Abstract
Kuchar 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 unitarily within the standard Fock space quantization of the scalar field. Classical scalar field evolution from an initial flat slice to an arbitrary (in general, curved) slice of the flat spacetime can be decomposed into (i) ‘time’ evolution in which the spatial Minkowskian coordinates of the initial slice serve as spatial coordinates of 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 Kuchar's ‘half parametrized formalism’ is well-defined.
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