Unitary evolution, canonical variables and vacuum choice for general quadratic Hamiltonians in spatially homogeneous and isotropic space-times

TL;DR

This paper investigates the conditions under which free scalar fields in homogeneous and isotropic space-times admit a unitary evolution, identifying unique canonical variables and vacuum states through asymptotic analysis and action angle variables.

Contribution

It establishes the precise conditions on Hamiltonians for unitary evolution, derives the unique canonical variables, and links vacuum choices with canonical transformations using advanced asymptotic methods.

Findings

Identifies conditions for unitary evolution in scalar field quantization.

Derives unique canonical variables that ensure unitary evolution.

Connects adiabatic vacuum conditions with canonical transformations and Hamiltonian diagonalization.

Abstract

Quantization of arbitrary free scalar fields in spatially homogeneous and isotropic space-times is considered. The quantum representation allowing a unitary evolution for the fields is taken as a requirement for the theory. Studying the group of linear canonical transformations, we show the relations between unitary evolution and choice of canonical variables. From these relations we obtain the conditions on the Hamiltonian such that there are canonical variables for which the field has unitary evolution. We then compute the linear transformation leading to these variables, also proving that they are unique. We obtain these results by developing the asymptotic analysis of the fields using the action angle variables, which proves to be a generalization of the usual Wentzel-Kramers-Brillouin approximation. These tools allow us to re-frame the adiabatic vacuum condition in a extensible format by using the action angle variables to relate these vacuum choices to those where the particle number density does not depend on angle (fast) variables. Finally, we develop a larger set of canonical variables relating the adiabatic vacuum conditions with the smearing of the quantum fields. This set of canonical variables also connects the adiabatic vacuum conditions with the instantaneous Hamiltonian diagonalization vacuum choice.