The self-coupled Einstein–Cartan–Dirac equations in terms of Dirac bilinears

Abstract

In this article we present the algebraic rearrangement, or matrix inversion of the Dirac equation in a curved Riemann-Cartan spacetime with torsion; the presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the Dirac field. We then demonstrate how the inversion leads to a reformulation of the fully non-linear and self-interactive Einstein-Cartan-Dirac field equations in terms of Dirac bilinears. It has been known for some decades that the Dirac equation for charged fermions interacting with an electromagnetic field can be algebraically inverted, so as to obtain an explicit rational expression of the four-vector potential of the gauge field in terms of the spinors. Substitution of this expression into Maxwell's equations yields the bilinear form of the self-interactive Maxwell-Dirac equations. In the present (purely gravitational) case, the inversion process yields two rational four-vector expressions in terms of Dirac bilinears, which act as gravitational analogues of the electromagnetic vector potential. These "potentials" also appear as irreducible summand components of the connection, along with a traceless residual term of mixed symmetry. When taking the torsion field equation into account, the residual term can be written as a function of the object of anholonomity. Using the local tetrad frame associated with observers co-moving with the Dirac matter, a generic vierbein frame can described in terms of four Dirac bilinear vector fields, normalized by a scalar and pseudoscalar field. A corollary of this is that in regions where the Dirac field is non-vanishing, the self-coupled Einstein-Cartan-Dirac equations can in principle be expressed in terms of Dirac bilinears only.

Gauge invariant formulation of the self-interacting Duffin-Kemmer-Petiau equations

“…In this letter we have given a gauge invariant reformulation of the Maxwell-DKP equations, in terms of the set of real bilinear DKP currents. Our work provides the basis for a systematic examination of classical solutions of the Maxwell-DKP equations under different spacetime symmetry group reductions [19] , and for the development of the bilinear method in an Einstein-Cartan setting [20] . We expect analogous methods to be applicable also to the 10 component DKP system.…”

On the Mathematics of Coframe Formalism and Einstein–Cartan Theory—A Brief Review

“…We will focus more on the geometrical side of these equations and we will not dwell on deepening matter interaction (couplings, symmetry breaking, etc. ), as done for instance in References [20,21,24,22,23].…”

“…ECSK theory with cosmological constant belongs to the Lovelock-Cartan family, which describes the most general action in four dimensions such that this action is a polynomial on the tetrads and the spin connection (including derivatives), is invariant under diffeomorphisms and local Lorentz transformations, and is constructed without the Hodge dual 23 .…”

“…Some classical works about ECSK theory and General Relativity with torsion, like References[11,12,13] 23. See Reference[25] for details.…”

Essential Fierz identities for a fermionic field

“…In fact, in every space-time event, the bilinear forms are algebraic quantities, in the sense that their definition does not depend on the dynamics (Dirac, Klein-Gordon equations) associated with the field. Moreover, the products of any two bilinear forms, when expressed in terms of linear combinations of all of them, satisfy certain algebraic relations, which are usually referred as Fierz identities or Pauli-Fierz-Kofink identities [3,4]. They are derived from the completeness relations that give the canonical basis of the exterior algebra in terms of the basis under consideration { } A A 1 16 G = (see, for instance, [5][6][7]).…”