Conformal scalar and torsion do not go together

TL;DR

This paper explores the conformal transformation properties of the VEP action, revealing that torsion and conformal invariance of scalar fields are incompatible when fermionic matter is included.

Contribution

It demonstrates that conformal invariance of scalar fields cannot be maintained in the presence of torsion induced by fermions within the VEP formalism.

Findings

Torsion appears in the dynamics of conformal scalar fields with fermions.

Conformal invariance of scalar equations breaks down when fermionic matter is present.

The conformal transformation of the spin connection is indeterminate off-shell.

Abstract

We investigate the conformal transformation of vierbein-Einstein-Palatini (VEP) action in terms of tetrads $e^I_\mu$ and spin connection $A^{IJ}_\mu$. The transformation of the spin connection is indeterminate off-shell unless equations of motion are satisfied. We construct the conformally invariant scalar field in the torsion-free VEP formalism. In presence of fermionic matter, torsion does not vanish, and shows up in the dynamics of conformal scalar, affecting the invariance. It is not possible to maintain conformal invariance of the scalar field equation when fermions are present.