Inequivalent representations in the functional integral framework

An important feature of Quantum Field Theory is the existence of unitarily inequivalent representations of canonical commutation relations. When one works with the functional integral formalism, it is not clear, however, how this feature emerges. By following the seminal work of M. Swanson on canonical transformations in phase-space path integral, we generalize his approach to coherent-state functional integrals which in turn will lead to a simplified formalism which makes the appearance of the inequivalent representations more transparent.