  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.
