A General Field-Covariant Formulation Of Quantum Field Theory

TL;DR

This paper develops a comprehensive, covariant framework for quantum field theory that treats all perturbative field variable changes as true variable transformations, simplifying the renormalization process and relating different variable frames.

Contribution

It introduces a general field-covariant approach to QFT that handles nonlinear field changes as true variable transformations, unifying renormalization across different variable frames.

Findings

Field changes can be expressed as source-dependent redefinitions of composite fields.

Renormalization in new variable frames can be derived from the old without recalculating.

The approach includes explicit examples and a linear transformation method.

Abstract

In all nontrivial cases renormalization, as it is usually formulated, is not a change of integration variables in the functional integral, plus parameter redefinitions, but a set of replacements, of actions and/or field variables and parameters. Because of this, we cannot write simple identities relating bare and renormalized generating functionals, or generating functionals before and after nonlinear changes of field variables. In this paper we investigate this issue and work out a general field-covariant approach to quantum field theory, which allows us to treat all perturbative changes of field variables, including the relation between bare and renormalized fields, as true changes of variables in the functional integral, under which the functionals Z and W = ln Z behave as scalars. We investigate the relation between composite fields and changes of field variables, and show that, if J are the sources coupled to the elementary fields, all changes of field variables can be expressed as J-dependent redefinitions of the sources L coupled to the composite fields. We also work out the relation between the renormalization of variable-changes and the renormalization of composite fields. Using our transformation rules it is possible to derive the renormalization of a theory in a new variable frame from the renormalization in the old variable frame, without having to calculate it anew. We define several approaches, useful for different purposes, in particular a linear approach where all variable changes are described as linear source redefinitions. We include a number of explicit examples.