Frame Covariant Formalism for Fermionic Theories

TL;DR

This paper develops a covariant formalism for fermionic quantum field theories using the Vilkovisky-DeWitt approach, enabling frame- and reparametrisation-invariant descriptions by constructing a suitable field-space supermanifold and metric.

Contribution

It introduces a novel, covariant framework for fermionic theories, defining a field-space supermanifold and metric to achieve invariance under reparametrisations.

Findings

Constructed a field-space supermanifold for fermionic fields.

Defined a covariant metric on the supermanifold.

Extended the Vilkovisky-DeWitt effective action to fermionic theories.

Abstract

We present a frame- and reparametrisation-invariant formalism for quantum field theories that include fermionic degrees of freedom. We achieve this using methods of field-space covariance and the Vilkovisky-DeWitt (VDW) effective action. We explicitly construct a field-space supermanifold on which the quantum fields act as coordinates. We show how to define field-space tensors on this supermanifold from the classical action that are covariant under field reparametrisations. We then employ these tensors to equip the field-space supermanifold with a metric, thus solving a long-standing problem concerning the proper definition of a metric for fermionic theories. With the metric thus defined, we use well-established field-space techniques to extend the VDW effective action and express any fermionic theory in a frame- and field-reparametrisation-invariant manner.