Russian Iz VUZ Izvestiya VUZ Matematika

Vol No pp UDC

SPINORS ON THE RIEMANNIAN

RF Bilyalov

Intro duction

In the most general mathematical form the spinors were intro duced by E Cartan in see

In the spinors were discovered again byvan der Waerden see in connection with Diracs

physical investigations VA Fock and DD Ivanenko implemented the spinors into the general

relativity see In L Rosenfeld suggested a general pro cedure for constructing the energy

momentum of material elds which stands in the righthand side of Einsteins equations

see L Rosenfelds metho d essentially involves Construction of Lie derivative

for tensor elds encounters no diculties but this is not true for spinor elds In the deduction

of spinor elds L Rosenfeld supp osed that a Lie of a formula for the energymomentum tensor

derivativeof spinor eld exists suchthat the partial derivativecommutes with the Lie derivative

A Lie derivative of spinor eld with resp ect to a Killing vector eld had b een rst constructed by

A Lichnerowicz see and then Y Kosmann p ostulated a formula for Lie derivative of spinor

by analogy with the Lie derivative of tensor eld see An explanation for this formula was

given in However Y Kosmans version of Lie derivative has not prop erty that the

of Lie derivatives is the Lie derivative with resp ect to commutator to say nothing of the prop erty

that the partial derivativecommutes with the Lie derivative

The problem on constructing the Lie derivative of a spinor eld was solved in where

this Lie derivativewas used to construct the energymomentum tensor of spinor elds in the space

time of on the of No ethers theorem In the metho d of induced

representations was used in order to expand the spinor representation of the Lorentz O

to a representation of the general linear group GL of fourdimensional space The space of

representation is also expanded this one is the spinor space multiplied by the space of symmetric

forms of typ e which determine the of signature With the use

of the constructed representation the spinors which earlier were treated as elements of a bundle

asso ciated with the principal bundle of orthonormal frames now are considered as elements of a

bundle asso ciated with the principal bundle of linear frames

In this article we generalize results from to arbitrary Riemannian manifolds We demon

strate that our construction of spinors on a Riemannian dep ends essentially on a choice

of sections of a principal bundle with total space GLn and structure group O p q n p q

In addition we prove that if the Lagrange function of spinor eld is invariant with resp ect to the

action of structure group on the asso ciated then the corresp onding theory of spinor

eld is gauge invariantincase the gauge transformation is a change of section

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