The time-dependent quantum harmonic oscillator revisited: Applications to Quantum Field Theory

TL;DR

This paper rigorously analyzes the unitary time evolution of infinite collections of time-dependent harmonic oscillators, with applications to quantum field theory and cosmological models, using novel techniques for transition amplitudes and semiclassical states.

Contribution

It introduces a rigorous framework for the unitary evolution of infinite harmonic oscillators and extends the analysis to quantum field theory and cosmological models.

Findings

Established conditions for unitary implementability of dynamics.

Constructed Schrödinger functional representations for infinite systems.

Identified semiclassical states in quantum field and cosmological models.

Abstract

In this article, we formulate the study of the unitary time evolution of systems consisting of an infinite number of uncoupled time-dependent harmonic oscillators in mathematically rigorous terms. We base this analysis on the theory of a single one-dimensional time-dependent oscillator, for which we first summarize some basic results concerning the unitary implementability of the dynamics. This is done by employing techniques different from those used so far to derive the Feynman propagator. In particular, we calculate the transition amplitudes for the usual harmonic oscillator eigenstates and define suitable semiclassical states for some physically relevant models. We then explore the possible extension of this study to infinite dimensional dynamical systems. Specifically, we construct Schroedinger functional representations in terms of appropriate probability spaces, analyze the unitarity of the time evolution, and probe the existence of semiclassical states for a wide range of physical systems, particularly, the well-known Minkowskian free scalar fields and Gowdy cosmological models.