Spinors in Terms of Arbitrary Frames. Covariant Derivative and the Lie

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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 by Allerton Press Inc Authorization to photo copy individual items for internal or p ersonal use or the internal or p ersonal use of sp ecic clients is granted by Allerton Press Inc for libraries and other users registered with the Copyright Clearance Center CCC Transactional Rep orting Service provided that the base fee of p er copy is paid directly to CCC Rosewo o d Drive Danvers MA .

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