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Title: Simplicity in simplicial phase space

Abstract

A key point in the spin foam approach to quantum gravity is the implementation of simplicity constraints in the partition functions of the models. Here, we discuss the imposition of these constraints in a phase space setting corresponding to simplicial geometries. On the one hand, this could serve as a starting point for a derivation of spin foam models by canonical quantization. On the other, it elucidates the interpretation of the boundary Hilbert space that arises in spin foam models. More precisely, we discuss different versions of the simplicity constraints, namely, gauge-variant and gauge-invariant versions. In the gauge-variant version, the primary and secondary simplicity constraints take a similar form to the reality conditions known already in the context of (complex) Ashtekar variables. Subsequently, we describe the effect of these primary and secondary simplicity constraints on gauge-invariant variables. This allows us to illustrate their equivalence to the so-called diagonal, cross and edge simplicity constraints, which are the gauge-invariant versions of the simplicity constraints. In particular, we clarify how the so-called gluing conditions arise from the secondary simplicity constraints. Finally, we discuss the significance of degenerate configurations, and the ramifications of our work in a broader setting.

Authors:
;  [1]
  1. MPI fuer Gravitationsphysik, Albert Einstein Institute, Am Muehlenberg 1, D-14476 Potsdam (Germany)
Publication Date:
OSTI Identifier:
21421168
Resource Type:
Journal Article
Journal Name:
Physical Review. D, Particles Fields
Additional Journal Information:
Journal Volume: 82; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevD.82.064026; (c) 2010 American Institute of Physics; Journal ID: ISSN 0556-2821
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; CONFIGURATION; GAUGE INVARIANCE; HILBERT SPACE; PARTITION FUNCTIONS; PHASE SPACE; QUANTIZATION; QUANTUM GRAVITY; SPIN; ANGULAR MOMENTUM; BANACH SPACE; FIELD THEORIES; FUNCTIONS; INVARIANCE PRINCIPLES; MATHEMATICAL SPACE; PARTICLE PROPERTIES; QUANTUM FIELD THEORY; SPACE

Citation Formats

Dittrich, Bianca, and Ryan, James P. Simplicity in simplicial phase space. United States: N. p., 2010. Web. doi:10.1103/PHYSREVD.82.064026.
Dittrich, Bianca, & Ryan, James P. Simplicity in simplicial phase space. United States. https://doi.org/10.1103/PHYSREVD.82.064026
Dittrich, Bianca, and Ryan, James P. 2010. "Simplicity in simplicial phase space". United States. https://doi.org/10.1103/PHYSREVD.82.064026.
@article{osti_21421168,
title = {Simplicity in simplicial phase space},
author = {Dittrich, Bianca and Ryan, James P},
abstractNote = {A key point in the spin foam approach to quantum gravity is the implementation of simplicity constraints in the partition functions of the models. Here, we discuss the imposition of these constraints in a phase space setting corresponding to simplicial geometries. On the one hand, this could serve as a starting point for a derivation of spin foam models by canonical quantization. On the other, it elucidates the interpretation of the boundary Hilbert space that arises in spin foam models. More precisely, we discuss different versions of the simplicity constraints, namely, gauge-variant and gauge-invariant versions. In the gauge-variant version, the primary and secondary simplicity constraints take a similar form to the reality conditions known already in the context of (complex) Ashtekar variables. Subsequently, we describe the effect of these primary and secondary simplicity constraints on gauge-invariant variables. This allows us to illustrate their equivalence to the so-called diagonal, cross and edge simplicity constraints, which are the gauge-invariant versions of the simplicity constraints. In particular, we clarify how the so-called gluing conditions arise from the secondary simplicity constraints. Finally, we discuss the significance of degenerate configurations, and the ramifications of our work in a broader setting.},
doi = {10.1103/PHYSREVD.82.064026},
url = {https://www.osti.gov/biblio/21421168}, journal = {Physical Review. D, Particles Fields},
issn = {0556-2821},
number = 6,
volume = 82,
place = {United States},
year = {Wed Sep 15 00:00:00 EDT 2010},
month = {Wed Sep 15 00:00:00 EDT 2010}
}