The symplectic geometry of a parametrized scalar field on a curved background

TL;DR

This paper rigorously analyzes the symplectic geometry of a parametrized massive Klein-Gordon field on curved spacetimes, establishing a geometric framework analogous to finite-dimensional systems.

Contribution

It provides a rigorous geometric and symplectic structure for the parametrized scalar field, including the extended phase space and constraint analysis.

Findings

Extended phase space is a weak-symplectic manifold.

Kuchar constraint defines a smooth constraint submanifold.

The geometric structure parallels finite-dimensional first-class systems.

Abstract

We study the real, massive Klein-Gordon field on a $C^\infty$ globally-hyperbolic background space-time with compact Cauchy hypersurfaces. In particular, the parametrization of this system as initiated by Dirac and Kucha\v{r} is put on a rigorous basis. The discussion is focussed on the structure of the set of spacelike embeddings of the Cauchy manifold into the space-time, and on the associated $e$-tensor density bundles and their tangent and cotangent bundles. The dynamics of the field is expressed as a set of automorphisms of the space of initial data in which each pair of embeddings defines one such automorphism. Using these results, the extended phase space of the system is shown to be a weak-symplectic manifold, and the Kucha\v{r} constraint is shown to define a smooth constraint submanifold which is foliated smoothly by the constraint orbits. The pull-back of the symplectic form to the constraint surface is a presymplectic form which is singular on the tangent spaces to the constraint orbits. Thus, the geometric structure of this infinite-dimensional system is analogous to that of a finite-dimensional, first-class parametrized system, and hence many of the results for the latter can be transferred to the infinite-dimensional case without difficulty.