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Title: Quantization of systems with temporally varying discretization. II. Local evolution moves

Several quantum gravity approaches and field theory on an evolving lattice involve a discretization changing dynamics generated by evolution moves. Local evolution moves in variational discrete systems (1) are a generalization of the Pachner evolution moves of simplicial gravity models, (2) update only a small subset of the dynamical data, (3) change the number of kinematical and physical degrees of freedom, and (4) generate a dynamical (or canonical) coarse graining or refining of the underlying discretization. To systematically explore such local moves and their implications in the quantum theory, this article suitably expands the quantum formalism for global evolution moves, constructed in Paper I [P. A. Höhn, “Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces,” J. Math. Phys. 55, 083508 (2014); e-print http://arxiv.org/abs/arXiv:1401.6062 [gr-qc]], by employing that global moves can be decomposed into sequences of local moves. This formalism is spelled out for systems with Euclidean configuration spaces. Various types of local moves, the different kinds of constraints generated by them, the constraint preservation, and possible divergences in resulting state sums are discussed. It is shown that non-trivial local coarse graining moves entail a non-unitary projection of (physical) Hilbert spaces and “fine grained” Dirac observables defined onmore » them. Identities for undoing a local evolution move with its (time reversed) inverse are derived. Finally, the implications of these results for a Pachner move generated dynamics in simplicial quantum gravity models are commented on.« less
Authors:
 [1]
  1. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5 (Canada)
Publication Date:
OSTI Identifier:
22305866
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 10; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DEGREES OF FREEDOM; EUCLIDEAN SPACE; GRAVITATION; HILBERT SPACE; QUANTIZATION; QUANTUM GRAVITY; VARIATIONAL METHODS