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

Sandro D. P. Vitenti|arXiv (Cornell University)|May 6, 2015

Quantum Electrodynamics and Casimir Effect5 citations

TL;DR

This paper establishes the conditions under which free scalar fields in spatially homogeneous and isotropic space-times admit a unitary time evolution via a specific choice of canonical variables. By analyzing time-dependent linear canonical transformations and using action-angle variable asymptotics, it derives a unique set of canonical variables that ensure unitary evolution, generalizing the adiabatic vacuum condition and linking it to instantaneous Hamiltonian diagonalization and field smearing.

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.

Motivation & Objective

• To identify the necessary conditions on a general quadratic Hamiltonian for which a unitary time evolution exists in spatially homogeneous and isotropic space-times.
• To determine the unique set of canonical variables that realize unitary evolution for such Hamiltonians.
• To generalize the adiabatic vacuum condition using action-angle variable asymptotics, linking it to particle number independence on fast variables.
• To connect the adiabatic vacuum choice with the instantaneous Hamiltonian diagonalization vacuum and field smearing properties.

Proposed method

• Analyzes time-dependent linear canonical transformations (LCTs) to relate unitary evolution to canonical variable choices.
• Applies asymptotic analysis using action-angle variables, generalizing the Wentzel-Kramers-Brillouin (WKB) approximation for time-dependent systems.
• Derives integral approximations for phase and amplitude corrections via integration by parts, retaining first-order non-oscillatory terms.
• Uses the frequency ν and background evolution to classify oscillatory vs. non-oscillatory contributions in perturbative expansions.
• Constructs a transformation to canonical variables that ensures unitary time evolution by eliminating non-oscillatory terms in the Hamiltonian evolution.
• Relates the resulting vacuum choice to the adiabatic vacuum via action-angle variable behavior and connects it to the instantaneous Hamiltonian diagonalization.

Experimental results

Research questions

• RQ1 Under what conditions on the Hamiltonian does a unitary time evolution exist for free scalar fields in spatially homogeneous and isotropic space-times?
• RQ2 What is the unique set of canonical variables that ensures unitary evolution for such systems?
• RQ3 How can the adiabatic vacuum condition be generalized using action-angle variables to describe particle number independence on fast variables?
• RQ4 How are the adiabatic vacuum, field smearing, and instantaneous Hamiltonian diagonalization related through the choice of canonical variables?
• RQ5 What role do non-oscillatory terms in the perturbative expansion play in determining the correct canonical variables for unitary evolution?

Key findings

• A unique set of canonical variables exists for which the time evolution of a free scalar field in a spatially homogeneous and isotropic background is unitarily implementable.
• The condition for unitary evolution is equivalent to the absence of non-oscillatory terms in the first-order perturbative expansion of the phase and amplitude corrections.
• The adiabatic vacuum condition is generalized through action-angle variables, where particle number density is independent of fast (angle) variables.
• The canonical variables derived via the asymptotic method correspond to the instantaneous Hamiltonian diagonalization vacuum, establishing a direct link between vacuum choices.
• The transformation to these canonical variables ensures that the Hamiltonian evolution is unitary, and the method provides a systematic way to compute it order-by-order in ν⁻¹.
• Non-oscillatory terms in the δφ correction (e.g., −∫( ̇ξ² / 2ν ) dt) are identified as critical for first-order accuracy and must be retained to achieve unitary evolution.