On the choice of coupling procedure for the Poincar\'e gauge theory of gravity

TL;DR

This paper addresses the ambiguity in the minimal coupling procedure within Poincaré gauge theory of gravity, proposing a modification that clarifies matter coupling and refines predictions, especially for Einstein--Cartan theory with fermions.

Contribution

It introduces a slight modification to the minimal coupling procedure in Poincaré gauge gravity, resolving ambiguities and justifying previous results without radically changing the theory's predictions.

Findings

The modification removes ambiguity in matter coupling.

Predictions of Einstein--Cartan theory with fermions are made unique.

Torsion singularities in the Proca field are shifted to different values.

Abstract

The gauge approach to the theory of gravity has been widely discussed as an alternative to standard general relativity. The Poincar{\'e} group, as a symmetry group of all relativistic theories in the absence of gravitation, constitutes the most natural candidate for a gauge group. Although the Poincar{\'e} gauge theory of gravity has been elaborated over the years and cast into a beautiful formal framework, some fundamental problems have remained unsolved. One of them concerns the inclusion of matter. The minimal coupling procedure, which is employed in standard Yang--Mills theories, appears to be ambiguous in the case of gravity. We propose a slight modification of this procedure, which removes the ambiguity. Our modification justifies some earlier results concerning the consequences of the Poincar{\'e} gauge theory of gravity. In particular, the predictions of Einstein--Cartan theory with fermionic matter are rendered unique. We recall the earlier proposed solution based on modified volume--forms. The advantage of our modification is that the predictions of the theory are not radically changed. Basically, this modification simply justifies the results that were obtained partly `by chance' in the hitherto prevailing accounts on the Einstein--Cartan theory. The only difference in the predictions, when compared to the standard treatment, concerns the Proca field in the presence of gravity. The `torsion singularities' which occur there are shifted towards other values of the field.