  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.
