Uniform discretizations: A new approach for the quantization of totally constrained systems

Miguel Campiglia, Cayetano Di Bartolo, Rodolfo Gambini, and Jorge Pullin
Phys. Rev. D 74, 124012 – Published 13 December 2006

Abstract

We discuss in detail the uniform discretization approach to the quantization of totally constrained theories. This approach allows to construct the continuum theory of interest as a well defined, controlled, limit of well behaved discrete theories. We work out several finite dimensional examples that exhibit behaviors expected to be of importance in the quantization of gravity. We also work out the case of BF theory. At the time of quantization, one can take two points of view. The technique can be used to define, upon taking the continuum limit, the space of physical states of the continuum constrained theory of interest. In particular we show in models that it agrees with the group averaging procedure when the latter exists. The technique can also be used to compute, at the discrete level, conditional probabilities and the introduction of a relational time. Upon taking the continuum limit one can show that one reproduces results obtained by the use of evolving constants, and therefore recover all physical predictions of the continuum theory. This second point of view can also be used as a paradigm to deal with cases where the continuum limit does not exist. There one would have discrete theories that at least at certain scales reproduce the semiclassical properties of the theory of interest. In this way the approach can be viewed as a generalization of the Dirac quantization procedure that can handle situations where the latter fails.

  • Received 6 October 2006

DOI:https://doi.org/10.1103/PhysRevD.74.124012

©2006 American Physical Society

Authors & Affiliations

Miguel Campiglia1, Cayetano Di Bartolo2, Rodolfo Gambini1, and Jorge Pullin3

  • 1Instituto de Física, Facultad de Ciencias, Iguá 4225, esq. Mataojo, Montevideo, Uruguay
  • 2Departamento de Física, Universidad Simón Bolívar, Aptdo. 89000, Caracas 1080-A, Venezuela
  • 3Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA

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Issue

Vol. 74, Iss. 12 — 15 December 2006

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