Functional evolution of quantum cylindrical waves

TL;DR

This paper investigates the unitarity of quantum cylindrical wave evolution in flat spacetime, revealing limitations in the functional evolution concept despite some well-defined formalisms.

Contribution

It demonstrates that while certain evolutions are unitarily implementable, generic spatial diffeomorphisms are not, challenging the viability of a Tomanaga-Schwinger type evolution.

Findings

Functional evolution of initial to final flat slices is unitarily implementable.

Generic spatial diffeomorphisms are not unitarily implementable.

Tomanaga-Schwinger type evolution is not viable for quantum cylindrical waves.

Abstract

Kucha{\v{r}} showed that the quantum dynamics of (1 polarization) cylindrical wave solutions to vacuum general relativity is determined by that of a free axially-symmetric scalar field along arbitrary axially-symmetric foliations of a fixed flat 2+1 dimensional spacetime. We investigate if such a dynamics can be defined {\em unitarily} within the standard Fock space quantization of the scalar field. Evolution between two arbitrary slices of an arbitrary foliation of the flat spacetime can be built out of a restricted class of evolutions (and their inverses). The restricted evolution is from an initial flat slice to an arbitrary (in general, curved) slice of the flat spacetime and can be decomposed into (i) `time' evolution in which the spatial Minkowskian coordinates serve as spatial coordinates on the initial and the final slice, followed by (ii) the action of a spatial diffeomorphism of the final slice on the data obtained from (i). We show that although the functional evolution of (i) is unitarily implemented in the quantum theory, generic spatial diffeomorphisms of (ii) are not. Our results imply that a Tomanaga-Schwinger type functional evolution of quantum cylindrical waves is not a viable concept even though, remarkably, the more limited notion of functional evolution in Kucha{\v{r}}'s `half parametrized formalism' is well-defined.