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

TL;DR

This paper explores the role of Ricci rotation coefficients in the Dirac equation within curved spacetime, linking them to Newman-Penrose coefficients and analyzing gauge symmetries in this formalism.

Contribution

It provides a detailed decomposition of Ricci rotation coefficients into spinors and establishes explicit gauge transformation laws, connecting them with Newman-Penrose formalism.

Findings

Identifies 8 key Ricci coefficient combinations in the Dirac equation.

Connects Ricci coefficients with Newman-Penrose spin coefficients.

Derives explicit gauge transformation laws for spin coefficients.

Abstract

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.