Extended Einstein-Cartan theory a la Diakonov: the field equations

TL;DR

This paper derives and extends a gravitational model based on the Einstein-Cartan theory within the Poincaré gauge framework, eliminating geometric variables in favor of a primordial spinor and exploring resulting nonlinear spinor equations.

Contribution

It derives the field equations for Diakonov's model and extends it by eliminating the Lorentz connection while maintaining covariance, revealing a nonlinear spinor equation.

Findings

Derived the first field equations for Diakonov's model.

Extended the model by eliminating the Lorentz connection.

Recovered a nonlinear Heisenberg-type spinor equation.

Abstract

Diakonov formulated a model of a primordial Dirac spinor field interacting gravitationally within the geometric framework of the Poincar\'e gauge theory (PGT). Thus, the gravitational field variables are the orthonormal coframe (tetrad) and the Lorentz connection. A simple gravitational gauge Lagrangian is the Einstein-Cartan choice proportional to the curvature scalar plus a cosmological term. In Diakonov's model the coframe is eliminated by expressing it in terms of the primordial spinor. We derive the corresponding field equations for the first time. We extend the Diakonov model by additionally eliminating the Lorentz connection, but keeping local Lorentz covariance intact. Then, if we drop the Einstein-Cartan term in the Lagrangian, a nonlinear Heisenberg type spinor equation is recovered in the lowest approximation.