SPINORS ON THE RIEMANNIAN MANIFOLDS 1. Introduction

Russian Mathematics (Iz. VUZ)
Vol. 46, No. 11, pp.6{23, 2002
Izvestiya VUZ. Matematika
UDC 530.12
SPINORS ON THE RIEMANNIAN MANIFOLDS
R.F. Bilyalov
1. Introduction
In the most general mathematical form the spinors were introduced by E. Cartan in 1913 (see 1]).
In 1928 the spinors were discovered again by van der Waerden (see 2]) in connection with Dirac's
physical investigations. V.A. Fock and D.D. Ivanenko implemented the spinors into the general
relativity (see 3]). In 1940 L. Rosenfeld suggested a general procedure for constructing the energymomentum tensor of material elds, which stands in the right-hand side of Einstein's equations
(see 4]). L. Rosenfeld's method essentially involves Lie derivative. Construction of Lie derivative
for tensor elds encounters no diculties, but this is not true for spinor elds. In the deduction
of a formula for the energy-momentum tensor of spinor elds L. Rosenfeld supposed that a Lie
derivative of spinor eld exists such that the partial derivative commutes with the Lie derivative.
A Lie derivative of spinor eld with respect to a Killing vector eld had been rst constructed by
A. Lichnerowicz (see 5]), and then Y. Kosmann postulated a formula for Lie derivative of spinor
by analogy with the Lie derivative of tensor eld (see 6]). An explanation for this formula was
given in 7]. However, Y. Kosman's version of Lie derivative has not property that the commutator
of Lie derivatives is the Lie derivative with respect to commutator, to say nothing of the property
that the partial derivative commutes with the Lie derivative.
The problem on constructing the Lie derivative of a spinor eld was solved in 8], 9], where
this Lie derivative was used to construct the energy-momentum tensor of spinor elds in the spacetime of general relativity on the basis of Noether's theorem. In 8], 9] the method of induced
representations was used in order to expand the spinor representation of the Lorentz group O(1 3)
to a representation of the general linear group GL(4) of four-dimensional space. The space of
representation is also expanded, this one is the spinor space multiplied by the space of symmetric
forms of type (2 0), which determine the quadratic form of signature (+ ; ; ;). With the use
of the constructed representation the spinors, which earlier were treated as elements of a bundle
associated with the principal bundle of orthonormal frames, now are considered as elements of a
bundle associated with the principal bundle of linear frames.
In this article we generalize results from 8], 9] to arbitrary Riemannian manifolds. We demonstrate that our construction of spinors on a Riemannian manifold depends essentially on a choice
of sections of a principal bundle with total space GL(n) and structure group O(p q), n = p + q.
In addition, we prove that, if the Lagrange function of spinor eld is invariant with respect to the
action of structure group on the associated spinor bundle, then the corresponding theory of spinor
eld is gauge invariant in case the gauge transformation is a change of section.
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