Perennials and the Group-Theoretical Quantization of a Parametrized Scalar Field on a Curved Background

TL;DR

This paper applies the perennial formalism to a parametrized Klein-Gordon field on curved backgrounds, constructing algebraic structures and analyzing time evolution via phase space shifts and spacetime symmetries.

Contribution

It introduces two algebraic frameworks for perennials in quantum field theory on curved spacetime, linking them to creation/annihilation operators and smeared fields, with explicit automorphism calculations.

Findings

Constructed ${ m S}_{ ext{can}}$ and ${ m S}_{ ext{loc}}$ algebras with Heisenberg structure

Described time evolution via transversal surfaces and spacetime isometries

Explicitly calculated automorphisms generated by time shifts

Abstract

The perennial formalism is applied to the real, massive Klein-Gordon field on a globally-hyperbolic background space-time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two different algebras ${\cal S}_{\text{can}}$ and ${\cal S}_{\text{loc}}$ of elementary perennials are constructed. The elements of ${\cal S}_{\text{can}}$ correspond to the usual creation and annihilation operators for particle modes of the quantum field theory, whereas those of ${\cal S}_{\text{loc}}$ are the smeared fields. Both are shown to have the structure of a Heisenberg algebra, and the corresponding Heisenberg groups are described. Time evolution is constructed using transversal surfaces and time shifts in the phase space. Important roles are played by the transversal surfaces associated with embeddings of the Cauchy hypersurface in the space-time, and by the time shifts that are generated by space-time isometries. The automorphisms of the algebras generated by this particular type of time shift are calculated explicitly.