Presymplectic Geometry And The Problem Of Time. Part 2

TL;DR

This paper explores the classical presymplectic approach to the Problem of Time in Quantum Gravity, focusing on three-space general relativity, constraints, observables, and path integral quantization.

Contribution

It introduces a presymplectic perspective on three-space general relativity, analyzing constraints, observables, and quantization methods in the context of the Problem of Time.

Findings

Comparison of constraints with standard ADM gravity

Re-derivation of ephemeris lapse identification

Path integral quantization of reduced phase space

Abstract

The Problem of Time in Quantum Gravity is analyzed from a classical presymplectic perspective. In the first part of the paper the Three Space Approach to General Relativity is introduced via the Barbour-Foster-\'O Murchadha action and the dynamics corresponding to a theory of relativistic gravity where spacetime is not an a priori requirement. We also look into the nature and physical interpretation of the constraints in this theory and compare them with those of Standard ADM General Relativity. We then study the presymplectic phase space of three space general relativity and discuss briefly the notion of observables and perennials of the system. We then move on to re-deriving the ephemeris lapse identification, and then discuss the notion of re-foliation invariance and its resolution in the conformal theory. Further, we study a new perspective of three space general relativity involving a Hamiltonian reduction of the phase space and subsequently, we discuss the path integral quantization of the same.