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

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

1995-10-16

arxiv.org/abs/gr-qc/9510028v1

J.Math.Phys. 37 (1996) 3505-3521

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

30 pages, revtex

Scientific paper

10.1063/1.531578

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.

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