Squeezing formalism and canonical transformations in cosmology

Abstract

Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameter-isations, but also generate the dynamics of the system. In the first part of this work we study the symplectic structure associated with linear canonical transformations. After reviewing salient mathematical properties of the symplectic group in a pedagogical way, we introduce the squeezing formalism, and show how any linear dynamics can be cast in terms of an invariant representation. In the second part, we apply these results to the case of cosmological perturbations, and focus on scalar field fluctuations during inflation. We show that di↵erent canonical variables select out di↵erent vacuum states, and that this leaves an ambiguity in observational predictions if initial conditions are set at a finite time in the past. We also discuss how the e↵ectiveness of the quantum-to-classical transition of cosmological perturbations depends on the set of canonical variables used to describe them. Keywords: quantum field theory on curved space, physics of the early universe, inflation