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

When one tries to take into account the nontrivial 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 one 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 us 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.

Figure 1

The complex time contour $\mathcal{C}$ in the Schwinger CTP formalism. Dots on the forward and the backward branches of the contour denote discrete time points.