Note on Canonical Quantization and Unitary Equivalence in Field Theory

TL;DR

This paper examines the differences between algebraic and Fock quantizations of a free scalar field, revealing that they are generally not unitarily equivalent, which has implications for field theory quantization.

Contribution

It demonstrates that algebraic and Fock quantizations of a scalar field are not always unitarily equivalent, challenging assumptions in field quantization methods.

Findings

Algebraic and Fock representations differ in general.

Unitary equivalence does not always hold in infinite-dimensional systems.

Implications for the choice of quantization in field theories.

Abstract

The problem of defining and constructing representations of the Canonical Commutation Relations can be systematically approached via the technique of {\it algebraic quantization}. In particular, when the phase space of the system is linear and finite dimensional, the `vertical polarization' provides an unambiguous quantization. For infinite dimensional field theory systems, where the Stone-von Neumann theorem fails to be valid, even the simplest representation, the Schroedinger functional picture has some non-trivial subtleties. In this letter we consider the quantization of a real free scalar field --where the Fock quantization is well understood-- on an arbitrary background and show that the representation coming from the most natural application of the algebraic quantization approach is not, in general, unitary equivalent to the corresponding Schroedinger-Fock quantization. We comment on the possible implications of this result for field quantization.