Functional integrals and inequivalent representations in Quantum Field Theory
Abstract
We discuss canonical transformations in Quantum Field Theory in the framework of the functionalintegral 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 coherentstate functional integrals. Specifically, in the case of linear canonical transformations, we derive the general functionalintegral representations for both transition amplitude and partition function phrased in terms of new canonical variables. By means of this, we show how in the infinitevolume 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.
 Publication:

Annals of Physics
 Pub Date:
 August 2017
 DOI:
 10.1016/j.aop.2017.05.022
 arXiv:
 arXiv:1703.04314
 Bibcode:
 2017AnPhy.383..207B
 Keywords:

 Quantum field theory;
 Functional integral;
 Canonical transformation;
 Coherent state;
 High Energy Physics  Theory;
 High Energy Physics  Phenomenology;
 Mathematical Physics;
 Mathematics  Functional Analysis;
 Quantum Physics
 EPrint:
 42 pages