Master Functional And Proper Formalism For Quantum Gauge Field Theory

TL;DR

This paper introduces a comprehensive, covariant formalism for quantum gauge theories using a master functional that extends the Batalin-Vilkovisky approach, enabling clearer handling of field redefinitions and proving renormalizability.

Contribution

It develops a new master functional framework that simplifies field redefinitions, gauge fixing, and proves gauge theory renormalizability in a covariant manner.

Findings

The master functional Omega unifies 1PI diagrams with proper fields.

Proper canonical transformations encode all perturbative field redefinitions.

The approach proves the renormalizability of gauge theories without anomalies.

Abstract

We develop a general field-covariant approach to quantum gauge theories. Extending the usual set of integrated fields and external sources to "proper" fields and sources, which include partners of the composite fields, we define the master functional Omega, which collects one-particle irreducible diagrams and upgrades the usual Gamma-functional in several respects. The functional Omega is determined from its classical limit applying the usual diagrammatic rules to the proper fields. Moreover, it behaves as a scalar under the most general perturbative field redefinitions, which can be expressed as linear transformations of the proper fields. We extend the Batalin-Vilkovisky formalism and the master equation. The master functional satisfies the extended master equation and behaves as a scalar under canonical transformations. The most general perturbative field redefinitions and changes of gauge-fixing can be encoded in proper canonical transformations, which are linear and do not mix integrated fields and external sources. Therefore, they can be applied as true changes of variables in the functional integral, instead of mere replacements of integrands. This property overcomes a major difficulty of the functional Gamma. Finally, the new approach allows us to prove the renormalizability of gauge theories in a general field-covariant setting. We generalize known cohomological theorems to the master functional and show that when there are no gauge anomalies all divergences can be subtracted by means of parameter redefinitions and proper canonical transformations.