Russian Mathematics Iz VUZ Izvestiya VUZ Matematika

Vol No pp UDC

SPINORS IN TERMS OF ARBITRARY FRAMES

COVARIANT DERIVATIVE AND THE LIE DERIVATIVE OF SPINORS

RF Bilyalov and BS Nikitin

Intro duction

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

rediscovered in byvan der Waerden see in connection with Diracs physical investigations

The spinors give a linear representation of the rotation group of a space of n dimensions and each

spinor is determined by comp onents n or The spinors of a fourdimensional space

app eared in the famous Dirac equation for an electron where the four wave functions are nothing

else but the comp onents of a spinor

The theory of the spinors in a Riemannian spacetime of the general theory of relativity was

develop ed in The symmetric energymomentum tensor for the spinor elds with arbitrary

Lagrange functions was found by L Rosenfeld in see Rosenfelds arguments when

No ether theorem for construction of the conservation laws were not covariant he applied the

in character the covariant form of the conservation laws was attained as a result of ingenious

transformations Acovariant construction of the symmetric energymomentum tensor for the spinor

elds was obtained in by applying the Lie derivative of a spinor which was suggested by

Y Kosmann see and considered in from the grouptheory viewp oint The grouptheory

approach to the Lie derivative of a spinor leads to the necessity to consider the spinors in terms of

arbitrary nonorthogonal frames By this reason a problem of developing spinor analysis in terms of

arbitrary frames arises and rst of all one arrives at the problem of construction of the covariant

and Lie derivatives of spinors in terms of arbitrary frames

The group Spin

Consider the Minkowski spacetime referred to an orthogonal Cartesian co ordinate system We

denote by its metric tensor In what follows

 

we assume that capital Latin indices run through values while all remaining indices

through Let us intro duce a system of four Dirac matrices with the complex

A

elements which satisfy the relations

B

I

where I stands for the unit matrix The Dirac matrices are determined up to the choice of

a basis in the dimensional complex space In the theoretical physics they usually are chosen as

c

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