2004, Vol.7, No.3, pp.250-262
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 are involved in the Dirac equation.
They are combined in two 4-vectors Ba(x) and Ca(x) under local
Lorentz group which has status of the gauge symmetry group.
In all orthogonal coordinates one of these vectors, "pseudovector"
Ca(x), vanishes identically.
The gauge transformation laws of the two vectors
are found explicitly. Connection of these Ba(x) and Aa(x)
with the known Newmann-Penrose coefficients is established.
General study of gauge symmetry aspects in Newmann-Penrose formalism is performed.
Decomposition of the Ricci object, "tensor" ,
into two "spinors"
and is done. At this Ricci rotation coefficients are divided into two groups:
12 complex functions
and 12 conjugated to them .
Components of spinor coincide with
12 spin coefficients by Newmann-Penrose
For listing these it is used a special letter-notation
Li , Ni , Mi,
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
and , and another as
of the latter in terms of 12 spin coefficients.
Key words: Dirac equation, Ricci and Newmann-Penrose coefficients, gauge symmetry
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