Poisson Bracket on the Space of Histories

Details Poisson Bracket on the Space of Histories Poisson Bracket on the Space of Histories

1993-08-30

arxiv.org/abs/hep-th/9308141v2

Annals Phys. 236 (1994) 374-391

Physics

High Energy Physics

High Energy Physics - Theory

25 pages REVTEX, CGPG-93/8-4 (minor changes made)

Scientific paper

10.1006/aphy.1994.1116

We extend the Poisson bracket from a Lie bracket of phase space functions to a Lie bracket of functions on the space of canonical histories and investigate the resulting algebras. Typically, such extensions define corresponding Lie algebras on the space of Lagrangian histories via pull back to a space of partial solutions. These are the same spaces of histories studied with regard to path integration and decoherence. Such spaces of histories are familiar from path integration and some studies of decoherence. For gauge systems, we extend both the canonical and reduced Poisson brackets to the full space of histories. We then comment on the use of such algebras in time reparameterization invariant systems and systems with a Gribov ambiguity, though our main goal is to introduce concepts and techniques for use in a companion paper.

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