A Master Functional For Quantum Field Theory

TL;DR

This paper introduces the master functional in quantum field theory, a new generating functional invariant under general field transformations, facilitating a more robust and scalar behavior-based formulation of quantum correlations and renormalization.

Contribution

It proposes the master functional as a new scalar-invariant generating functional, extending the formalism with proper fields and improved transformations for better quantum field analysis.

Findings

Master functional is invariant under general field transformations.

Proper formulation allows calculations without passing through traditional functionals.

Classical action matches the master functional's classical limit.

Abstract

We study a new generating functional of one-particle irreducible diagrams in quantum field theory, called master functional, which is invariant under the most general perturbative changes of field variables. The usual functional Gamma does not behave as a scalar under the transformation law inherited from its very definition as the Legendre transform of W = ln Z, although it does behave as a scalar under an unusual transformation law. The master functional, on the other hand, is the Legendre transform of an improved functional W with respect to the sources coupled to both elementary and composite fields. The inclusion of certain improvement terms in W and Z is necessary to make the new Legendre transform well defined. The master functional behaves as a scalar under the transformation law inherited from its very definition. Moreover, it admits a proper formulation, obtained extending the set of integrated fields to so-called proper fields, which allows us to work without passing through Z, W or Gamma. In the proper formulation the classical action coincides with the classical limit of the master functional, and correlation functions and renormalization are calculated applying the usual diagrammatic rules to the proper fields. Finally, the most general change of field variables, including the map relating bare and renormalized fields, is a linear redefinition of the proper fields.