**NONLINEAR
PHENOMENA IN COMPLEX SYSTEMS**

An Interdisciplinary Journal
* 2004, Vol.7, No.3, pp.250-262*

**Ricci Coefficients in Covariant Dirac Equation,
Symmetry Aspects and Newmann-Penrose Approach.**

*V.M. Red'kov*
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 *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 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
*L*_{i} , *N*_{i} , *M*_{i},
.
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
detailed elaboration
of the latter in terms of 12 spin coefficients.

*Key words: *
Dirac equation, Ricci and Newmann-Penrose
coefficients,
gauge symmetry

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Last updated: *November 8, 2004*