Classical Histories in Hamiltonian Systems

TL;DR

This paper explores the advantages of a history-based approach to classical Hamiltonian systems, particularly in addressing spacetime issues, by generalizing the derivation of geometrodynamics within the history phase space framework.

Contribution

It demonstrates that a history approach offers benefits over equal-time methods for classical spacetime discussions, extending the derivation of geometrodynamics.

Findings

History approach clarifies spacetime issues in classical systems

Generalizes derivation of geometrodynamics from first principles

Highlights advantages of history phase space in classical theory

Abstract

The incompatibility between the treatment of time in the classical and in the quantum theory results in the so-called problem of time in canonical quantum gravity. For this reason, attempts have been made to devise algorithms of quantization which accomodate the covariance of the classical theory from the outset. One of the most prominent of these attempts is based on the notion of continuous histories (Isham and Linden) in the context of the consistent histories approach to quantum theory (Griffiths, Omnes, Gell-Mann and Hartle). By the term continuous histories it is implied that the canonical fields and the symplectic structure of the theory depend on time as well as space. The aim of this thesis (in the form it was submitted to the University of London, February 2000) is to show that, even at the purely classical level, a history approach has several advantages (compared to its equal-time counterpart) when it comes to discussing spacetime issues. This is illustrated here by reframing and generalizing the derivation of geometrodynamics from first principles (Hojman, Kuchar, Teitelboim) in the language of the history phase space.