Canonical transformations and squeezing formalism in cosmology

Abstract

Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameterisations, 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 different canonical variables select out different 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 effectiveness of the quantum-to-classical transition of cosmological perturbations depends on the set of canonical variables used to describe them.

arXiv comment82 pages, 2 figures, matches published version

References in corpus (24)

• Semiclassicality and Decoherence of Cosmological Perturbations
• The Real Symplectic Groups in Quantum Mechanics and Optics
• A note on inflation and transplanckian physics
• Bouncing Cosmologies: Progress and Problems
• Higher order corrections to primordial spectra from cosmological inflation
• On the Dependence of the Spectra of Fluctuations in Inflationary Cosmology on Trans-Planckian Physics
• On matching conditions for cosmological perturbations
• Quantum-to-classical Transition of Cosmological Perturbations for Non-vacuum Initial States
• Trans-Planckian Issues for Inflationary Cosmology
• Quantum Discord of Cosmic Inflation: Can we Show that CMB Anisotropies are of Quantum-Mechanical Origin?
• Inflation, holography, and the choice of vacuum in de Sitter space
• Inflationary Perturbations: the Cosmological Schwinger Effect
• Stochastic inflation in phase space: Is slow roll a stochastic attractor?
• Observational issues in loop quantum cosmology
• Squeezed States in the de Sitter Vacuum
• Obstructions to Bell CMB Experiments
• Tensor Perturbations in Quantum Cosmological Backgrounds
• Bell's Inequality Violation (BIQV) with Non-Negative Wigner Function
• Bell's Inequalities for Continuous-Variable Systems in Generic Squeezed States
• Novel quantum initial conditions for inflation
• A Primer of Group Theory for Loop Quantum Gravity and Spin-foams
• Quantum Cosmic No-Hair Theorem and Inflation
• Applications of the group SU(1,1) for quantum computation and tomography
• The Berry phase in inflationary cosmology

Cited by in corpus (34)

• Non-Gaussian tail of the curvature perturbation in stochastic ultra-slow-roll inflation: implications for primordial black hole production
• Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation
• Minimal decoherence from inflation
• Cosmological Krylov Complexity
• Discord and Decoherence
• Benchmarking the cosmological master equations
• Quantum aspects of chaos and complexity from bouncing cosmology: A study with two-mode single field squeezed state formalism
• On the choice of the collapse operator in cosmological Continuous Spontaneous Localisation (CSL) theories
• Four-mode squeezed states: two-field quantum systems and the symplectic group $\mathrm{Sp}(4,\mathbb{R})$
• Bipartite temporal Bell inequalities for two-mode squeezed states
• Hamiltonian formalism for cosmological perturbations: the separate-universe approach
• Decoherence out of fire: Purity loss in expanding and contracting universes
• Numerical stochastic inflation constrained by frozen noise
• Comparing quantumness criteria
• Does inflation squeeze cosmological perturbations?
• In-in formalism for the entropy of quantum fields in curved spacetimes
• Real-space quantum-to-classical transition of time dependent background fluctuations
• Quantum initial conditions for inflation and canonical invariance
• Cosmological Bell Tests with Decoherence Effects
• Four-mode squeezed states in de Sitter space: A study with two field interacting quantum system
• Quantum initial conditions for curved inflating universes
• Stochastic inflation and non-perturbative power spectrum beyond slow roll
• Hot entanglement? -- Parametrically coupled quantum oscillators in two heat baths: instability, squeezing and driving
• Itô, Stratonovich, and zoom-in schemes in stochastic inflation
• Classical and Quantum Dynamics of Gyroscopic Systems and Dark Energy
• Hamiltonian formalism for cosmological perturbations: fixing the gauge
• Inflationary power spectrum from the Lanczos algorithm
• Ambiguous power spectrum from a quantum bounce
• Entanglement and correlations between local observables in de Sitter spacetime
• The relation between the symplectic group $Sp(4,\mathbb{R})$ and its Lie algebra: its application in polymer quantum mechanics
• Cosmological Geometric Phase From Pure Quantum States: A study without/with having Bell's inequality violation
• Effective dynamics of quantum fluctuations in field theory: with applications to cosmology
• Unitarily inequivalent quantum cosmological bouncing models
• Separate universe in multifield inflation: a phase-space approach