special case. That is the model to
be accepted here, assuming that classical theory, as it
describes the rational world, is universally valid. Quantum
phenomena only emerge in systems where interaction with
the vacuum produces significant perturbations. The most
basic ingredient of a cosmologically reasonable model
of the vacuum is therefore an ability to predict quantum
behaviour for sub-atomic particles.
THE VACUUM INTERFACE
The minimum statement consistent with all relevant theories
is that the physical vacuum represents an element of PCT
symmetry in four-dimensional space. Literally this defines
the vacuum as an interface, either between two universes
or between two regions of opposite chirality in the same
universe. The latter more economical situation is the more
attractive. The experimentally observed structure of the
vacuum would then represent the faint echo of another
enantiomeric world from across the interface.
Progressively smaller particles experience, to an increasing
extent, the effects of interacting with the hidden world
beyond the interface. An observer keeping track of the
particle is not aware of this hidden interaction and finds
that the motion becomes inexplicably more erratic. The
differential equation to model the motion is found to
represent a wave packet rather than a classical particle.
The mathematical description of the particle’s progress is
precise, but the physical interpretation is incomplete. The
crucial result is that the particle in the quantum region
does not behave differently from classical particles. Its
progress follows the same logic and causality, but since
its equations of motion are formulated with neglect of a
vital segment of its total environment, they appear more
complicated than necessary. This anomalous behaviour
decreases rapidly with increasing aggregation. The genesis
of the postulated dual system is like the spontaneous
separation of phases that occurs on the cooling of a twocomponent homogeneous fluid. This happens through
symmetry breaking down when the interaction between like
entities becomes dominant. The phase separation occurs
in four dimensions and three-dimensional observations
cannot penetrate the dividing surface. A useful analogy
is to consider a bilayer of two-dimensional worlds. They
are everywhere in contact but oblivious of each other. The
interface provides the contact with the third dimension.
The three-dimensional vacuum is to be visualized in strict
analogy with this two-dimensional interfacial surface and
there is no freedom of motion, either towards or away from
the interface. To cross the interface it is necessary to move
into a third dimension, which is not an allowed operation in
101
two dimensions. The only way to detect the presence of the
interface is by quantum interaction which has little effect
on massive entities, but influences microscopic particles
dramatically. Likewise, a massive three-dimensional universe
is everywhere in contact with the three-dimensional vacuum
interface. In order to cross the interface it is necessary to
proceed along a fourth dimension.
but is simply used here as an instructive two-dimensional
analogue. However, this surface is difficult to visualise as
it cannot be embedded in R3, three-dimensional euclidean
space. What is proposed instead is to consider as a model
of the physical universe some three-manifold which,
like the Möbius strip, is a non-orientable and one-sided
elliptical manifold.
The three-dimensional analog requires a four-dimensional
twist or curvature of three-dimensional space that
closes the universe onto itself and turns left-handed into
right-handed objects. Considered as a single universe in
three-dimensional space, chirality is preserved throughout.
However, the interface created by the curvature separates
regions of space with opposite chiralities. This interface
cannot be crossed in three-dimensional motion, but allows
interaction between entities near the interface to give
rise to the quantum effects, (see figure 2 of our universe
model). By the principle that the boundary of a boundary
is zero, an interface in four-dimensional space has no
two-dimensional boundary and the postulated vacuum
must be three dimensional as observed.
Figure 2. Positron Spin ( /2) in L-Universe and Electron
Spin (–  /2) in a R-Universe or both are in one
Universe with two enantiomeric regions.
The exact nature of the difference between the complementary
universes makes interesting speculation, but it is useful to
think of this as a difference in chirality. It is known from
the spontaneous separation of enantiomers how chirality
can be the driving force of phase separation. It is well
known that objects with two-dimensional chirality can be
identical when analysed in three dimensions. Rotations
in the plane of these two-dimensional objects can never
bring them into coincidence, but a simple rotation about
an axis in the plane readily achieves this. The chirality
is removed by a three-dimensional operation.
Now consider a row of entities of the L-form only, along
the two-dimensional surface of a Möbius strip and with
orientation in the surface preserved. A two-dimensional
world, populated by objects of the same chirality is
obtained. However, the paper can be considered as an
interface which separates different worlds on the opposite
sides of the sheet.
One finds that it separates enantiomeric forms. The inversion
of chirality is brought about by the Möbius twist, which is
a three-dimensional operation. The Möbius ribbon is not
proposed to represent the true topology of the universe,
102
The present proposal pictures a universe based on the
orientable double cover of period 2π. The postulated
interface, called the vacuum, is closed in four dimensions
with period π, and corresponds to the relativistic
hypersurface which is the locus of light signals and
populated by bosons only.
The normal to the surface oriented in space is itself oriented
in time, the fourth dimension of the present argument. This
reduces quantum behaviour to the fluctuation along a time
coordinate, between regions of three-dimensional space.
It must be emphasized that this proposed model is
nowhere in conflict with either relativistic or quantum
theories, and is fully consistent with both. It has the merit
of simplicity and provides the logical structure to relate
quantum effects directly to the macroscopic physical
world, (see appendix 2).
APPLICATION OF THE MODEL
It is of interest to examine how the model deals with some
of the vexing problems of quantum theory, like the nonlocality embodied in the EPR argument. It is now proposed
that the behaviour of the particle remains rational also in
the quantum region, near the interface, and hence there
must be a persistent causal relationship of both individuals
from a correlated pair with their common origin. This
relationship persists until the first of the two individuals
becomes involved in a unique encounter. Until that point,
however, synchronised observation of the pair cannot but
reveal correlated behaviour. The degree of correlation is
neither a function of their separation nor dependent on
an exchange of information.
The model also provides a simple account of the missing
anti-matter. It is not difficult to identify anti-matter
with the enantiomeric matter introduced above. This
means that the vacuum separates the material and antimaterial worlds. However, exploration of the universe
never reveals the gradual change in chirality along
the curvature in four dimensions, and all matter is
perceived to be of the same chirality. Encounters across
the vacuum interface brings matter into violent contact
with anti-matter, and a stable universe is therefore
only possible if it is in strict equilibrium with itself.
To have a cosmic potential on opposite sides of the
interface in balance would certainly require well-tuned
characteristics of certain relevant physical parameters. It
probably requires a specific value of the fine-structure
constants, and could be at the root of the famous large
number coincidences. This mercifully eliminates
concepts like the anthropic principle and many-world
theories. It neatly places radiation in the vacuum where
it belongs, between matter and anti-matter. As across
any other interface that separates phases in equilibrium,
constant seepage must occur. This provides a plausible
origin of cosmic rays as chance excursions across the
interface, and could also account for the isotropic 3K
microwave background.
It is also necessary to consider the time-reversibility of
quantum theory and the arrow of time in the macroscopic
world. To jump into time it is necessary to cross the vacuum
interface, in either direction. Because of the curvature
of space, any displacement in the three-dimensional
universe is a small step in the same direction, and hence
a positive displacement in time. A time axis always points
directly into the interface and time flows towards the site
directly on the other side. The two-dimensional world
model assists to demonstrate how this argument defines
the arrow of time. Unfortunately, it also shows how time
travel is self-destructive through an encounter with antimatter. The quantum particle makes a small hop into
time, but bounces back with time-reversal and random
perturbation of its space coordinates. In contra-distinction
to classical particles it therefore manages a displacement
in space without an inevitable time advancement. It can
even appear to be in two places at the same time as in
two-slit diffraction experiments. Time-reversibility and
uncertainty principle are implied at the same time. That
is the price for not seeing the other side.
The vacuum, considered as an interface, is empty. It is
no longer required that it accommodates all the Dirac
oceans of negative quantum states. These states are on
the opposite side of the interface and they need not be
filled. As an example, the negative states for electrons
occur in the positron-rich world on the other side of the
vacuum interface. Electrons from this side are prevented
from dropping down into their negative-energy states by
the interfacial surface potential. Photons at the interface
can still be considered as representing electron-positron
pairs as before. The same holds for the rest of the quantum
field entities. The two individuals of a virtual pair are
now actually associated with different, symmetry-related
time regimes, giving substance to the definition of an
anti-particle as a particle moving backwards in time.
The most significant result of the model is perhaps the
way it distinguishes between weak and electromagnetic
interactions. In order to demonstrate the difference it
is noted again that quantum behaviour is a function of
aggregation. A massless photon at the interface interacts
equally with the material and anti-material worlds. It is the
archetype of a non-classical particle: its interaction with
exactly one half of its total environment is of necessity
ignored by any observer, and it appears to propagate like
a harmonic wave, without being a wave. The particulate
nature of the photon is demonstrable in experiments where
forced confrontation with matter so dominates its behaviour
that the influence of the anti-matter is effectively masked.
The photoelectric effect illustrates this well.
Its interaction with the other side is scaled down. It is not
at the interface in the same sense as a photon and coexists
with the positron, its reflection across the interface. It moves
like a wave-packet and interacts more strongly with its own
matter. The vector boson that mediates the weak interaction
also has mass, of about 100 GeV and, unlike the photon,
is therefore not identical with its anti-particle and reflects
across the vacuum interface that contains the element of
PCT symmetry. The weak interaction likewise has reflection
symmetry only across the vacuum interface, manifesting
itself asymmetrically on both sides of the interface. This
requires β-decay to be unsymmetrical.
It remains to explain why the vacuum interface is stable,
yet separates two interactive states of matter. The answer
is that the three-dimensional interface, postulated here,
separates the two layers in time rather than space. This
prevents macroscopic interaction, provided the equilibrium
is maintained. Interaction between the opposite sides is
confined to quantum events which represent penetration of
rarefied particles into the time barrier and into the domain of
influence of the world beyond. In the appendix 2 and 3, we
examine the cosmological implications of this vacuum.
103
CONCLUSION
The implications of the vacuum interface model have
striking parallels with gauge theory. Local gauge
invariance, under an appropriate symmetry group, implies
a transformation that links an internal property like the
phase of a wavefunction or isotropic spin to a gauge field.
Breaking of the gauge symmetry has important physical
consequences. This locks the phases of the wavefunctions
together over macroscopic distances and destroys the
gauge symmetry. Gauge theory is presently being extended
successfully to incorporate the strong interaction. The
quest is to find that symmetry group which contains all
the necessary subgroups to define the gauge symmetry
of each force separately. The full symmetry occurs only
at the grand unification energy, which is so high that all
forces are equivalent. At lower energy the symmetry
breaks spontaneously and the different forces separate
into different symmetry species, as observed.
In the formalism of the present argument a free particle at the
vacuum interface has a gauge connection with the quantum
potential. The symmetry group of this gauge field contains
the unified quantum and classical theories. The contiguous
worlds that meet at the interface, like the atomic lattice at
superconductivity, provide the mechanism to create a Higgs
field which breaks the symmetry and produces the massive
worlds of classical theory, with lower symmetry.
a black hole. At an intermediate level, an increased h
would also produce radiation of constant energy quanta
at lower frequency. This represents an intrinsic red shift
of the type observed for quasar.
APPENDIX 1:A NEW INSIGHT INTO THE
NEGATIVE-MASS AND THE ACCELERATING
CHIRAL UNIVERSE
The discovery of acceleration of the universe expansion
in recent astrophysics research prompts the author to
think that Newton’s gravitation law can be generalized
to accommodate the antimatter: While the force between
matters (antimatters) is attractive, the force between
matter and antimatter is a repulsive one. A paradox
of negative-mass in gravity versus a basic symmetry
( m → − m ) based on quantum mechanics is discussed
in sufficient detail so that the new postulate could be
established quite naturally. Corresponding modification
of the theory of general relativity is also proposed. If
we believe in the symmetry of particle and antiparticle
as well as the antigravity between them, it might be
possible to consider a new scenario of the expansion of
universe which might provide some new insight into the
interpretation of cosmological phenomena including the
accelerating universe observed.
Symmetry of space-time inversion
Rigorous formulation of this theory requires definition of
the full symmetry group, containing also the symmetry
subgroups of the known forces of nature. Progressive
symmetry breaking produces each of the forces in turn
until, at the lowest level of mass separation, gravitational
effects appear for the first time. The model therefore
contains not only the seeds of grand unification, but
also the mechanism for spontaneous separation of matter
and anti-matter into different time domains. The three
stronger forces are defined in quantum-mechanical terms
and gravitation follows classical mechanics.
Quantum effects are proposed to have their origin in
the gradient at the vacuum interface. As the curvature
of space-time is distorted by large masses, so would the
gradient be enhanced, and quantum effects are predicted
to become more pronounced in the vicinity of high-density
material. Likely candidates are neutron stars, pulsars,
quasars and black holes. The effect would manifest itself
through a higher value of h, and hence more pronounced
quantum-mechanical uncertainty. More massive particles
will show quantum behaviour, and in the limit of infinite
gradient, the interface is punctured and uncertainty
becomes total (∆E⋅∆t≥h). This situation corresponds to
104
Starting from RQM, we consider the wavefunction (WF)
of a freely moving (along x axis) particle:
ψ ∼ exp [i( p x − E t ) / ], (1.1)
where p is the momentum and E(>0) the total energy.
But what is the WF ψc of an antiparticle? Before 1956, it
was assumed to be a consequence of the operation of a
so-called charge-conjugate transformation C which can
bring a charged particle ( say an electron with charge –e)
to its antiparticle (say the positron with charge e) [2]:
ψ c = Cψ ∼ ψ ∗ ∼ exp [i(− p x + E t ) / ]. (1.1*)
We see that the negative-energy –E<0 emerges immediately
due to the basic operators in quantum mechanics (QM):
pˆ R = −i
∂ ˆ
∂
, E R = i . ∂x
∂t
(1.2)
The negative-energy difficulty at the level of QM was
remedied to some extent by the so-called “hole theory
for positron” (which is obviously impossible for the
Klein-Gordon particle) and solved formally at the level
of quantum field theory (QFT) by the redefinition of
creation (annihilation) operators.
For example, the Schrodinger equation is nonrelativistic.
But t he fol low i ng couple d Sch ro d i nger l i ke
equation
Since the discovery of parity violation, i.e., the violation of
space-inversion P(x→–x) symmetry in 1956-1957, physicists
realize that not only P but also C transformations are not
conserved in the weak-interaction processes. So Eq. (1.1)
is not applicable in general and the WF of antiparticle
should be redefined as:
ψ c = CPTψ ∼ exp [−i( p x − E t ) / ], ψ → Tψ = ψ ( x , −t ) ∼ exp [i(− p x − E t ) / ]. (1.4)
Note that: First, the name “time-reversal” is actually a
misnomer [3, 4]. What the transformation (1.4) means
is merely a reversal of motion (p→–p). Second, the
correctness of definition of antiparticle WF (1.3) depends
on the validity of the CPT theorem which in turn is
ensured by basic principles of SR and QFT. Third, as
the complex-conjugate operations in C and T cancel
each other, what a combined CPT transformation in (1.3)
means is merely a sign change of coordinates (x, t) in
comparison with Eq. (1) [1, 2]. But the original meaning
of C, P and T implies that Eq. (1.3) should describe an
antiparticle with the same p and E(>0) as that of the
particle described by (1.1). Hence for antiparticles, we
should forget the “hole theory” and use the following
operators instead of (1.2):
∂
∂ (1.5)
, Eˆ L = − i 
.
∂x
∂t
In fact, Eq. (1.5) had been proven to be the direct and
unique outcome of the full solutions to the EPR paradox
and Klein Paradox [4-6].
pˆ L = i
Fourth, once we accept Eqs. (1.3) and (1.5), the CPT
theorem becomes a new fundamental postulate, i.e., a basic
symmetry which can be stated in the following form:
Under the (newly defined) space-time inversion denoted
by PT , meaning merely x→–x, t→–t, the theory of RQM
remains invariant with its concrete solution, e.g., a particle
WF transforming to its antiparticle WF (denoted by C)
automatically. It means that our postulate reads:
PT = C (1.6)
(1.7)
is just the relativistic Klein-Gordon equation
(i
(1.3)
where the so-called “time-reversal transformation” T is
defined as:
∗
i ∂ ϕ = mc 2ϕ − 2 ∇ 2 (ϕ + χ ),
 ∂t
2m

2
2
∂
i ∂t χ = − mc χ + 2m ∇ 2 ( χ + ϕ ),
∂ 2
) ψ = − c 2  2 ∇ 2 ψ + m 2c 4ψ , ∂t
(1.8)
with relation first pointed out by Feshbach and Villars
in 1958:
ϕ = (ψ + i  ψ / mc 2 ) / 2,

2
 χ = (ψ − i  ψ / mc ) / 2.
(1.9)
Now we see that under the space-time inversion (x→–x,
t→–t) and the transformation:
ϕ (− x , −t ) → χ ( x , t ), χ (− x , −t ) → ϕ ( x , t ), (1.10)
the Eq. (1.7) does remain invariant while a particle WF
(1.6) with | ϕ |>| χ | (due to E<0, see (1.8) turning to its
antiparticle WF (1.9)) with | χc |>| ϕ c | (due to, E<0,
Ec = –E>0, see (1.9)).
Symmetry of mass inversion
Alternatively, we can restate the above basic symmetry
in the following way: Under the mass inversion:
m → − m, ϕ ( x , t ) → χ ( x , t ), χ ( x , t ) → ϕ ( x , t ), (1.11)
the theory, e.g., Eq. (1.6), remains invariant. Although
transformation (1.6) is equivalent to transformation
(1.11), they share different advantages. The former is
relevant to unobservable coordinates (x, t) and so is
more essential in RQM and equivalent to even more
abstract symmetry of i versus –i (see Eq. (1.1) versus
(1.3)), while the latter is relevant to observable mass
m and so is easily to be generalized to the case of
classical theory.
Here, we’d better use the following working rule: to deal
with particle (matter) and antiparticle (antimatter) on an
equal footing, a classical theory must be invariant under
the mass-inversion transformation m→–m. Note that:
First, m is always positive. Second, being the external
field, the electric-(magnetic) field strength E(B) undergoes
105
no change in the transformation. Third, in RQM like
(1.7) or (1.8), the motion equation for antiparticle is
the same one as that for particle. This is because each
particle state like (1.1) contains its hidden antiparticle
field under the condition |ϕ|>|χ| whereas an antiparticle
state like (1.3) contains its hidden particle field under
the condition |χc|>|ϕc|. Fourth, to clarify further why we
prefer the new postulate (1.6) instead of C transformation
and CPT theorem, we wish to emphasize an important
difference between a postulate (law) and a theorem.
All quantities in a theorem must be defined in advance
separately and unambiguously and the outcome of the
theorem is actually contained in its premise implicitly.
For example, the definitions of C, P and T are all clear in
mathematics and the validity of CPT theorem is ensured
by the basic principles of SR and QFT. Once C, P and
T are not conserved in experiments, they cease to be
meaningful as physical transformations. In this situation,
the CPT theorem immediately exhibits itself as a new
postulate (1.12) in which the definition of transformation
of particle to antiparticle is just contained in the same
equation. In general, a postulate or law can often (not
always) accommodate a definition of physical quantity,
and the validity of the postulate (law) together with the
definition must be verified by experiments. Hence the
establishment of a law is a process “from particular to
general” or an outcome of “analysis and induction method”.
By contrast, to prove a theorem from well-established
theories is a process “from general to particular” or a
consequence of “deduction method”.
As is well known, the EP served as a starting point in
establishing the theory of general relativity (GR). The
possible invalidity of EP in the presence of antimatter
implies that GR is dealing with the gravitation of pure
matter without the coexistence of antimatter. Indeed, let
us look at the Einstein field equation [9]:
For example, the definition of gravitational mass m is
contained in the gravitational law. The definition of inertial
mass m is contained in Newton’s law. What we have done
is a similar thing - the definition of particle-antiparticle
transformation C is contained naturally in a new postulate
(1.6) - not one that comes from elsewhere.
(where the superscript c means antimatter,) since under
c
c
→ −Tµν .
the mass inversion, Tµν → −Tµν
and Tµν
Notice that the form of the energy-momentum tensor
is the same for both matter and antimatter. We stress
once again that the distinction between m and –m is
merely relative, not absolute. So in the whole universe
eff
c
(matter+antimatter) we have Tµν
= Tµν − Tµν
≡ 0 and
the Einstein equation is
Generalization of Einstein field equation in general
relativity
Consider a positronium and an atom of matter. If Newton
Equation is correct, there will be no gravitational force
between them. This means that the gravitational mass m
(grav.) of positronium is zero! However, the energy or the
relevant inertial mass m (inert.) of positronium is definitely
nonzero. Hence we see that in the case of coexistence of
particles and antiparticles, the equivalence principle (EP)
in the (weak) sense that [8]
m (grav.) = m (inert.) 106
(1.13)
On the left side, the Ricci tensor Rµv, curvature scalar R
and the metric tensor gµv are all functions of coordinates
xµ. While on the right side, the energy-momentum tensor
Tµv is introduced to describe the existence of matter in
the vicinity (a macroscopic small volume) of xµ. Then
under a transformation of mass inversion m→–m to
reflect that of matter to antimatter, Tµv should change its
sign due to its proportionality to the mass m. Hence Eq.
(1.13) changes sign on the right side whereas not on the
left side. This reflects the fact that GR is a classical field
theory and so cannot treat the matter and antimatter on
an equal footing.
To keep Eq. (1.13) invariant under the mass inversion, we
manage to modify its right side by a generalization as:
eff
c
Tµν → Tµν
= Tµν − Tµν
,
Rµν −
1
g R≡ 0
2 µν
(1.14)
(1.15)
APPENDIX 2: THE COSMOLOGICAL CONSTANT
PROBLEM
Let’s outline shortly the cosmological constant
problem.
Consider Einstein equation with ∆ -term (  = c = 1 ):
(1.12)
cannot be valid.
1
Rµν − gµν R = −8π GTµν 2
1
Rµν − gµν R = 8π GTµν + Λgµν 2
(2.1)
Here Tµv is energy-momentum tensor of the matter and Λ
is some constant parameter having the dimension [cm-2].
In the used unit system the Newtonian gravitational
constant
G  lP2  2, 5 ⋅ 10 −66 cm 2 (2.2)
and according to experimental data the mean energy
density today is of the order
Tµν  ρ1  108 cm −4 → 8π GTµν  5 ⋅ 10 −57 cm −2 (2.3)
In the vacuum interface (see figure 2), the Einstein
equation is described by
em
Rµν = 8π GTµν
+ Λ P gµν (2.8)
It is clear that the interaction of fields doesn’t changes
qualitatively the estimation (2.7). From (2.7) and (2.3) we
see that the contribution to the right hand side of Eq. (2.1)
estimated in the framework of canonical quantum field
theory is larger about 10120 times in comparison with the
experimental estimations.
and
Λ  10 −56 cm −2 (2.4)
Thus, if Einstein equation (1) is used for description of
the today dynamics of our R-Universe, the quantities
in its right hand side are of the same order indicated in
(2.3) and (2.4). In our model, for the L-Universe, we have
Tµv < 0, Λ < 0.
Now let us estimate the possible value of the right hand
side of Eq. (2.1) in the framework of canonical quantum
field theory. For simplicity consider energy-momentum
tensor in quantum electrodynamics in flat spacetime:
1
1
i
Tµν = − ( Fµλ Fνλ − ηµν F 2 ) + (ψγ ( µ ∇ν ) − ∇(νψγ µ )ψ ) (2.5)
4
4
2
Casimir effect, predicted in [1] and experimentally
verified in [2], shows for reality of zero-point energies.
Moreover, the attempts to drop out zero-point energies by
appropriate normal ordering of creating and annihilating
operators in energy-momentum tensor fail for many of
reasons (the discussion of this problem see, for example,
in [3]). Thus, at estimating vacuum expectation value of
energy-momentum tensor (2.5), it should not be performed
normal ordering of creating and annihilating operators
in (2.5). Thus we obtain for vacuum expectation value of
tensor (5) in free theory:
Tµν
0
=

d 3 k  kµ kν
3 
k0

∫ (2π )
−2
k0 = k
kµ kν
k
o
k 0 = m2 + k 2




(2.6)
Here m is the electron mass. The first item in (2.6) gives
the positive contribution but the second item gives the
negative contribution since these items give the boson and
fermion contributions to vacuum energy, respectively. If
integration in (2.6) is restricted by Planck scale, kmax ∼ lP−1 ,
then from (2.6) and (2.2) it follows:
8π G Tµν
0
 lP−2  1066 cm −2  Λ P (2.7)
APPENDIX 3: COSMOLOGICAL IMPLICATIONS
It is instructive to examine some of the cosmological
implications of the present proposal. When light traverses
intergalactic space it displays a Doppler frequency shift,
invariably interpreted in terms of receding sources, and
therefore assumed to imply an expanding universe. This
interpretation is not inconsistent with the present model,
but neither is it a necessary consequence. Curvature of
three-dimensional space along a time coordinate implies
that a distant source is separated from an observer in both
space and time. During transit, the photon moves towards
an observer ahead of it in time and therefore appears to lag
as if its source was receding. The observed red shift, as
before, is a function of separation, but the proportionality
constant does not necessarily relate distance to rate of
recession, but rather to a time separation interval, ∆t,
)
∆t, zc = rH = r 1 t − 1 ( t + ∆t  , where t=r/c. Hence:
)
z = 1 − t ( t + ∆t .
This formulation allows the calculation of a Hubble
radius rather than a Hubble age of the universe. As
∆t → ∞ , H → 1 t 0 = c r0 , where the maximum value
of the interval r0 = ct0 is interpreted as an upper bound
to some radius of the universe. This effective radius
corresponds to 4x109 parsec. Converted to a dimensionless
distance, one has
N1 = r0 m e c 2 e 2 = t 0 m e c3 e 2 ≈ 10 40 .
Using this value with the dimensionless mass, or number
of baryons, of the universe, N3 = 1080, a mass density
of ρ = N 3 N13 = 10 −40 is calculated. This matches the
third large number
N 2 = e 2 / (4πε 0 Gm p me ) = 2.3 × 1039 ,
107
the gravitational coupling constant, as required by Mach’s
principle, without invoking anthropic principles. There is
no reason why any of these large numbers should change
with time.
The foregoing does not demand a static universe. As
time flows, observers move along the interface and
therefore experience a gradually changing curvature.
Time evolution amounts to the threading of the interface
through the universe, and the consequent changes in
configuration are like pseudo-rotation, which increases
the isotropic appearance of the universe. The resulting
relative motion would probably be not unlike that of an
expanding universe.
In-so-far as an abstract surface like P2 does not exist in
less than four dimensions, the vacuum interface must
likewise be four-dimensional. Its three-dimensional aspect
differentiates between the chiral forms of matter, and by
analogy, the four-dimensional vacuum could conceivably
differentiate between opposite directions in time. A closed
journey along the interface would gradually turn positive
into negative time flow as was shown for the chirality
of matter, suggesting that time has no unique beginning.
The same conclusion is reached by modern quantum
cosmologies. Hawking’s idea that the universe is finite
but has no boundary in imaginary time, may indeed be
fully consistent with the four-dimensional chiral structure
arrived at here.
The model of the universe with two chiral spaces
The model of the homogenous and isotropic universe
with two spaces is considered. The background space is a
coordinate system of reference and defines the behaviour
of the universe. The other space characterizes the gravity
of the matter of the universe produced by electromagnetic
waves. In the presented model, the first derivative of the
scale factor of the universe with respect to time is equal
to the velocity of light. The density of the matter of the
universe changes from the Plankian value at the Planck
time to the modern value at the modern time. The model
under consideration describes the whole universe from the
Planck time to the modern time and avoids the problems
of the Friedmann model such as the flatness problem and
the horizon problem.
As known [14, 15], the Friedmann model of the universe
has fundamental difficulties such as the flatness
problem and the horizon problem. These appear to
be a consequence of that the space of the Friedmann
universe, on the one hand, is defined by the gravity
of the matter of the universe, and on the other hand,
108
is a coordinate system of reference. The solution of
the problem is to introduce the background space as
a coordinate system of reference. In this case, the
background space defines the behaviour of the universe,
and the other space characterizes the gravity of the
matter of the universe.
Let us consider the model of the homogenous and isotropic
universe with two chiral spaces. Let us introduce the
background space as a coordinate system of reference.
Then the evolution of the universe is described as a
deformation of the background space. Let us take the
homogenous and isotropic background space, with the
spatial interval of its metric is given by
ds 2 =
a 2 dl 2
2
1 + kl 2  4 


(3.1)
Suppose that the background space is defined by the total
mass of the universe including the mass of the matter
and the energy of gravity produced by electromagnetic
waves
Gik = Tiktot = Tikem = Tik + tik . (3.2)
Let us consider the case when the total mass of the universe
is equal to zero. It means that the Maxwell tensor vanishes,
Tikem = 0, but with det F k ≠ 0 .
i
Tiktot = Tik + tik = 0 (3.3)
Then eq. (3.2) take the form
Gik = 0. (3.4)
The solution of the equations (3.4) gives
d 2a
=0
(3.5)
da
= c. dt
(3.6)
dt 2
Thus the second derivative of the scale factor of the
universe with respect to time is equal to zero, and the
first derivative of the scale factor is equal to the velocity
of light. It should be noted that the scale factor of the
universe coincides with the size of the horizon
a = ct. (3.7)
In the model (3.4), the laboratory coordinate system is
synchronous. In the laboratory coordinate system, the
background space is described by the flat metric
ds 2 = c 2 dt 2 − a 2 dl 2 . (3.8)
Thus we arrive at the Milne model [16] in which the size
of the universe being the maximum distance between the
particles coincides with the scale factor of the universe and
coincides with the size of the horizon. In the universe with
one space, the Milne model is derived from the condition
that the density of the matter tends to zero ρ → 0. Here
the Milne model describes the background space of the
universe, with the total mass of the universe being equal
to zero mtot = 0.
Let us determine the relationship between the lifetime of
the universe and the Hubble constant. Since the Hubble
constant is
H=
1 da
,
a dt
(3.9)
so from (3.6), (3.7), (3.9) one can obtain
t=
1
.
H
(3.10)
Allowing for (3.7) and (3.10), from (3.11) it follows that
the mass of the matter changes with time as
m=
c 2 a c 3t
c3
=
=
, G
G GH
(3.12)
and the density of the matter, as
ρ=
3c 2
3
3H 2
=
=
2
2
4 πG
4 πGa
4 πGt
(3.13)
According to (3.12), growth of the mass of the matter
takes place during all the evolution of the universe. At
the Planck time tp1, the mass of the matter is equal to
1/ 2
the Planck mass mPl = (c / G ) . At present, the mass
56
of the matter is m0 ≈ 1.4 ⋅ 10 g , and the density of the
-29
-3
matter is ρ0 ≈ 3.2 ⋅ 10 g cm . Thus the model of the
universe (3.3)-(3.6) provides growth of the mass of the
matter from the Plankian value to the modern one.
We have considered the model of the homogenous and
isotropic universe with two spaces, with the behaviour of
the universe is defined by the background space. Unlike
the Friedmann model, the presented model gets rid off
the flatness and horizon problems.
Remind [14, 15] that the horizon problem in the Friedmann
universe is that two particles situated within the horizon
at present were situated beyond the horizon in the past.
In the universe under consideration, all the particles
are situated within the horizon during all the evolution
of the universe, since the size of the universe being the
maximum distance between the particles coincides with
the size of the horizon. Hence the presented model avoids
the horizon problem.
Let us estimate the size of the universe at the Planck
time and at present. Remind that the size of the universe
coincides with the scale factor of the universe. According
5 1/ 2
to (3.7), at the Planck time t Pl = (G / c ) , the scale
factor of the universe is equal to the Planck length
TPl = lPl = (G / c3 )1/ 2 . According to (3.7), (3.9), for the
modern Hubble constant H 0 ≈ 3 ⋅ 10 −18 c-1 , the modern
scale factor of the universe is a0 ≈ 10 28 cm .
Remind [14, 15] that the essence of the flatness problem in
the Friedmann universe is impossibility to get the modern
density of the matter starting from the Planck density of
the matter at the Planck time. In the presented theory, the
density of the matter of the universe changes from the
Planckian value at the Planck time to the modern value
at the modern time. Hence the flatness problem is absent
in the presented theory.
Let us determine the relationship between the mass of the
matter and the scale factor of the universe at t = const.
The total mass of the universe is equal to zero, given the
mass of the matter is equal to the energy of its gravity
In order to resolve the above problems of the Friedmann
universe an inflationary episode is introduced in the
early universe [14, 15]. Since the presented model
describes the universe from the Planck time to the
modern time and avoids the above problems of the
Friedmann universe, there is no necessity to introduce
the inflationary model.
m=
Gm
c2 a
.
(3.11)
109
REFERENCES
[9]
N.A. Bahcall, J.P. Ostriker, S. Perimutter and P.J.
Steinhardt. “The cosmic triangle: revealing the state
of the universe”. Science. Vol. 284, pp. 1481-1488.
May 28 1999.
[10]
W. Freedman. “The Hubble constant and the
expanding universe”. American Scientist, Vol. 91,
pp. 36-43. Jan-Fe, 2003.
[1]
H. Bondi. “Negative mass in general relativity”.
Rev. Mod. Phys. Vol. 29, pp. 423-428. 1957.
[2]
J.D. Bjorken and S.D. Drell. “Relativistic Quantum
Mechanics”. McGraw-Hill Book Company. 1964.
[3]
J.J. Sakurai. “Modern Quantum Mechanics”. John
Wiley & Sons Inc. 1994.
[11]
M. Livio. “Moving right along”. Astronomy.
pp. 34-39. July 2002.
[4]
G.J. Ni and S.Q. Chen. “Advanced Quantum
Mechanics”. Chinese Ed. Press of Fudan University.
2000. English Ed. Rinton Press. 2002.
[12]
P. Davies. “Seven wonders”. New Scientist.
September 21 2002.
[5]
G.J. Ni, H. Guan, W.M. Zhou and J. Yan.
“Antiparticle in the light of Einstein-PodolskyRosen paradox and Klien paradox”. Chin. Phys.
Lett. Vol. 17, pp. 393-395. 2000.
[13]
gravitación”. Ingeniare. Rev. chil. ing. Vol. 16 1,
pp. 6-23. 2008.
[6]
G.J. Ni. “Ten arguments for the essence of special
relativity”. Proceedings of the 23rd Workshop on
High-energy Physics and Field Theory, pp. 275-292.
Edit: I.V. Filimonova and V.A. Petrov. Protvino,
Russia. June 2000.
[14]
A.D. Dolgov, Ya.B. Zeldovich and M.V. Sazhin.
“Cosmology of the early universe”. Moscow Univ.
Press. Moscow. 1988.
[15]
A.D. Linde. “Elementary particle physics and
inflationary cosmology”. Nauka. Moscow. 1990.
[7]
L. Smolin. “Three Roads to Quantum Gravity”.
Basic Books, p. 149. 2001.
[16]
Ya. B. Zeldovich and I.D. Novikov. “Structure and
evolution of the universe”. Nauka. Moscow. 1975.
[8]
S. Weinberg. “Gravitation and cosmology”. John
Wiley. 1972.
[17]
L. Landau and E.M. Lifshitz. “The classical theory
of fields”. 4th Ed. Pergamon. Oxford. 1976.
110
H. Torres-Silva: A new relativistic field theory of the electron
A NEW RELATIVISTIC FIELD THEORY OF THE ELECTRON
UNA NUEVA TEORÍA RELATIVÍSTICA DE CAMPO PARA EL ELECTRÓN
H. Torres-Silva1
RESUMEN
En este trabajo se presenta un examen cualitativo sobre una nueva Teoría General Relativística para el electrón, con la
obtención de la ecuación de Dirac a partir de los campos electromagnéticos con el campo eléctrico paralelo al campo
magnético. El principio rector es el de la relatividad general, y la principal hipótesis es que de las ecuaciones fundamentales
se desprende la teoría de Dirac y la teoría de Maxwell - Lorentz como de dos casos especiales cuidando la coherencia
y compatibilidad entre las condiciones en las que las ecuaciones fundamentales se reducen a la ecuación de Dirac y las
ecuaciones de Maxwell - Lorentz. Se espera que la presente investigación arroje alguna luz sobre las desconcertantes
dificultades a las que nos encontramos en la comprensión del comportamiento de un electrón exclusivamente en función
de la ecuación de Dirac y las ecuaciones de Maxwell - Lorentz. Más allá de esto, se puede investigar la posibilidad de
que otras partículas elementales se puedan regir por las mismas ecuaciones fundamentales bajo variadas condiciones
restrictivas.
Palabras clave: Ecuación de Dirac, tensor de materia, sistema Einstein-Maxwell.
ABSTRACT
In this paper we present a qualitative discussion of a new General Relativistic Field Theory for the electron, obtaining
the Dirac equation from electromagnetic fields with the electric field parallel to the magnetic field. The guiding principle
is that of general relativity, and the main hypothesis is that the fundamental equations embrace the Dirac theory and the
Maxwell-Lorentz theory as of two special cases respectively. We concern ourselves with the consistency and compatibility
among those conditions under which the fundamental equations are reduced to the Dirac equation and the MaxwellLorentz equations. We expect that the present investigation will shed some light on those perplexing difficulties which we
encounter in comprehending the behavior of an electron solely according to the Dirac equation and the Maxwell-Lorentz
equations. Beyond this, we aim to investigate the possibility that other elementary particles are governed by the same
fundamental equations under varied restrictive conditions.
Keywords: Dirac equation, matter tensor, Einstein-Maxwell system.
INTRODUCTION
Einstein’s Dream
Albert Einstein spent several years of his life trying to
develop a theory which would relate electromagnetism
and gravity to a common “unified field”. Hence the name
unified field theory.
1
After Einstein finished his first article on the unified field
theory in 1922, despite criticism he spent much of the
second half of his life pursuing the development of the
unified field theory besides the discussion of completeness
of quantum mechanics. In the first several years, he was
very optimistic, thought success would come soon, but he
found it was full of difficulties afterwards. He considered
mathematical tools in being was not sufficient, then turned
111
to study mathematics, but never obtained any result with
real physical sense. Because Einstein wanted to found an
encompassing mathematical construct that would unite
not only gravitational field but also electromagnetic field
under a single set of equations, but then the task has
become even more difficult, with the discovery of two
other basic field: the weak interaction field and strong
interaction field. Most physicists thought Einstein’s quest
was hopeless, and in fact he never succeeded. But Einstein
was convinced such a basic harmony and simplicity
existed in nature, he kept his chin up, went ahead along
his own road. Because he was apart from the mainstream
of physical research - quantum field theory, he was very
alone in his old age, but he was fearless. He still prepared
to keep on his mathematical calculation of unified field
theory on his sickbed until the day before his death. He
said with a sigh before his death: I cannot finish this
work, it will be forgotten, but it will be rediscovered
in the future. Einstein did manage to develop a theory
which “wrapped” electromagnetism and gravitation into
a common metric tensor. In one of his formulations of a
unified field theory (called Einstein-Schrodinger Theory),
gravitation was wrapped into the symmetric part of the
metric tensor, while electromagnetism was wrapped into
the antisymmetric part of the metric tensor. This wrapping
is possible because electromagnetism and gravity share
some mathematical similarities. They both have a stressenergy tensor. The electric charge is analogous to the
gravitational mass. The magnetic moment is analogous to
the angular momentum moment. The electric potential and
electric field are analogous to the gravitational potential
and gravitational field, respectively. Finally, the magnetic
field is analogous to the magneto-gravitic field.
The mathematical wrapper which Einstein developed
exploits this analogy. However, the analogy between
electromagnetism and gravity breaks down at higher
field strengths when nonlinear field effects set in. As a
result, Einstein-Schrödinger theory correctly describes
electromagnetism and gravity at low field strengths where
they are not coupled to each other. However, it does not
describe the interactions between electromagnetism and
gravitation which occur at higher field strengths. Thus,
Einstein-Schrödinger theory achieved an approximate
mathematical unification, but no real physical unification
of electromagnetism and gravity. In this sense, it did not
really achieve its objective.
Kaluza and Klein developed an alternative wrapper for
electromagnetism and gravitation. Instead of wrapping
electromagnetism into the antisymmetric part of the
metric tensor, they retained a symmetric metric tensor
but added a fifth dimension. They were able to show that
112
Maxwell’s Laws and General Relativity can be expressed
in terms of their five-dimensional metric tensor. Again,
this exploits the analogies between electromagnetism
and gravity.
The problem with Einstein’s unified field theory and
Kaluza-Klein’s unified field theory is that they don’t address
the fundamental issue. They still treat gravitation and
electromagnetism as two completely separate interactions.
Neither theory can tell you how a gravitational field is
fundamentally produced by a charged particle (electron).
Today, the search for a unified field theory has been
replaced by loftier goals. Physicists are now looking for a
so-called Theory of Everything (TOE) which will unify
not only electromagnetism and gravity, but also the nuclear
interactions and other potential physical forces such as
inflation and “dark energy”. At the time of Einstein, modern
particle physics had not yet been developed and the strong
and weak nuclear interactions were not well understood.
Within of the unified program a fundamental question
was if gravitational fields did play an essential part in the
structure of the elementary particles of matter (electron).
The first unimodular theory was developed by Einstein in
1919, assuming as source the Maxwell tensor, where the
quantum electron theory was not reproduced. Thus, as stated
by Einstein in 1919, “we cannot arrive at a theory of the
electron [and matter generally] by restricting ourselves to
the electromagnetic components of the Maxwell-Lorentz
theory, as has long been known” [6].
Motivation
In the beginning of this century, Lorentz, Poincaré,
Abraham, Mie and others attempted to show that the
constitution of an electron be explained as a field of
electromagnetic nature. In order to make the motion of
the electron special-relativistic, however, it was necessary
to consider a mechanical core (Poincaré) or to introduce
some nonlinearity (Mie) in the electromagnetic field under
consideration [1-3]. To overcome these difficulties appeared
to have completely been resolved with the Dirac equation
for the electron discovered in 1928. It has conventionally
been believed that the information of an electron near its
core is fully provided by the Dirac equation. The notion
of the electron formed by the conventional interpretation
of the Dirac equation is hardly acceptable as rational and
feasible. Instead, in an accompanying paper, it was shown
that the Dirac equation has a solution that indicates that an
electron is a field localized in space and deformable and
that the motion of this “elementary field” is determinate
and causal. Moreover, is shown, it is possible to regard the
field governed by the Dirac equation as if electromagnetic.
This possibility is not surprising if we note that the
intrinsic magnetic moment of an electron is a notion to be
comprehended only in the context of Faraday-MaxwellLorentz’s theory of electricity and magnetism. Thus we
are led to infer the following: An electron is a localized
field of which some part remote from its center may well
be regarded as normal electromagnetic, and some other
part near its center is governed by the Dirac equation
derived from parallel fields. The connection between
the two parts must be continuous and gradual, and there
is no clear-cut border between them. A real electron, as
a whole, must be a unified field governed by a common
set of partial differential equations. It is important to
anticipate the possibility that those fundamental equations
governing the field be reduced to the Maxwell-Lorentz
equations under a restrictive condition and to the Dirac
equation under another restrictive condition. Although
with the early theories of Einstein and others [6-7], there
is no deductive way of giving the fundamental equations,
it is not difficult to anticipate the following:
a. The electronic mass has its representation in the Dirac
equation, but not in the Maxwell-Lorentz equations.
On the other hand, the electronic charge is seen in
the Maxwell-Lorentz equations, but not in the Dirac
equation for a free electron. We infer from these
observations that the electronic mass and charge
are approximate substitutes of field variables that
are functions of time and space in the fundamental
equations. Only because the variables are comparatively
less variants, they may be replaced with constants as
depending on conditions of observation.
b. The Maxwell-Lorentz equations are covariant under
the Lorentz transformation, however, the covariance
of the Dirac equation under the same transformation
is conditional. Besides, the field variables in MaxwellLorentz equations and those in the Dirac equation
are apparently of different characteristics under the
Lorentz transformation. In order to embrace those two
sets of equations as of special cases, the fundamental
equations must be formed in a geometrical frame less
restrictive than the Euclidean, i.e., as covariant for
observers in varied conditions [8-9].
These difficulties are overcomes with our Maxwell’s
Equations with parallel electromagnetic fields, (see
accompanying paper). Considering those observations in
the above, it is significant to recall the demonstration of the
similarity between the Dirac field and the electromagnetic
field. In the demonstration, we see a clue to electing a set
of fundamental equations that are reducible to both the
Dirac equation and the Maxwell-Lorentz equations. In
paper (Physical interpretation of the Dirac equation with
electromagnetic mass), we considered the Dirac equation
for a free electron derived from Maxwell’s equations when
the electric field is parallel to the magnetic field.
The Nature of the Investigation
If one accepts as valid the principle of relativity, i.e.,
the principle of covariance of the laws under coordinate
transformations, the choice of a proper scheme of
geometry is an essential part of the task of constructing
the fundamental equations concerned. In this respect,
it is significant to recall that the Dirac equation is not
completely covariant under the Lorentz transformation.
It appears that the range of the meaning implied by the
Dirac equation can no longer be confined in the Euclid
space. This situation suggests first that the scheme of
geometry be properly generalized and then that the Dirac
equation be modified accordingly. We expect, in this way,
that the fundamental equations thus found will be able
to embrace the Dirac equation and the Maxwell-Lorentz
equations as of two special cases respectively.
In a geometrical scheme more general than the Euclidean,
each component of the metric tensor gij is a function of
space-time coordinates. Therefore, it seems to be sensible
to expect that any matter field, with no exception, is
accompanied by a gravity field. The fundamental equations
govern simultaneously the matter field and the metric
1
field is the equation Rij − gij R = − kTij proposed earlier
2
by Einstein [10-11] . The left hand side of the equation
is sometimes called the Einstein tensor Gij. One might
surmise that a matter field determines uniquely the Einstein
tensor of the space where the matter field is located. Thus
it appears that the Einstein tensor can be the representation
or the image of the matter field. But the uniqueness of
the relation between a matter field and the resulting
Einstein tensor is unknown. We note that the equations
to be found must be regarded only as of an approximate
means of representing the reality concerned. (None of
the equations utilized in physics may escape this fate.)
Therefore, even when a field, e.g., of an electron, governed
by those equations has a singularity that implies a strong
distortion of space curvature, one can not immediately
conclude that the real field, expected to be represented
by the solution, has the same singularity.
The field equations of general relativity are rarely used
without simplifying assumptions. The most common
application treats of a mass, sufficiently distant from
other masses, so as to move uniformly in a straight line.
All applications of special relativity are of this type,
in order to stay in Minkowski space-time. A body that
113
moves inertially (or at rest) is thus assumed to have fourdimensionally straight world lines from which they deviate
only under acceleration or rotation. The well-known
Minkowski diagram of special relativity is a graphical
representation of this assumption and therefore refers to
a highly idealized situation, only realized in isolated free
fall or improbable regions of deep intergalactic space.
In the real world the stress tensor never vanishes and so requires
a non- vanishing curvature tensor under all circumstances.
Alternatively, the concept of mass is strictly undefined in
Minkowski space-time. Any mass point in Minkowski space
disperses spontaneously, which means that it has a space-like
rather than a time-like world line. In perfect analogy a mass
point can be viewed as a local distortion of space-time. In
euclidean space it can be smoothed away without leaving
any trace, but not on a curved manifold. Mass generation
therefore resembles distortion of a euclidean cover when
spread across a non-euclidean surface. A given degree of
curvature then corresponds to creation of a constant quantity
of matter, or a constant measure of misfit between cover and
surface, that cannot be smoothed away.
Here, a strain field appears in the curved surface. At any
point on the curved manifold the gradient of the strain field
is perpendicular to the tangent vector and coincides with
the axis of the local light cone. To relieve the stress, the
natural response of the mass point is displacement along
the stress gradient and hence it traces out a time-like world
line at constant spatial coordinates. This displacement,
along the time coordinate only, is the arrow of time,
which appears as a direct consequence of the curvature
of space. There is no time in euclidean space.
The primary cause of mass generation by space curvature
is elimination of the strict orthogonality between time
and space coordinates which allows the strain field (mass
point) to acquire complementary time-like and space-like
attributes. This is the mechanism envisaged by Corben
[4] as a model for creating mass through relativistically
invariant self-trapping of a free bradyon and a free tachyon,
(time-like and space-like waves).
The essence of the argument advanced here is that real
world-space is not euclidean and that space is generally
curved into the time dimension, consistent with the
theory of general relativity. The curvature may not be
sufficient to become obvious in a local context. However,
it is sufficient to break the time-reversal symmetry that
seems to characterize the laws of physics. Not only does
it cause perpetual time with respect to all mass, but
actually identifies a fixed direction for this It creates an
arrow of time and thereby eliminates an inconsistency
114
in the logic of physics: how reversible microscopic laws
can underpin an irreversible macroscopic world. General
curvature of space breaks the time-reversal symmetry
and produces chiral space, manifest in the right-hand
force rule of electromagnetism. The fact that most other
fundamental laws of physics do not refer the chirality of
space, nor the arrow of time, confirms that the curvature
on a local scale is barely detectable.
The one exception to apparent time-reversal symmetry
is the law of entropy.
It has been stated [1] that “...the second law of thermodynamics
is excluded from the classication fundamental due to its
statistical nature”. This is an unconvincing explanation and
the curved-space argument provides a better mechanism
for entropy production. In any curved-space manifold
gradient vectors drive time-like displacement of separate
particles along non-parallel world lines. Even among pairs
of stationary particles three-dimensional line elements
therefore do not remain invariant over a period of time. An
initially stationary array of non-interacting particles (ideal
gas) spontaneously generates relative internal (zero point)
motion leading to chaotic distribution in a container, or
spontaneous dispersal in the open. Where local interactions
constrain dispersal, zero-point vibration develops. This
intrinsic microscopic instability, caused by the curvature
of space, is the source of entropy.
The conclusions reached here are clearly related to those of
Prigogine [5] who deduced that the irreversible creation of
matter generates cosmological entropy and that the arrow
of time is provided by the transformation of gravitational
energy into matter. The difference is that Progonine’s
result was obtained by incorporating the second law
of thermodynamics into the relativistic field equations,
whereas the present model makes no assumption about
macroscopic behaviour.
Theses observations, usual in classical mechanics, are
significant in evaluating Einstein’s attempt recollected
in the following.
Recollection of Einstein’s Attempt
It seems that Einstein devoted the last twenty years, at least,
of his life to the attempt of materializing his deterministic
view of particles. Einstein explicitly used the term “unified
field theory” (gravitation-electromagnetism) in the title of
a publication for the first time in 1925. Ten more papers
appeared in which the term is used in the title, but Einstein
had dealt with the topic already in half a dozen publications
before 1925 . In total he wrote more than forty technical
papers on the subject. This work represents roughly a
fourth of his overall oeuvre of original research articles,
and about half of his scientific production published
after 1920. As is well known, however, the endeavor
was not fruitful. In retrospect, the cause of his difficulty
appears to be in his interpretation of Schrödinger’s wave
equation. A clue to knowing Einstein’s interpretation in
question is found in an essay published by him in 1936
(Einstein, 1936). His interpretation of wave mechanics
may be summarized as follows:
i) The wave function does not in any way describe the
condition of a single system; it relates rather to many
systems, an ensemble of systems, in the same sense
as of statistical mechanics, so Schrödinger’s equation
determines the time variation that is experienced by
an ensemble of systems.
ii) Quantum mechanics will not be the point of
departure in the search of the foundation of quantummechanical phenomena, just as one cannot go from
thermodynamics to the foundation of mechanics, so
there must be a field theory that results in a way of
representing particles and the representation must be
free of singularities. The foundation of the theory is
given by the differential equations of the field, and
the theory leads also to quantum mechanics in the
same way as classical mechanics of particles leads
to thermodynamics.
Einstein emphasized often that the field in question must
be free of singularities. His reasoning seems to be based
on the following two observations: Conventional wave
functions in quantum mechanics are free of singularities.
On the other hand, in his general theory of relativity
completed in 1916, the differential equations of the metric
space completely replace the Newton theory of the motion
of celestial bodies, if the masses are substituted with
singularities of the field; those equations contain the law
of force as well as the law of motion while eliminating
inertial systems. His theory with Tij = TijMaxwell , however,
does not explain quantum-mechanical phenomena, and is
not satisfactory (unimodular theory). Considering these
two facts, Einstein had a conjecture that a satisfactory
theory be obtained by modifying the general theory of
relativity so that the singularities do not arise in a field
determined by the differential equations of the metric space.
He assumed that the desirable modification be made by
eliminating the symmetry condition of the metric tensor
from the general theory of relativity completed in 1916.
According to Einstein, equations of such complexity as
expected can be found only through the discovery of a
logically simple mathematical scheme that determines
the equations of physics completely or almost completely.
Once one has a proper mathematical scheme, one requires
only little knowledge of physical facts for setting up a
proper theory.
In 1948, near the end of his life, Einstein thought that he had
success in formulating a satisfactory scheme of geometry
in which the metric tensor is no longer symmetric. He
hoped that this geometry could provide the framework
in which the new theory of physics be established.
Unfortunately, however, the result was disappointing;
a stationary field free from singularities could never
represent a mass different from zero. We thus recognize
that Einstein’s view of conventional quantum mechanics
is partially right, and a causal and determinative law is
underlying conventional quantum-mechanical phenomena
of the electron. Considering this, it appears to be a serious
misjudgment of Einstein to attribute immediately the
cause of singularities to the symmetry condition of the
metric tensor in the Riemann geometry.
Now, we can say that the general solution of a partial
differential equation contains a set of functions whose
forms are not determined by the equation but by initial
and boundary conditions. A physically significant solution
is a particular solution that satisfies proper initial and
boundary conditions. It is a significant event in the history
of physics that Einstein had persistently failed to recognize
the significance of initial and boundary conditions in
interpreting physical laws. We see the same failure in
Dirac’s interpretation of the Dirac equation for the electron
if we not considerer that the Dirac equation is derived
from chiral electromagnetic fields with E || B.
FUNDAMENTAL EQUATIONS
In the following investigation, the variables are in general
defined as tensors in a four-dimensional Riemannian space.
The mathematical treatment of them follows the ordinary
rule of tensor calculus. For the convenience of reference,
the mathematical symbols employed are mostly similar to
those in (Møller, [9-11]), unless otherwise specified.
Those equations are mutually coupled, and the strong
tendency of the electron to be a localized and stable field
must be effected by the characteristics of those equations
and proper boundary and initial conditions.
For formulating the fundamental equations, it is customary
to rely on Hamilton’s principle of variation of deriving
covariant equations from a Lagrangian function. But the
choice of the Lagrangian function is arbitrary, and so is of
variation methods. There is no assurance of uniqueness of
115
the result. As Eddington remarked earlier [12], the physical
significance of the method is unknown and doubtful,
particularly when we have no means of evaluating those
resulting equations immediately and directly in comparison
with empirical information. Our experience in this field of
physics is yet naive; instead of taking any axiomatic approach,
it seems to be desirable to continue an effort of reflecting
on the physical reality via equations known thus far. The
guiding principle is that of general relativity, and the main
hypothesis is that the fundamental equations embrace the
Dirac equation and the Maxwell-Lorentz equations as of
two special cases respectively. Although we do not intend
to compare solutions of the fundamental equations directly
with empirical information, we concern ourselves with the
consistency and compatibility among those conditions under
which the fundamental equations are reduced to the Dirac
equation and the Maxwell-Lorentz equations. We expect
that the present investigation will shed some light on those
perplexing difficulties which we encounter in comprehending
the behavior of an electron solely according to the Dirac
equation and the Maxwell-Lorentz equations. Beyond this,
we have an ambition to investigate the possibility that other
elementary particles are governed by the same fundamental
equations under varied restrictive conditions.
We expect that those equations in the above will eventually
be reduced to the Dirac equation and also to the MaxwellLorentz equations, and write for Fij
1 ∂
(
∂x
−g
1 ∂
−g
− gF
(
j
)−g
ij
− gF *
∂x j
ij
)−g
∂η
∂x
ij
j
∂ξ
∂x j
=0
=0
(1)
(2)
In these equations, g is the determinant of the metric tensor
gij; Fij is an antisymmetric tensor and F*ij is conjugate to
Fij; ξ and η are scalars. One might ask why these equations
are fundamental. The answer is simple: Firstly, these
equations are covariant in the Riemannian sense; secondly,
i
by considering the current select for gij ∂η ∂x j and by
∂ξ
i
assuming ξmagnet
for gij
, these equations can be as
∂x j
the Riemannian generalization of the Maxwell-Lorentz
equations. However, we do not immediately relate these
equations to the Maxwell-Lorentz equations; a physical
consideration is needed prior to doing so.
116
−Q y
0
Qx
− Qx
0
Py
Pz
− Px 

− Py 
.
− Pz 
0 
F *ij = g ik g jm F *km
(3)
=
1
− gg ik g jmδ kmst F st
2
(4)
where δkmst is the Levi-Civita symbol, we have
The equations for the matter field and those for the
metric tensor field are intimately coupled together. In a
conventional sense, however, we may call the following
the equations for the matter field:
Qz
Considering
The Matter Field
ij
 0

 −Qz
ij
F =
 Qy
 P
 x
 0

 Pz
F *ij = 
 − Py

 − Qx
− Pz
Py
0
− Px
Px
0
−Q y
−Qz
Qx 

Qy 
× −g ,
Qz 
0 
 0 − Pz′ Py′ −Qx′ 


0 − Px′ −Qy′ 
 Pz′
ij
F* =
.
0
−Qz′ 
 − Py′ Px′
 Q′ Q′ Q′
0 
 x
y
z
(5)


From here on, we shall often write P for (Px, Py, Pz) and Q
for (Qx, Qy, Qz) simply for the sake of convenience, although
they are not three-vectors. We note that in general.
 
P ≠ P´,
 
Q ≠ Q´ (6)
 
Q = Q´ = B (7)
However, if
 
P = P´ = E,
We have a nice approximation. We expect the equivalence
between the two sets of equations will be established, if the
metric tensor field be properly evaluated in the following.
The Metric Tensor Field
As is well known, Einstein in 1916 proposed an equation
for the metric tensor [6].
1
Rij − gij R = − kTij 2
(8)
where Rij is the contracted curvature tensor, R is the
curvature scalar, and Tij is the energy-momentum
tensor of the matter field. Einstein gave this equation by
considering that the only fundamental tensors that do
not contain derivatives of gij beyond the second order are
functions of gij and the Riemann-Christoffel curvature
tensor and that the equation is analogous to the Poisson
equation for the gravitational field to the non-relativistic
limit. It seems that Einstein proposed this equation for
the purpose of solving cosmological problems, i.e., the
structure of the universe as a whole [6]. Therefore, Tij is
expected to be a known tensor supplied from the data of
astronomical observation of the average mass distribution.
Schwarzschild showed that the equation with Tij = 0 has
a particular solution that expresses properly the gravity
field induced by a material point [12].
Only when Eq. (8) is considered simultaneously with
Eqs. (1) and (2), the equation for an elementary particle
may be solved. If it is noticed that Eq. (8) alone consists
of ten simultaneous partial differential equations of the
second order, the analytical treatment of those equations
concerned is an extremely difficult task. Moreover, it
was not completely known how Tij is to be constructed
in terms of Fij , η and ξ in the Einstein epoch. (As
noted a short time ago, the variation method is not a
decisive one.) Our present purpose is to show that the
Dirac equation and the Maxwell-Lorentz equations,
which are covariant only in the Euclidean sense, are
both attainable by linearization of the same one set
of nonlinear equations covariant in a non-Euclidean
sense. From this viewpoint, we consider that it may
not be necessary that the fundamental equations are
immediately covariant in the Riemannian sense. There
may be schemes of geometry that are more general than
the Euclidean and less than the Riemannian. It is noted
that, because of the restrictive conditions, viz., Eq. (8),
Einstein’s geometry is less general than the Riemannian
[12]. According to Einstein, the Einstein tensor, the left
hand side of Eq. (8), should vanish in a space empty of
matter. On the other hand, in the Riemann geometry, it
does not vanish in general. (In the Riemann geometry,
the idea of matter does not exist.) However, the covariant
divergence of the Einstein tensor vanishes always in the
Riemann geometry as well as in [10]. As noted earlier,
Einstein chose Eq. (8) as one of the possibly simplest
equations. Our conjecture is even when we adopt
Einstein’s equation the obtained equation is adequate
completely for describing the field extremely near the
center of the core of an electron. Instead of taking an
axiomatic approach, it is essential to study carefully the
circumstances under which the present investigation is
motivated.
Later Einstein [6] attempted to investigate the structure
of an elementary particle as based on the same equation.
There, however, he did not pay much attention to Tij. He
simply speculated that the matter field is an electromagnetic
field, using a unimodular theory with Tij = TijMaxwell and
the magnetic field B perpendicular to electric field E,
(E ⊥ B). Contrary to Einstein conjecture, in our present
problem in which an electron is considered to be a small
universe, we considerer E••B, ie, we suppose that the
electron –positron equation is the Dirac equation if only if
it is derived from electromagnetic fields with E••B, inserted
1
in the original Einstein equation Rij − gij R = − kTij ,
2
Maxwell
ij
with Tij = Tij
. That means F = iF *ij , where
E = iB
i
i
, given by
i = −1 , and select
, ξmagnet
∂F µν
∂x
ν
∂F µν
∂x
ν
=
imc µ
4π µ
i
Je = −
E = select
c
 e
(9)
=
4π µ imc µ
i
J =
B = ξmagnet
c m
 m
(10)
(For the specific demonstration see the article Maxwell’s
theory with chiral current)
Thus, contrary to Einstein equation for the electron
(unimodular theory) [13], i.e., the equations for the matter
field and those for the metric tensor, do not contain
Planck’s constant h, the electronic mass m and charge e,
our equation (8), (9) and (10) contain h, m, e, which are
essential to obtain the Dirac equation.
With equations (9-10), it’s possible to show that an electron
is like a toroid with E••B, spin ½, without radiation and
rp = T =  / 2mc (figure 1).
rp
Figure 1. Electron model
117
Assumptions as regards the metric tensor field
According to the above results and considerations, we
assume that the field in question is spatially localized.
In the space outside the field, the Riemann-Christoffel
tensor is negligibly small. It would be more reasonable to
regard a part of the space as outside when the RiemannChristoffel tensor is negligibly small in that part. Also we
note that, owing to the other bodies of matter contained
in the universe, the tensor in question does not completely
vanish at any point of the space. But our interest is in the
local field, the electron. Hence, we ignore the curvature
of the global scale, and may consider an inertial frame
of reference outside the electron. (Einstein, perhaps due
to his esteem of Ernst Mach, did not necessarily seem
to think that the field of an electron can be completely
closed and sustained by itself [12, 13]). If we consider
an electron fixed to an inertial frame of reference, the
electron appears to be free from the influence of the
external universe. Classically, if we consider the internal
structure of the electron, the situation is not necessarily so
simple. It seems possible that a portion of the electron is
in acceleration relative to the inertial frame reference in
the same way as a portion of a spinning top resting as a
whole on the inertial frame is. Such a classical-mechanical
structure is inconceivable. However, it is sensible then to
assume that the electron has a stable structure with its own
permanent gravity field, as independent of the influence
of the external universe.
[5]
S. Weinberg. “A Unified Physics by 2050”. Sci.
Am. Vol. 281 Nº 6, pp. 68-75. December, 1999.
[6]
Essential Part in The Structure of the Elementary
Particles of Matter?” The Principle of Relativity.
Dover, pp. 191-198. 1952.
[7]
P.G. Bergmann and R. Thomson. Spin and Angular
Momentum in General Relativity. Phys. Rev. Vol. 2
Nº 89, pp. 400-407. 1953.
[8]
J. N. Goldberg, Conservation Laws in General
Relativity, Phys. Rev. Vol. 2 Nº 111, pp. 315-320.
1958.
[9]
C. Møller. “On the Localization of the Energy
of a Physical System in the General Theory of
Relativity”. Ann. Physics. Vol. 4, pp. 347-371.
1958.
[10]
C. Møller. “Further Remarks on the Localization
of the Energy in the General Theory of Relativity”.
Ann. Physics. Vol. 12, pp. 118-133. 1961.
[11]
C. Møller. Conservation Laws and Absolute
Parallelism in General Relativity. Mat. Fys. Skr.
Danske. Vid. Selsk. Vol. 1 Nº 10, pp. 1-50. 1961.
[12]
A.S. Eddington. The Mathematical Theory of
Relativity. Cambridge University Press, Cambridge.
1921.
[13]
A. Einstein. “Die Kompatibilität der Feldgleichungen
in der einheitlichen Feldtheorie”. Preuss. Akad.
Wiss. Berlin, Phys. Math. Klasse, Sitzber, pp. 18-23.
1930.
CONCLUSION
Thus we are presented a new theory called “Teoría Total
Simplificada” (TTS) based on chiral electrodynamic which
reproduces at the first time the Dirac equation for the electron
unifying the gravity with electromagnetism [14-16].
REFERENCES
[1]
W.G. Dixon, Special Relativity. University Press,
Cambridge. 1978.
[2] R. Adler, M. Bazin and M. Schiffer, Introduction
to General Relativity. McGraw-Hill, NY. 1965.
[3]
M. Friedman. Foundations of Space-time theories.
Princeton U.P., Princeton, NJ. 1983.
[4]
E. Witten. “Duality, Spacetime and Quantum
Mechanics”. Physics Today. Vol. 50, pp. 28-33.
1997.
118
[14] H. Torres-Silva. “Electrodinámica quiral: eslabón
Nº 1, pp. 6-23. 2008.
[15]
H. Torres-Silva. “The close relation between the
Maxwell system and the Dirac equation when
the electric field is parallel to the magnetic field”.
2008.
[16]
H. Torres-Silva. “Chiral field ideas for a theory
of matter”. Ingeniare. Rev. chil. ing. Vol. 16 Nº 1,
pp. 36-42. 2008.
H. Torres-Silva: Chiral waves in a metamaterial medium
CHIRAL WAVES IN A METAMATERIAL MEDIUM
ONDAS QUIRALES EN UN MEDIO METAMATERIAL
H. Torres-Silva1
RESUMEN
En este trabajo se estudia la refracción anómala en el borde de un medio metamaterial con fuerte quiralidad. El hecho de que
para una onda monocromática el vector de Poynting es antiparalelo a la dirección de la velocidad de fase conduce a relevantes
propiedades que pueden tener ventajas en el diseño de novedosos dispositivos y componentes a frecuencias de microondas.
Palabras clave: Ondas quirales, metamateriales.
ABSTRACT
In this paper we study the anomalous refraction at the boundary of a metamaterial medium with strong chirality. The fact
that for a time-harmonic monochromatic plane wave the direction of the Poynting vector is antiparallel with the direction
of phase velocity, leads to exciting features that can be advantageous in the design of novel devices and components at
microwaves frequencies.
Keywords: Chiral waves, metamat erial.
INTRODUCTION
Composite materials in which both permittivity and
permeability possess negative values at some frequencies
has recently gained considerable attention. This idea was
originally initiated by Veselago in 1967, who theoretically
studied plane wave propagation in a material whose
permittivity and permeability were assumed to be
simultaneously negative. Recently have been constructed
such a composite medium for the microwave regime, and
experimentally the presence of anomalous refraction in
this medium is verified [1]. Previous theoretical study
of electromagnetic wave interaction with omega media
using the circuit-model approach had also revealed the
possibility of having negative permittivity and permeability
in omega media for certain range of frequencies [2]. That
is important for design of circulary polarized antennas
The anomalous refraction at the boundary of such a
medium with a conventional medium, and the fact that for
a time-harmonic monochromatic plane wave the direction
1
of the Poynting vector is antiparallel with the direction of
phase velocity, can lead to exciting features that can be
advantageous in design of novel devices and components.
For instance, as a potential application of this material, the
idea of compact cavity resonators in which a combination of
a slab of conventional material and a slab of metamaterial
with negative permittivity and permeability. The problems
of radiation, scattering, and guidance of electromagnetic
waves in metamaterials with negative permittivity and
permeability, and in media in which the combined paired
layers of such media together with the conventional media
are present, can possess very interesting features leading
to various ideas for future potential applications such as
phase conjugators, unconventional guided-wave structures,
compact thin cavities, thin absorbing layers, high-impedance
surfaces, to name a few. In this talk, we will first present a
brief overview of electromagnetic properties of the media
with negative permittivity and permeability, and we will
then discuss some ideas for potential applications of these
materials. In this work we discuss the chiral waves in
metamaterial media.
119
CHIRAL WAVES
In classical electrodynamics, the response (typically
frequency dependent) of a material to electric and
magnetic fields is characterized by two fundamental
quantities, the permittivity ε and the permeability µ.
The
field
 permittivity relates the
 electric displacement


D to the electric field E through D = ε E, and the

permeability

 µ relates the magnetic field B and H
by B = µ H . If we do not take losses into account and
treat ε and µ as real numbers, according to Maxwell’s
equations, electromagnetic waves can propagate through
a material only if the index of refraction n, is real.
Dissipation will add imaginary components to ε and
and µ cause losses, but for a qualitative picture, one can
ignore losses and treat ε and µ as real numbers. Also,
strictly speaking, ε and µ are second-rank tensors, but
they reduce to scalars for isotropic materials. In a medium
with ε and µ both positive, the index of refraction is
real and electromagnetic waves can propagate. All our
everyday transparent materials are such kind of media.
In a medium where one of the ε and µ is negative but
the other is positive, the index of refraction is imaginary
and electromagnetic waves cannot propagate. Metals and
Earth’s ionosphere are such kind of media. In fact, the
electromagnetic response of metals in the visible and
near-ultraviolet regions is dominated by the negative
epsilon concept [1-3].
Although all our everyday transparent materials have both
positive ε and positive µ, from the theoretical point of view,
in a medium with ε and µ both negative, electromagnetic
waves can also propagate through. Moreover, if such
media exist, the propagation of waves through them
should give rise to several peculiar properties. This was
first pointed out by Veselago over 30 years ago when
no material with simultaneously negative ε and
 ε was

known [4]. For example, the cross product of E and H
for a plane wave in regular media gives the direction of
both
flow, and the electric
 field
 propagation and energy

E , the magnetic field H , and the wave vector k form
a right-handed triplet of vectors.
 In contrast, in a medium
with ε and µ both negative, E x H for a plane wave still
gives the direction of energy flow, but the wave itself that
is, the phase velocity

 propagates in the opposite direction,
i.e.,
wave
vector
lies in the opposite direction of E x
k

H for propagating
 waves. In this case,
 electric field E ,
magnetic field H , and wave vector k form a left-handed
triplet of vectors.
Such a medium is therefore termed left-handed medium
[5]. In addition to this ‘‘left-handed’’ characteristic, there
120
are a number of other dramatically different propagation
characteristics stemming from a simultaneous change of
the signs of ε and µ, including reversal of both the Doppler
shift and the Cerenkov radiation, anomalous refraction,
and even reversal of radiation pressure to radiation
tension. However, although these counterintuitive
properties follow directly from Maxwell’s equations,
which still hold in these unusual materials. Such type of
left-handed materials have never been found in nature
but such media can be prepared artificially, they will
offer exciting opportunities to explore new physics and
potential applications in the area of radiation-material
interactions. Following the suggestion of Pendry, Smith
and co-workers reported that a medium made up of an
array of conducting nonmagnetic split ring resonators
and continuous thin wires can have both an effective
negative permittivity ε and negative permeability
µ for electromagnetic waves propagating in some
special direction and special polarization at microwave
frequencies [5]. This is the first experimental realization
of an artificial preparation of a left-handed material,
where on the one hand, the permittivity of metallic
particles is automatically negative at frequencies less
than the plasma frequency, and on the other hand, the
effective permeability of ferromagnetic materials for
electromagnetic waves propagating in some particular
direction and polarization can be negative at frequency
in the vicinity of the ferromagnetic resonance
frequency, which is usually in the frequency region
of microwaves. However this configuration exhibit
chirality and a rotation of the polarization so the
analysis of metamaterial presented by several authors
provides a good but not exact characterization of the
metamaterial [6]. The evidence of chirality behavior
suggests that if it is included in the conditions to obtain
a metamaterial behavior of a medium futher progress
will be obtained. In this short paper, we propose to
investigate the conditions to obtain a metamaterials
having simultaneously negative ε and negative µ and very
low eddy current loss. As a initial point, we considerer
a media where the electricpolarization P depends not

M
only on the electric field E , and the magnetization

depends not only on the magnetic field H , and we may
have, for example, constitutive relations given by the
Born-Federov formalism [7].
 
 
 
D(r , ω ) = ε (ω )( E (r , ω ) + T (ω )∇ × E (r , ω )) (1)
 
 
 
B(r , ω ) = µ (ω )( H (r , ω ) + T (ω )∇ × H (r , ω )) (2)
H. Torres-Silva: Chiral waves in a metamaterial medium
The pseudoscalar T represents the chirality of the
material and it has length units [7]. In the limit T → 0,
the constitutive relations (1) and (2) for a standard linear
isotropic lossless dielectric with permittivity ε and
permeability µ are recovered.
According to Maxwell’s equations, electromagnetic
waves propagating in the direction of magnetization
in a homogeneous magnetic material is either positive
or negative transverse circularly polarized. If the
composite can truly be treated as a homogeneous
magnetic system in the case of grain sizes much
smaller than the characteristic wavelength, electric
and magnetic fields in the composite should also be
either positive or negative circularly polarized and can
be expressed as
 
− j ( k z −ω t )
E ± (r , t ) = Eˆ 0 ( ± )e ± 0 (3)
 
− j ( k z −ω t )
H ± (r , t ) = Hˆ 0 ( ± )e ± 0 (4)
 

where E0± = E0 ( xˆ ± yˆ ) , and ∇ × E ± (r , t ) =  k± E ± , k± ≥ 0
is the chiral wave number.
In this case of right polarized wave we can see that the
effective permittivity εp and the effective permeability
µp are obtained from
with µ p = µ (1 − k+ T ) and where k eff and ω are related
by keff 2 = ω 2 (ε p µ p ) . Equations (5) and (6) are exact
in principle assuming that nonlocal effects can be
neglected. This assumption is appropriate in many
cases. But in some cases, nonlocal effects can be
significant and cannot be neglected, as has been shown
in the past. In such cases, Eqs. (5) and (6) shall be not
exact. For simplicity, in this paper we have assumed
that nonlocal effects can be neglected and hence Eqs.
(5) and (6) shall be valid. In Eqs. (3 and (4) the sign of
the effective wave number can be positive or negative
depending on the product k+T and the energy flow. For
convenience we assume that the direction of energy
flow is in the positive direction of the z axis, but the
sign of k eff still can be positive or negative. In the case
of right polarization, if 1 ≥ k+ T ≥ 0, the phase velocity
and energy flow are in the same directions, and from

E
Maxwell’s equation, one
 can see that the electric

and magnetic field and H and the wave vector keff will
form a right-handed triplet of vectors. This is the usual
case for right-handed materials. In contrast, if k+ T ≥ 1
the phase velocity
flow are in opposite

 and energy
directions, and E , H , and keff will form a left-handed
triplet of vectors. This is just the peculiar case for left
handed materials where the effective permittivity εeff
and the effective permeability µ eff are simultaneously
negative. So, for incident waves of a given frequency
v, we can determine whether wave propagation in the
composite is right handed or left handed through the
relative sign changes of k eff..
Based on Eqs. (5)-(6), we have computed T. and ε/εp or
 
 
 − jk z 
− jk z 
∫ D(r ,ω )e + f dr = ε ∫ (E (r ,ω ) + T ∇xE )e + dr (5)
 − jk z 
= ε ∫ (1 − k+ T ) Ee + dr
with ε p = ε (1 − k+ T ) and k+ T ≥ 1.
 
− jk+ z
close to µ P ε P , the value of T. is quite large, indicating
a strong spatial dispersion. Hence the singular point is
the very point of traditional limitation. However, with
κ / µP ε P continuously increasing, the spatial dispersion
strength falls down very quickly. Therefore, if κ. is not
Similary, we have
∫ B(r ,ω )e
µ/µp. versus κ / µ P ε P , as shown in Fig. 1 When κ. is very
 

− jk z 
dr = µeff ∫ H (r , ω )e + dr
(6)
 
− jk z 
= µ (1− k+ T ) ∫ H (r , ω )e + dr
around µ P ε P , e.g. κ < 0.7 µ P ε P or κ > 1.3 µ P ε P ,
we need not take nonlinear terms into consideration at
all. Hence the strong spatial dispersion and nonlinearity
cannot put the upper limitation to chirality parameters
either.
121
dispersion, the Pasteur model is meaningful. Neither
spatial dispersion nor energy will hinder chirality to be
stronger, but we cannot realize strong chirality only by
increasing the spatial dispersion. The necessary condition
of strong chiral medium is that the chirality and spatial
dispersion are of conjugated types. We remark that strong
chiral media have found wide applications in the negative
refraction and supporting of backward waves, useful in
metamaterial substrates.
30
20
10
0
–10
–20
ACKNOWLEDGEMENT
–30
0
1
2
3
Figure 1. The strength relationship of chirality and spatial
dispersion. T versus κ / µ P ε P The point of
κ / µP ε P = 1 is singularity, corresponding
The author acknowledges discussions with colleagues
in the school of electrical and electronic engineering
EIEE – Universidad de Tarapacá. This work have been
supported by the project N° 8721-06 of the Universidad
de Tarapacá.
infinite spatial dispersion coefficient T. When
κ / µP ε P > 1, T becomes negative for keeping
the positive rotation term coefficients with
negative µ and ε.
REFERENCES
[1]
S. Tretyakov, A. Sihvola, and L. Jylha. Photonics and
Nanostructures - Fundamentals and Applications.
Vol. 3, p. 107. 2005.
CONCLUSIONS
[2]
From figure 1, it is clear that enhancing spatial dispersion
will not lead to strong chirality and will reach the traditional
limitation point. This is why we have never succeeded in
realizing strong chirality no matter how to improve the
asymmetry and spatial dispersion.
M.M.I. Saadoun and N. Engheta. “Theoretical study
of electromagnetic properties of non-local omega
media”. Chapter 15. Progress in Electromagnetic
Research (PIER) Monograph series, vol. 9, A.
Priou, (Guest Editor), pp. 351-397. 1994.
[3]
R.A. Shelby. “Experimental verification of a
negative index of refraction”. Science. Vol. 292
Nº 5514, pp. 77-79. 6 April, 2001.
[4]
V.G. Veselago. “The electrodynamics of substances
with simultaneously negative values of epsilon
and mu”. Soviet Physics Uspekhi. Vol. 10 Nº 4,
pp. 509-514. 1968.
[5]
D.R. Smith. “Composite medium with simultaneously
negative permeability and permittivity”. Phys. Rev.
Lett. Vol. 84 Nº 18, pp. 4184-4187. 1 May, 2000.
[6]
D.R. Smith. Physical Review B. Vol. 65, p. 195104.
2002.
[7]
H. Torres Silva. “Chiro-Plasma Surface Waves”.
Advances in complex Electromagnetic Materials,
Kruwer Academic Publishers. Netherlands, p. 249.
1997.
Fortunately, as pointed out earlier, the strong chirality
does not require strong spatial dispersion. Hence the
most important difference between strong and weak
chirality is that T. and κ. have opposite signs, which
necessarily leads to negative ε and µ. Here, κ. stands
for chirality and T. is the chiral coefficient of the first
order for spatial dispersion. Strong chirality roots
from using one type of spatial dispersion to get the
conjugate stereoisomer, or chirality. It is an essential
condition for supporting the backward eigenwave in
strong chiral medium.
In conclusion, a strong chiral medium behaves like
Veselago’s medium. Under the weak spatial dispersion,
the energy is always positive for chiral medium. We
show that strong chirality does not equal strong spatial
dispersion, which occurs only around a singular point.
Even in this small region with very strong spatial
122
UNIVERSIDAD DE TARAPACÁ
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nombres de los autores, la dirección (postal y electrónica) del autor encargado del manuscrito, el título del artículo,
los nombres y las direcciones de los árbitros, la fecha de envío a los árbitros, la recomendación de los árbitros, la
decisión tomada luego de la evaluación y la fecha de aceptación o de rechazo del artículo.
El editor enviará el artículo a dos o tres árbitros con el objeto de que éste sea debidamente evaluado. Los árbitros
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Los árbitros considerarán la solidez del diseño experimental, verificarán que las conclusiones estén de acuerdo con
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toda la literatura pertinente; asimismo, se considerará la calidad de la redacción.
Se usará en esta etapa, el sistema de árbitros desconocidos, donde los árbitros conocen la identidad del autor, pero el
autor desconoce la identidad de los árbitros.
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Después de evaluar las recomendaciones de los árbitros, el editor tomará una de las siguientes decisiones:
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los cambios no conllevan modificaciones significativas de la redacción, el editor leerá el artículo y añadirá sus
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la versión final del artículo, el editor confirmará al autor su aceptación, indicándole en qué número de la revista se
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Ejemplos de cambios menores: errores tipográficos, páginas sin numerar, artículos citados en el texto que no
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la redacción.
Rechazo: El editor devolverá el artículo con la evaluación de los árbitros e informará sus razones para no publicarlo.
Esta decisión será casi siempre final.
Ejemplos de motivos de rechazo: contenido del artículo no apropiado para la revista, violaciones crasas de las
normas de publicación, artículos carentes de significación, mérito científico o tecnológico.
SELECTION PROCESS FOR ARTICLES SUBMITTED FOR PUBLICATION
IN THE INGENIARE. REVISTA CHILENA DE INGENIERÍA
1. Preliminary Selection and Peer Evaluation
The editor will first verify that the content of the article is appropriate for the magazine and the manuscript is set in
accordance with the journal’s standards. The editor may reject, at his discretion, submissions that do not comply with
the general directions, are poorly written or do not posses sufficient scientific or technological merit.
Articles that pass the preliminary selection are sent for peer review by specialists and scientists of renown research in
areas related to the article. Upon assessing the article, peer reviewers write a report including comments and suggestions
recommending its acceptance or rejection.
2. About the Evaluation by Peers Reviewers
As part of the evaluation of the article, reviewers will consider the soundness of the experimental design, verifying
that the conclusions coincide with the statistical and experimental data. Reviewers will also make sure and that the
author(s) consulted all literature pertinent to the topic discussed. Wording and writing quality will also be assessed.
During this process the identity of the reviewers will not be known by the author(s), but the latter’s will be available
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After assessing the reviewers’ recommendations, the editor will make a decision based on the following options:
Accepting the article, subject to minor revisions: The editor will return the article to the author(s) with a list of
suggestions for minor corrections, including his own as well as those made by the reviewers. Once the modified
manuscript is received by the editor, this will confirm the acceptance of the article, indicating the publication date.
Minor revisions in an article include: typographical errors, page numbering, articles cited in the text that are not in
the bibliography and vice versa, slight discrepancies between the summaries and the abstract, moderate corrections
to the text.
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for major corrections that must be made before the article is accepted for publication.
Major revisions include: analyzing data using other statistical tests, adding or redoing figures and tables,
repetition of experiments, rewriting the discussion of the problem using additional literature and substantial
changes to the text.
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decision will be final (in most cases).
Reasons for rejection of an article include: inappropriate content, manuscripts not complying with the norms for
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People interested in publishing papers in “Ingeniare. Revista chilena de ingeniería” must send articles which comply with the
publication norms detailed below. The topics must relate to the following areas of engineering: Electronics, Electricity, Computing,
Mechanics, Industry, Acoustic, Metallurgy and Engineering Education. The author’s rights may not be granted to third parties. If
necessary, the editor reserves the right to carry out minor modifications for editing, in order to achieve a better presentation of the
work. Papers may be presented in English (if possible), Portuguese and in Spanish.
Four types of contributions are regularly considered:
Papers. Presentation of significant research, development, or application. Brief Papers. Concise descriptions of a contribution to a
specific aspect of design, realization, or operation. Letters. Significant remarks of interest to engineers, and comments on previously
published papers. In addition, special papers (tutorials, surveys, and perspectives on the current trends) are solicited. Authors are
encouraged to contact the Editor or Co-Editor before submitting such papers.
Papers and Brief Papers go through the same review process. Letters go through a shorter review process to facilitate rapid
publication.
PUBLICATION CHARACTERISTICS
The Advisory Editor Committee will recommend the publication of papers whose content will be the author’s full responsibility.
Originals accepted for publication will not be returned. Editing or rejections will be notified in due time. The Advisory Editor
Committee may consider papers presented in national and international conferences and scientific meetings.
The main title of the article should be in English and Spanish. It must be short and it must clearly state the topic of the work. Abstracts
must be brief, containing clear and precise ideas. It must contain keywords which define the content of the paper (minimum of five and
maximum of ten).
The text should begin with a summary of not more than 250 words. It must briefly state: 1) what has been done in this work, 2) how it
was done (only if it is important to be detailed), 3) main results obtained, 4) relevance of the results. A summary is an abbreviated but
comprehensive presentation of the article and it must inform about the objective, methodology and the results of the work described
in the paper. A translation of the summary into English, headed by the word abstract, must be included immediately after Spanish
resumen. The introduction must be presented in about a page and a half and it must guide the reader towards an understanding of
the problem presented. It should also include information about the nature of the problem, references to previous works, purpose and
meaning of the paper.
The body of the paper will contain fundamental information about the work. Information must be clearly presented. Language
must be objective and impersonal. The author must see to it that the article complies with the norms of publication, language register
and specific terms accepted by the scientific community. The introduction and the body of the article must be written in two 8 cm.
columns, with a 1.05 cm. space between columns.
Conclusions must be clear, stating what is shown in the work as well as its relevance. They must also state advantages, limitations and
application results.
References will be written with numbers in parenthesis and they will be listed at the end of the paper as follows:
Books: [N°] Author’s First Initial Name, Last Name. “Title”. Editorial name. Number of Edition. City, Country. Volume, pp. Pages
number. Date. Articles of magazines: [N°] Author’s First Initial Name, Last Name. “Title”. Magazine name. Volume. Number, pp.
Pages number. Date. Electronics references: [N°] Author’s Name. “Title”. Site Update, pp. Pages number. Date of visit. URLs.
Individualization: Footnotes, written with arabic numbers, will state: 1) date of reception of the original, people involved in the
intellectual work, materialy or financially. If necessary the mention whether the paper is part of a study, a thesis, a project, etc., 2)
name of the author, profession, institution, author and institution’s electronic and mail address.
Aknowledgements: Will be included before the bibliography, identifying individuals and institutions that gave intellectual or financial
support to the research.
All equations, pictures and photographs must bear a number that will be used for identification purposes throughout the text.
Pictures and photographs must also include an explanatory caption.
Periodicity: “Ingeniare. Revista chilena de ingeniería” is published periodically, is printed in three issues per volume annually.
Reception of papers: Manuscripts should be in Microsoft Word or LaTeX format and submitted in electronic form, via e-mail or in optical
or magnetic media, according to given instructions. Address: Editor Committee, “Ingeniare. Revista chilena de ingeniería”, Casilla 6-D,
Arica-Chile, Sud América. E-mail: [email protected]
Papers should not be longer than 15 pages and must comply with the following format: Paper: letter size 21,59 cm x 27,94 cm;
Margins: top 2 cm, bottom 2 cm, left 2,54 cm and right 2 cm; Title of the article, centered in bold type, Times New Roman 10 pt.;
two blank spaces before the initial paragraph; Subtitles should be at the left margin in black letters with Times New Roman 10 pt.; The
body of the text must be written with Times New Roman 10pt., one blank space between paragraphs.
NORMAS DE PUBLICACIÓN
Los autores interesados en publicar artículos en la “Ingeniare. Revista chilena de ingeniería”, deben enviar sus trabajos ajustados a
las normas de publicación que se detallan más abajo. Los temas deben enmarcarse dentro de las siguientes áreas de la ingeniería:
Electrónica, Eléctrica, Computación, Mecánica, Industrias, Acústica, Metalurgia y Enseñanza de la Ingeniería. El trabajo sometido no
debe tener “Derechos de Autor” otorgados a terceros. El editor se reserva el derecho a realizar modificaciones menores de edición, para
una mejor presentación del trabajo. Se podrá presentar trabajos en idioma inglés (de preferencia), español y portugués.
Se consideran cuatro tipos de contribuciones:
Trabajos in extenso (papers). Presentaciones relativas a investigación significativa, desarrollo o aplicación de sistemas tecnológicos.
Trabajos condensados (brief papers). Descripciones concisas de contribución específica a investigación significativa, desarrollo o
aplicación de sistemas tecnológicos. Comunicaciones (letters). Observaciones de interés para los lectores (ingenieros) y/o comentarios
acerca de publicaciones previas. Trabajos especiales. Trabajos tales como tutoriales, encuestas y visiones de las tendencias actuales
en ingeniería son bienvenidas. Se invita a los autores de tales trabajos a contactarse con el Editor antes de presentarlos para publicación.
Los trabajos in extenso y condensados (papers & brief papers) se someten al mismo procedimiento de revisión. Las comunicaciones
(letters) son revisadas en un proceso más breve, para facilitar su pronta publicación.
CARACTERISTICAS DE LAS PUBLICACIONES
El Comité Editor Asesor será el encargado de autorizar la publicación de los trabajos, cuyo contenido será de responsabilidad
exclusiva de los autores. Los originales de los artículos aceptados para publicar no serán devueltos. Las modificaciones o rechazos se
indicarán con notas explicativas. El Comité Editor Asesor podrá considerar trabajos presentados en congresos y reuniones científicas
nacionales e internacionales.
El título principal debe estar escrito en inglés y español, indicando claramente la materia del artículo. Éste deber ser breve, pero
preciso en la idea que represente. Además, debe contener un número suficiente de palabras clave que definan el contenido del artículo
(mínimo cinco y máximo diez). El texto comienza con un resumen de no más de 250 palabras, donde debe precisarse brevemente: 1) lo
que el autor ha hecho, 2) como lo hizo (sólo si es importante detallarlo), 3) los resultados principales, 4) la relevancia de los resultados.
El resumen es una representación abreviada, pero comprensiva del artículo y debe informar sobre el objetivo, la metodología y los
resultados del trabajo descrito. A continuación del resumen, debe incluirse su traducción al idioma inglés, encabezado por la palabra
abstract.
La introducción deberá orientar al lector respecto al problema presentado e incluir: la naturaleza del problema, los antecedentes o
trabajos previos, el propósito o significancia del artículo. Ésta deberá desarrollarse en una página y media, como máximo. El cuerpo
contendrá, en detalle, la información fundamental del artículo. Deberá asimismo considerar el objeto de la información, la que deberá
ser entregada en forma clara y eficiente. La redacción de los trabajos será de carácter objetivo e impersonal. El autor cuidará que la
forma se ajuste a las normas de presentación, corrección en el lenguaje y uso de terminologías aceptadas por organismos científicos.
La introducción y el cuerpo deberán ser escritos en doble columna de 8 cm cada una, con un espacio de 1,5 cm entre columnas.
Las conclusiones tendrán que ser claramente definidas y deberán cubrir lo siguiente: lo que se demuestra en el trabajo, su relevancia,
ventajas y limitaciones y aplicación de los resultados.
Las referencias se indicarán con número entre paréntesis y se listarán al final de la publicación, de la siguiente forma:
Libros: [N°] Iniciales del Nombre del autor (es), Apellido del autor (es). “Título”. Nombre de la Editorial. Número de Edición. Ciudad,
País. Volumen, pp. Páginas consultadas. Año de publicación.
Artículos de revistas: [N°] Iniciales del Nombre del autor (es), Apellido del autor (es). “Título del artículo”. Nombre de la revista.
Volumen. Número, pp. páginas entre las que se encuentra el artículo. Mes y año.
Textos electrónicos: [N°] Iniciales del Nombre del autor (es), Apellido del autor (es). Título del artículo, pp. Páginas consultadas. Fecha
de actualización. Fecha de consulta. Dirección web.
Individualización: Al pie de la página, mediante números arábigos, deben señalarse: institución a la que pertenecen los autores, dirección
postal y electrónica de él o los autores.
Agradecimientos: Se ubicarán antes de las referencias bibliográficas, señalando a las personas o instituciones que colaboraron en el
trabajo intelectual, material o financiero.
Todas las ecuaciones, figuras y fotografías deberán tener un número que las identifique, al que se hará referencia en el texto. Las figuras
y fotografías deberán ser nítidas y tener, además, una leyenda explicativa al pie de las mismas.
Periodicidad: “Ingeniare. Revista chilena de ingeniería”, edita tres números al año (cuatrimestral).
Recepción de colaboraciones: Los manuscritos deberán estar en formato Microsoft Word o LaTeX, y enviarse por algún medio
electrónico (correo electrónico, medio de almacenamiento óptico o magnético), de acuerdo a las instrucciones presentadas. Éstas
deben ser enviadas a: Comité Editor, “Ingeniare. Revista chilena de ingeniería”, Casilla 6-D, Arica - Chile, Sudamérica. E-mail:
[email protected]
Los artículos no podrán tener una extensión mayor de 15 páginas y deberán respetar el siguiente formato: papel tamaño carta, 21,59x27,94
cm; márgenes: superior 2 cm, inferior 2 cm, izquierdo 2,54 cm, derecho 2 cm; título del artículo, centrado en negrita, con letra mayúscula
tipo Times New Roman 12 pt.; títulos del texto, centrados en negrita, con letra Times New Roman 10 pt., dejando dos líneas en blanco
antes del párrafo; texto del cuerpo con letra Times New Roman 10 pt., dejando una línea en blanco entre párrafos.

Teoría Total simplificada, Revista Chilena de Ingeniería, Vol. 16, Nº1

Transcription

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