[Paper Review] Unitary evolution and uniqueness of the Fock quantization in flat cosmologies with compact spatial sections

Laura Castelló Gomar, Jerónimo Cortez+3 more|arXiv (Cornell University)|Dec 22, 2012

Advanced Operator Algebra Research2 references3 citations

TL;DR

This paper establishes the uniqueness of the Fock quantization for scalar fields with time-dependent mass in flat cosmologies with compact spatial sections—specifically the three-torus—by requiring invariance under spatial isometries and unitary quantum dynamics. It proves that these two criteria uniquely select both a canonical field pair and a Fock representation, resolving long-standing ambiguities in cosmological quantum field theory.

ABSTRACT

We study the Fock quantization of scalar fields with a time dependent mass in cosmological scenarios with flat compact spatial sections. This framework describes physically interesting situations like, e.g., cosmological perturbations in flat Friedmann-Robertson-Walker spacetimes, generally including a suitable scaling of them by a background function. We prove that the requirements of vacuum invariance under the spatial isometries and of a unitary quantum dynamics select (a) a unique canonical pair of field variables among all those related by time dependent canonical transformations which scale the field configurations, and (b) a unique Fock representation for the canonical commutation relations of this pair of variables. Though the proof is generalizable to other compact spatial topologies in three or less dimensions, we focus on the case of the three-torus owing to its relevance in cosmology, paying a especial attention to the role played by the spatial isometries in the determination of the representation.

Motivation & Objective

• To resolve the ambiguity in Fock quantization of scalar fields with time-dependent mass in cosmological spacetimes.
• To identify criteria that uniquely select a Fock representation when standard symmetries (like time-translation invariance) are broken due to cosmic expansion.
• To demonstrate that spatial isometry invariance and unitary evolution jointly eliminate both field variable ambiguity and Fock representation ambiguity.
• To establish a robust, unique quantum framework applicable to cosmological perturbations and effective field theories in flat FRW spacetimes.
• To validate the physical consistency of the selected quantization across different physical systems, including massive scalar fields and cosmological perturbations.

Proposed method

• Formalize the field dynamics as a Klein-Gordon equation with time-dependent mass on a flat spacetime with compact three-torus spatial topology.
• Apply time-dependent canonical transformations to generate equivalent field variable pairs, introducing a class of field descriptions parameterized by such transformations.
• Impose the requirement that the vacuum state be invariant under the isometries of the three-torus (e.g., translations and rotations in the toroidal geometry).
• Demand that the Heisenberg evolution of the field operators be unitarily implementable in the quantum theory, ensuring probabilistic consistency.
• Use the interplay between spatial isometry invariance and unitary dynamics to constrain the allowed complex structures, thereby fixing the Fock representation.
• Prove that only one such complex structure satisfies both conditions, leading to a unique Fock representation up to unitary equivalence.

Experimental results

Research questions

• RQ1 Can a unique Fock quantization be selected for scalar fields with time-dependent mass in flat cosmologies with compact spatial sections?
• RQ2 To what extent do spatial isometries and unitary dynamics constrain the choice of field variables and Fock representations?
• RQ3 Does the combination of spatial symmetry invariance and unitary evolution uniquely determine the vacuum and field quantization in the absence of time-translation invariance?
• RQ4 How does the time-dependent mass affect the existence and uniqueness of the Fock representation under these criteria?
• RQ5 Can this criterion be applied consistently across different physical systems, such as cosmological perturbations and effective field theories in FRW spacetimes?

Key findings

• The requirement of invariance under the spatial isometries of the three-torus uniquely selects a specific canonical pair of field variables among all those related by time-dependent canonical transformations.
• The requirement of unitary quantum dynamics uniquely selects a single Fock representation for the canonical commutation relations of the chosen field variables.
• The resulting Fock quantization is unitarily equivalent to the standard representation for a massless field, even when applied to a massive, time-dependent system.
• The uniqueness result holds under a mild condition: the time-dependent mass must have a second derivative that is integrable over compact time intervals.
• The criterion applies to any non-infinitesimal time domain, making it robust for cosmological applications.
• The selected quantization is consistent across different physical systems, including scalar and tensor cosmological perturbations, and is unitarily equivalent to previously proposed quantizations in gauge-invariant formulations.