Ricci Coefficients in Covariant Dirac Equation, Symmetry Aspects and Newman-Penrose Approach

Details Ricci Coefficients in Covariant Dirac Equation, Symmetry Aspects and Newman-Penrose Approach Ricci Coefficients in Covariant Dirac Equation, Symmetry Aspects and Newman-Penrose Approach

2011-09-05

arxiv.org/abs/1109.0819v1

NPCS (Nonlinear Phenomena in Complex Systems). 2004, Vol. 7, no 3. p. 250-262

Physics

Mathematical Physics

17 pages

Scientific paper

The paper investigates how the Ricci rotation coefficients act in the Dirac equation in presence of external gravitational fields described in terms of Riemannian space-time geometry. It is shown that only 8 different combinations of the Ricci coefficients \gamma_{abc}(x) are involved in the Dirac equation. They are combined in two 4-vectors B_{a}(x) and C_{a}(x) under local Lorentz group which has status of the gauge symmetry group. In all orthogonal coordinates one of these vectors, "pseudovector" C_{a}(x), vanishes identically. The gauge transformation laws of the two vectors are found explicitly. Connection of these B_{a}(x) and A_{a}(x) with the known Newman-Penrose coefficients is established. General study of gauge symmetry aspects in Newman-Penrose formalism is performed. Decomposition of the Ricci object, "tensor" \gamma_{abc}(x), into two "spinors" \gamma(x) and \bar{\gamma}(x) is done. At this Ricci rotation coefficients are divided into two groups:12 complex functions \gamma(x) and 12 conjugated to them \bar{\gamma}(x). Components of spinor \bar{\gamma}(x) coincide with 12 spin coefficients by Newman-Penrose. The formulas for gauge transformations of spin coefficients under local Lorentz group are derived. There are given two solutions to the gauge problem: one in the compact form of transformation laws for spinors \gamma(x) and \bar{\gamma}(x), and another as detailed elaboration of the latter in terms of 12 spin coefficients.

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