Elementary Derivation of the Dirac Equation. X

Elementary Derivation of the Dirac Equation. X
Hans Sallhofer
Forschungs- und V ersuchsanstalt der A ustria M etall A G , B raunau, A ustria
Z. N aturforsch. 41a, 4 6 8 -4 7 0 (1986); received D ecem ber 20, 1985
Final considerations o f the isom orphism betw een electrodynam ics an d wave m echanics.
1. The M axwell-Dirac Isomorphism
rot E
The preceding nine papers o f ours were concerned
with the formal connection between electrodynamics
and wave mechanics, a problem which has attracted
the attention of a num ber of other authors too
[1 -6 ]. We were able to demonstrate the isomorphism
of the two theories in the dom ain of differential
equations of the first order. There we ultimately
started from the completely source-free electro­
dynamics, i.e.
+—^ / / - 0
c dt
r o t//-— J -£ = 0
c dt
div
eE
=0
div n H = 0
d iv £ = 0
= [ ( y V + / 4 >04) !P = 0 ].
(5)
In words: M ultiplication of source-free electro­
dynamics by the Pauli-vector yields wave mechan­
ics.
rs
This derivation of wave mechanics appears, from
r o t E + —---- H = 0 . d i w e E = 0 , div E = 0 ,
c dt
the electrodynamical point of view, as a transform a­
(1)
tion of the solutions by which the num ber of
r o t / / - — J - £ = 0 , div y. H = 0 , div H = 0 .
unknown com ponent functions is reduced from six
c dt
to four. In addition, the Pauli-vector generates those
Skalar-m ultiplication of these equations by the
symmetries which yield the well-known ease of
Pauli-vector leaves the last four unchanged, while
solution so characteristic of wave mechanics. We
the first two go over into
may therefore say:
i (<r • E)
Wave mechanics is a solution-transform of elec­
V(2) trodynamics. Here one has to bear in m ind that the
(o-H )
well-known circulatory structure of the wave func­
This coincides, after separating out the time depen­
tions, manifest in D irac’s hydrogen solution, is not
dence, with the Dirac am plitude equation
introduced just by the Pauli-vector. The corre­
(4 )
sponding
angular or spin m om ent exists a priori in
i (hco —0 + moCz
^ = 0 . source-free electrodynamics and is described its
y ■V+
fl CO — 0 — M q C‘
tic
momentum balance [7]
The electrodynamical as well as the wave-mechani­
cal components are interconnected, as are refraction
V + div 7 = 0 ,
(6)
dt
( e,/ u) and potential <2 >, by simple linear relations.
A short and elegant derivation of this isomorphism
with the m om entum density V and the stressis reproduced in the last section.
tensor T.
Summing up we can say that under the sufficient­
ly general assumption of periodic time dependence
2. Epistological Perspectives
the following connection exists between electro­
dynamics and wave mechanics:
The epistological perspectives of the MaxwellR ep rin t requests to D r. Flans Sallhofer, Bahnhof-Str. 36,
A-5280 B raunau, A ustria.
Dirac isomorphism are in accord with the tenets of
logical positivism. The latter doubts, as one knows,
the existence of particles or particle like matter. It
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469
H. Sallhofer • Elementary D erivation o f the Dirac Equation. X
regards them as illusions, as “things o f the im agina­
tion” according to Mach [8-11]: “N ot each existing
scientific theory arises (as) naturally and unaffect­
edly. When, e.g., chem ical, electrical, optical phe­
nomena are explained through atoms, then the
auxiliary concept o f atom s has not resulted from the
principle of continuity, it rather has been invented
especially for that purpose. We can nowhere observe
atoms, they are things of the im agination, as all
substances are. And to atoms even such properties
are partially attributed which contradict all those
observed so far. M ay the atomic theories still be
suited for describing a num ber of facts, the natural
scientists who took N ew ton’s rules of philosophizing
to heart will accept any such theory only as a
provisional aid and will strive for a replacem ent by
a more natural notion.”
An alternative to particle m atter is field m atter, in
case of the Maxwell-Dirac isom orphism an electro­
dynamic kind of m atter: Two photon fields “circle
around each other, their centers of gravity forming a
Kepler system. In the neighbourhood of the geo­
metric center of this hom eopolar system the field
strengths reach high intensities and convey the im ­
pression of being m aterialized. Here, the hydrogen
atom is a standing disturbance of a special con­
figuration, in essence a disturbance like stationary
light.
“To atoms even such properties are partially
attributed which contradict all those observed so
far.” Here Mach addresses the wellknown contra­
dictions inherent in the particle oriented atom ism of
his time. His critique fits, however, in a completely
analogous way the wave models of Schrödinger and
dirac originating from some tim e after him, because
this wave picture o f the twenties was somehow
turned back by the conventional Copenhagen inter­
pretation to the Rutherford-Bohr model o f the
atom. The particles o f R utherford’s model, “things
of the imagination, as all substances are”, which
seemed to have been overcome in the wave model,
recurred by Bom’s interpretation.
There is, e.g. the orbiting electron which in viola­
tion of electrodynamics — the most reliable of all
physical theories — is not allowed to radiate, despite
its continually accelerated revolution. Then, how­
ever, in jumping between different states, it is
obliged to radiate: an absurdity which has been
passed on from the old mechanical models to the
undulatory ones. These ultimately connect, in an
artificial way, the purely electromagnetic effects of
spectral light with the purely mechanical causes of
Bohr’s model. The Maxwell-Dirac isomorphism, on
the other hand, connects the spectral light with a
source-free standing electrodynamic disturbance as
its cause, thus avoiding the contradiction.
3. A Short Version of the M axwell-Dirac
Isomorphism
If one wanted to describe the hydrogen gas by
means of electrodynamics one should probably start
from the firmly established experience that the
hydrogen gas may absorb and reem it electro­
magnetic energy, and that w ithout external inter­
vention there is no indication for the gas to contain
electric charges. Thus we consider the hypothesis
which visualizes the gas as a charge-free electro­
magnetic field as the starting point with the least
num ber of assumptions; and so we try to char­
acterize the field by the covariant Maxwell system
rot E + — — H =
0
c ot
,
e ö
rot H ------— E = 0
C 0/
divfi£=0 ,
(7)
div n H = 0 ;
and the condition of being charge-free by means of
the requirement
d i v £ = 0 besides div H = 0 .
(8)
Because of this we get for (7) the equations
U
0
rot £ H---- — H = 0 ,
c ot
e
div E = 0 ,
(9)
0
rot H ------— E = 0 ,
c dt
div H = 0
with
E ± grad e ,
( 10)
H I . grad n .
Scalar m ultiplication of the rot equations in (9) by
the Pauli-vector, using the algebraic relation
(a ■V) (<r • A) = div A + / a • rot A
(11)
together with the two div equations, transforms that
system into
(a ■V) (a
(<!?)(<, • £ )
c ot
a - E ) = 0,
+ ü | - („.«)=
c ot
E _L grad e,
H !_ grad n
0
,
(1 2 )
H. Sallhofer • Elementary D erivation o f the Dirac Equation. X
470
finally yields the am plitude equation
In matrix notation this reads
a
v _fel
\®
tu/
E _L grad e,
® \ ± !
/ i l | c S(
<l9)
(13)
H I . g ra d //.
Denoting the quantity on which the differential
operators act by
that is
iH iE 3
iE + ~ iE 3
H_
Hi
i ( a ■E)
(er H )
H+ -
h
r < '
> 1
vi
3
^ 3
n
n
n
(14)
with X± = X \ ± i X 2, and considering the well-known
connection
(15)
= y
between the Pauli and the Dirac matrices, we get
for (13) the system
r v —' e t
® \J_A
c 5f_
0
Its agreement with the D irac am plitude equation
/ lh co — 0 + Wo c 2
yV+—
c ti \
®
®
,
\
, r //=0
(20)
ti co — & — m Cq
is obvious.
The Eqs. (17), as well as (19) or (20), show in
addition that the electrodynamical and the wave
mechanical field components are connected by sim­
ple linear relations, the same holding true for the
refraction (e,yu) in relation to the potential <P.
This isomorphism can be checked easily and
directly because the eight Eq. (9) may be combined
into two systems of four equations each, in the
following way:
± i (rot H ) 3 + i c ~ 1 e E 3 + div H
=0,
± i (rot H ) 1 + i c - 1 e E\ - (rot H ) 2 + c _I e E 2 = 0,
^ =
0
,
(16)
± / div E
—(rot E ) 3
—c - 1
/ / / / 3
= 0,
± i (rot E ) 2 ± / c~ 1 fj. H 2 + (rot E ) x + c ~ 1 n H\ = 0.
E _L grad e,
H I . grad pi.
Here one has to bear in mind that each of both
columns matrix (14) that is
(21)
(18)
Inserting here the first or the second wave function
of (17) into the first system (upper signs) or the
second one (lower signs), respectively, the wave
function of (16) ends up immediately, in both cases,
and we are back to D irac again.
With this proof of the theorem that wave me­
chanics represents a specialized electrodynamics, a
first stage in our investigations into the phenom­
enological meaning of the Schrödinger function has
come to an end. The result seems unam biguous and
incompatible with the current doctrine which rests
on a particle interpretation.
[1] J. R. O pp en h eim er, Phys. Rev. 38,725 (1931).
[2] H. E. Moses, Sup. N uovo Cim . 7 (Ser. X., No. 1), 1
(1958).
[3] T. O hm ura, Prog. T heor. Phys. 16,684 (1956).
[4] R. H. G ood Jr., L ectures in T heoretical Physics,
Vol. I, Interscience Publ., 1959, p. 30.
[5] J. E. E dm onds Jr., Lett. N uovo Cim . 13, 185 (1975).
[6] W. H eitler, Q u a n tu m T heory o f R adiation, 2nd Ed.,
O xford U niversity Press, O xford 1944, p. 1.
[7] H. Sallhofer, Z. N atu rfo rsch . 39 A, 142 (1984).
[8] Ernst M ach, D ie M echanik in ih rer Entw icklung, 8th.
ed., 466, F. A. B rockhaus, L eipzig 1921.
[9] E. M ach, D ie A nalyse d e r E m pfindungen, 9th ed.,
254, G ustav F ischer, Jena 1922.
[10] E. M ach, Populär-w issenschaftliche V orlesungen, 5th
ed., 115, 237, 238, 244, Johann A m brosius Barth,
Leipzig 1923.
[11] E. M ach, Bosoton studies in the p h ilo so p h y o f science,
86, 87, 94, 100, 101, 102, D. R eidel P ublishing C om ­
pany, D o rtrec h t 1900.
iE 3
i E^ + E 2
1
1
tf/ =
H3
H x+ / H 2
and
=
—i E 3
H x- i H 2
- h 3
(17)
independently reprents a system of functions solv­
ing (16). From this, a separation of the time
dependence according to
_ ... „ —/to /