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Symmetry 2011, 3(2), 134-154; doi:10.3390/sym3020134
Article

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

Centre for Time, School of Philosophical and Historical Enquiry, University of Sydney, Australia
Received: 21 February 2011 / Revised: 21 March 2011 / Accepted: 28 March 2011 / Published: 31 March 2011
(This article belongs to the Special Issue Quantum Symmetry)
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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.
Keywords: problem of time; canonical quantisation; symplectic reduction; Hamiltonian constraints; quantum gravity problem of time; canonical quantisation; symplectic reduction; Hamiltonian constraints; quantum gravity
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Thébault, K.P.Y. Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity? Symmetry 2011, 3, 134-154.

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