On operator fields in the bundle of Dirac spinors.pdf

a r X i v : 0 8 0 2 .1 4 9 1 v 1 [ m a t h .D G ] 1 1 F e b 2 0 0 8 ON OPERATOR FIELDS IN THE BUNDLE OF DIRAC SPINORS. R. A. Sharipov Abstract. Operator fields in the bundle of Dirac spinors and their conversion to spatial fields are considered. Some commutator equations are studied with the use of the conversion technique. 1. Introduction. The bundle of Dirac spinors is used for describing particles with half-integer spin in general relativity and in quantum field theory. It is a special four-dimensional complex vector-bundle over the space-time manifold M . Let’s remind that the space-time manifold M itself is a four-dimensional real manifold equipped with a Minkowski type metric g of the signature (+,?,?,?). Apart from g, the space- time manifoldM is equipped with two other geometric structures — the orientation and the polarization. The orientation distinguishes right quadruples of tangent vectors from left ones, while the polarization distinguishes future and past half light cones in tangent spaces at each point of M . The bundle of Dirac spinors is denoted DM . It is equipped with four basic spin-tensorial fields in addition to g. They are presented in the following table. Symbol Name Spin-tensorial type d Skew-symmetric metric tensor (0, 2|0, 0|0, 0) H Chirality operator (1, 1|0, 0|0, 0) D Dirac form (0, 1|0, 1|0, 0) γ Dirac γ-field (1, 1|0, 0|1, 0) The metric tensor g itself is interpreted as a spin-tensorial field of the spin-tensorial type (0, 0|0, 0|0, 2). In this paper, saying an operator field, we assume a spin-tensorial field F of the spin-tensorial type (1, 1|0, 0|0, 0). In the coordinate form it is presented by a matrix 2000 Mathematics Subject Classification. 53B30, 81T20, 81R25. Typeset by AMS-TEX 2 R. A. SHARIPOV F ab , where a and b are two spinor indices. Each operator field F in the bundle of Dirac spinors has a unique presentation of the following form: F ab = u δ a b + v H a b + 3∑ k=0 γakb uk + + 3∑ k=0 4∑ c=1 Hac γ ck b vk + 3∑ p=0