Polarization-free Quantization of Linear Field Theories

TL;DR

This paper introduces a polarization-free quantization method for linear quantum field theories that is explicit, adaptable to curved spacetimes, and bridges traditional Fock and algebraic approaches.

Contribution

It presents a new quantization scheme that avoids polarization choices, explicitly constructs quantum state spaces, and supports a broad class of symplectomorphisms, especially useful in curved spacetimes.

Findings

Provides an explicit construction of quantum states without polarization.

Embeds all Fock representations from reasonable polarizations.

Demonstrates applicability to a Klein-Gordon field on cosmological spacetime.

Abstract

It is well-known that there exist infinitely-many inequivalent representations of the canonical (anti)-commutation relations of Quantum Field Theory (QFT). A way out, suggested by Algebraic QFT, is to instead define the quantum theory as encompassing all possible (abstract) states. In the present paper, we describe a quantization scheme for general linear (aka. free) field theories that can be seen as intermediate between traditional Fock quantization and full Algebraic QFT, in the sense that: * it provides a constructive, explicit description of the resulting space of quantum states; * it does not require the choice of a polarization, aka. the splitting of classical solutions into positive vs. negative-frequency modes: in fact, any Fock representation corresponding to a "reasonable" choice of polarization is naturally embedded; * it supports the implementation of a "large enough" class of linear symplectomorphisms of the classical, infinite-dimensional phase space. The proposed quantization (like Algebraic QFT) is notably meant for use on curved spacetimes, where the lack of a preferred choice of polarization makes the introduction of a Fock representation problematic. Accordingly, we illustrate it in the simple case of a free Klein-Gordon field on a spatially-compact, cosmological spacetime.