Particle creation, classicality and related issues in quantum field theory: I. Formalism and toy models

TL;DR

This paper develops a formalism for analyzing quantum harmonic oscillators with time-dependent frequencies, addressing conceptual issues like classicality and particle content, and applies it to toy models with exact and approximate solutions.

Contribution

It introduces a parametrization of the wave function using an excitation parameter to generalize particle and related concepts in time-dependent quantum systems.

Findings

Exact solutions for a toy model are derived.

Approximate solutions are obtained for adiabatic and non-adiabatic cases.

Numerical results are compared with analytic approximations.

Abstract

The quantum theory of a harmonic oscillator with a time dependent frequency arises in several important physical problems, especially in the study of quantum field theory in an external background. While the mathematics of this system is straightforward, several conceptual issues arise in such a study. We present a general formalism to address some of the conceptual issues like the emergence of classicality, definition of particle content, back reaction etc. In particular, we parametrize the wave function in terms of a complex number (which we call excitation parameter) and express all physically relevant quantities in terms it. Many of the notions -- like those of particle number density, effective Lagrangian etc., which are usually defined using asymptotic in-out states -- are generalized as time-dependent concepts and we show that these generalized definitions lead to useful and reasonable results. Having developed the general formalism we apply it to several examples. Exact analytic expressions are found for a particular toy model and approximate analytic solutions are obtained in the extreme cases of adiabatic and highly non-adiabatic evolution. We then work out the exact results numerically for a variety of models and compare them with the analytic results and approximations. The formalism is useful in addressing the question of emergence of classicality of the quantum state, its relation to particle production and to clarify several conceptual issues related to this. In Paper II (arXiv:0708.1237), which is a sequel to this, the formalism will be applied to analyze the corresponding issues in the context of quantum field theory in background cosmological models and electric fields.