Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?

TL;DR

This paper examines the interpretation of quantum Hamiltonian constraints in canonical gravity, questioning assumptions about quantisation, reduction, and their commutation, and explores modern approaches to their quantisation.

Contribution

It critically analyzes the assumptions behind quantising Hamiltonian constraints and proposes refined interpretations within modern canonical gravity frameworks.

Findings

Quantum Hamiltonian constraints lack a straightforward reduction procedure.

Quantisation may not commute with reduction in canonical gravity.

Interpretation of quantum constraints requires careful conceptual analysis.

Abstract

Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that `quantisation commutes with reduction' and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity.