Dirac equation with self interaction induced by torsion

Abstract

A formulation of Dirac equation with non linear self interacting term induced by torsion is studied in Minkowski space-time. The equation is made explicit in Cartesian coordinates by a null tetrad frame whose corresponding van der Waerden matrices are the Pauli matrices. Plane waves are shown to be solution. They are subject to a constraint relation that leave only two of them arbitrary. They are determined in a special case. In the two dimensional space-time one is left with a non linear Dirac equation whose solution is ensured by existing results. Analytical solutions are determined in form of standing waves. In the mass less case there are solutions that propagate with periodic "burst" of amplitude. The paper is also an improvement of a previous study.

Einstein-Cartan-Dirac equations in the Newman-Penrose formalism

“…It can be seen upon the substitutions λ = 0.5 (equivalently a = √ 2) and m = 1 (equivalently m 0 = −1), that this is a generalisation of the equations for A(z) and B(z) in [21](see section III). A similar solution is found in [16], with a(l 1 ) = a(L pl ) and b(l 2 ) = b(λ c ). In terms of the spinor components:…”

“…For previous work on plane wave solutions of the non-linear Dirac equation see [16,19]. Our work in this section provides a more detailed analysis.…”

Non linear spin 1 field equation in curved space-time: separation and solution in static RW metric

“…Another open problem of interest is of determining explicit solution of the field equation, at least for the lowest spin values, in some specific space time model. (For recent models see e.g., [8]; for the non linear Dirac equation see e.g., [15,16,17]).…”