Poisson Bracket on the Space of Histories

TL;DR

This paper extends the Poisson bracket to the space of histories in gauge systems, exploring the resulting algebraic structures and their implications for path integrals and decoherence in time reparameterization invariant contexts.

Contribution

It introduces a novel extension of the Poisson bracket to the space of histories, including gauge systems, and discusses algebraic structures relevant for path integration and decoherence.

Findings

Extended Poisson brackets define Lie algebras on history spaces.

Applied to gauge systems, both canonical and reduced brackets are extended.

Discussed implications for systems with Gribov ambiguities and reparameterization invariance.

Abstract

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.