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When one tries to take into account the non-trivial vacuum structure ofQuantum Field Theory, the standard functional-integral tools such as generatingfunctionals or transitional amplitudes, are often quite inadequate for suchpurposes. Here we propose a generalized generating functional for Green'sfunctions which allows to easily distinguish among a continuous set of vacuathat are mutually connected via unitary canonical transformations. In order tokeep our discussion as simple as possible, we limit ourselves to QuantumMechanics where the generating functional of Green's functions is constructedby means of phase-space path integrals. The quantum-mechanical setting allowsto accentuate the main logical steps involved without embarking on technicalcomplications such as renormalization or inequivalent representations thatshould otherwise be addressed in the full-fledged Quantum Field Theory. Weillustrate the inner workings of the generating functional obtained bydiscussing Green's functions among vacua that are mutually connected viatranslations and dilatations. Salient issues, including connection with QuantumField Theory, vacuum-to-vacuum transition amplitudes and perturbation expansionin the vacuum parameter are also briefly discussed.

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