Generalized generating functional for mixed-representation Green's functions: A quantum mechanical approach

TL;DR

This paper introduces a generalized generating functional for Green's functions in quantum mechanics, enabling the distinction of multiple vacua connected by canonical transformations, with potential applications to quantum field theory.

Contribution

It proposes a new generalized generating functional that distinguishes among different vacua connected by unitary transformations, simplifying the analysis of non-trivial vacuum structures.

Findings

Functional distinguishes vacua connected by translations and dilatations.

Framework clarifies vacuum transition amplitudes and perturbation expansions.

Simplified quantum mechanics setting highlights core ideas without field-theoretic complexities.

Abstract

When one tries to take into account the non-trivial vacuum structure of Quantum Field Theory, the standard functional-integral tools such as generating functionals or transitional amplitudes, are often quite inadequate for such purposes. Here we propose a generalized generating functional for Green's functions which allows to easily distinguish among a continuous set of vacua that are mutually connected via unitary canonical transformations. In order to keep our discussion as simple as possible, we limit ourselves to Quantum Mechanics where the generating functional of Green's functions is constructed by means of phase-space path integrals. The quantum-mechanical setting allows to accentuate the main logical steps involved without embarking on technical complications such as renormalization or inequivalent representations that should otherwise be addressed in the full-fledged Quantum Field Theory. We illustrate the inner workings of the generating functional obtained by discussing Green's functions among vacua that are mutually connected via translations and dilatations. Salient issues, including connection with Quantum Field Theory, vacuum-to-vacuum transition amplitudes and perturbation expansion in the vacuum parameter are also briefly discussed.