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

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.

Unitarily inequivalent vacua and long-range forces: Phenomenology with scalar boson mass-shift

“…In fact, issues with boundary conditions in general situations involving unitarily inequivalent vacua have been shown to require quite subtle treatments under the functional approach. 18,19 Hence, instead of using the aforementioned functional technique, we proceed with the more standard diagrammatic approach for calculating potentials from the Lorentzinvariant Feynman amplitudes. Although this method is fully equivalent to the above functional approach, 17 its physical basis arises much more transparently in the actual calculation.…”

Neutrino Dynamics in a Quantum-Corrected Schwarzschild Spacetime

“…Although some basic puzzles in the physics of neutrinos have been resolved, especially since the solar neutrino problem [4][5][6][7] has been understood and explained in terms of the neutrino oscillation mechanism [8,9] together with the MSW effect [10,11], some fundamental questions remain unanswered, such as whether neutrinos are Dirac or Majorana fermions. 1 Another important open issue concerns the impossibility of relating the flavor and the mass bases by means of unitary transformations in the framework of quantum field theory [13][14][15][16][17][18].…”