Formal commutators of the gravitational constraints are not well defined: A translation of Ashtekar’s ordering to the Schrödinger representation

John L. Friedman and Ian Jack
Phys. Rev. D 37, 3495 – Published 15 June 1988
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Abstract

Ashtekar’s factor ordering of the gravitational constraint equations, using spinorial variables, formally implies (in his σ representation) a certain symmetric factor ordering of the Hamiltonian constraint in the Schrödinger (metric) representation. This ordering is among those for which the constraints were believed not to close, and a straightforward formal computation of the commutator fails to give closure. However, with an alternative formal computation (equivalent, in the Schrödinger representation, to Ashtekar’s spinorial computation) the constraints close. More generally, there is no well-defined formal factor-ordering problem in quantum gravity: different formal computations of the same commutator yield different results. If the constraints are regularized by point splitting, using a fixed flat background metric, their commutators have, in general, no well-defined coincidence limit. An alternative, covariant, point-splitting prescription that uses the exponential map is available, but adopting it implies that all orderings of the momentum constraint are equivalent; all orderings of the Hamiltonian constraint would then lead to a closed commutator algebra.

  • Received 18 November 1987

DOI:https://doi.org/10.1103/PhysRevD.37.3495

©1988 American Physical Society

Authors & Affiliations

John L. Friedman

  • Department of Physics, University of Wisconsin, Milwaukee, Wisconsin 53201

Ian Jack

  • Department of Physics, The University, Southampton, SO9 5NH England

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Issue

Vol. 37, Iss. 12 — 15 June 1988

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