Poisson Bracket on the Space of Histories

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.

arXiv comment25 pages REVTEX, CGPG-93/8-4 (minor changes made)

Cited by in corpus (15)

• Quantum Observables and Recollapsing Dynamics
• Observables, gravitational dressing, and obstructions to locality and subsystems
• Covariant phase space, constraints, gauge and the Peierls formula
• A general framework for gravitational charges and holographic renormalization
• Algebra of diffeomorphism-invariant observables in Jackiw-Teitelboim Gravity
• Mass Superselection, Canonical Gauge Transformations, and Asymptotically Flat Variational Principles
• Comments on Microcausality, Chaos, and Gravitational Observables
• Geometric Wave Equations
• Unambiguous Phase Spaces for Subregions
• The Holographic Dual of the Entanglement Wedge Symplectic Form
• Islands and Uhlmann phase: Explicit recovery of classical information from evaporating black holes
• From Peierls brackets to a generalized Moyal bracket for type-I gauge theories
• Green hyperbolic complexes on Lorentzian manifolds
• A parametrix for quantum gravity?
• A new antisymmetric bilinear map for type-I gauge theories