# Functional integrals and inequivalent representations in Quantum Field Theory

## TL;DR

This paper explores how canonical transformations in Quantum Field Theory affect functional integrals and representations, revealing their role in phase transitions and illustrating with models like the van Hove and Bogoliubov systems.

## Contribution

It derives functional-integral representations for linear canonical transformations and clarifies their impact on inequivalent representations in Quantum Field Theory.

## Key findings

- Canonical transformations induce transitions between inequivalent representations.
- The energy gap relates to continuous phase transitions.
- Illustrations include the van Hove and Bogoliubov models.

## Abstract

We discuss canonical transformations in Quantum Field Theory in the framework of the functional-integral approach. In contrast with ordinary Quantum Mechanics, canonical transformations in Quantum Field Theory are mathematically more subtle due to the existence of unitarily inequivalent representations of canonical commutation relations. When one works with functional integrals, it is not immediately clear how this algebraic feature manifests itself in the formalism. Here we attack this issue by considering the canonical transformations in the context of coherent-state functional integrals. Specifically, in the case of linear canonical transformations, we derive the general functional-integral representations for both transition amplitude and partition function phrased in terms of new canonical variables. By means of this, we show how in the infinite-volume limit the canonical transformations induce a transition from one representation of canonical commutation relations to another one and under what conditions the representations are unitarily inequivalent. We also consider the partition function and derive the energy gap between statistical systems described in two different representations which, among others, allows to establish a connection with continuous phase transitions. We illustrate the inner workings of the outlined mechanism by discussing two prototypical systems: the van Hove model and the Bogoliubov model of weakly interacting Bose gas.