Poisson Bracket on the Space of Histories

TL;DR

This work develops Lie-bracket structures on spaces of histories, extending the Poisson bracket to the history space $\mathcal{H}$ and to spaces of Lagrangian histories, with explicit constructions that respect the equations of motion and gauge structure. It introduces a gauge-breaking framework that can handle Gribov ambiguities, and defines a robust space of evolutions $\mathcal{E}$ for pull-backs of the brackets, enabling canonical and Dirac-like quantization schemes. The paper also shows how these history-based algebras translate to quantization in the Heisenberg picture and relate to covariant formulations via spaces of histories and Lagrangian histories, setting the stage for a unified generalized Peierls algebra in a companion work. Overall, it provides a concrete, local-to-global method to define and quantize algebras on histories, including constrained and gauge systems, while clarifying the roles of $\mathcal{H}$, $\mathcal{S}$, $\mathcal{E}$, and $\mathcal{L}$ in connecting canonical and path-integral perspectives.

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.