1. Edge modes of gravity. Part III. Corner simplicity constraints
- Author
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Marc Geiller, Laurent Freidel, Daniele Pranzetti, Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon, and École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)
- Subjects
High Energy Physics - Theory ,Nuclear and High Energy Physics ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,Computer Science::Digital Libraries ,01 natural sciences ,Regularization (mathematics) ,General Relativity and Quantum Cosmology ,symmetry: algebra ,topological ,phase space ,Poincare ,tetrad ,0103 physical sciences ,BF model ,Models of Quantum Gravity ,lcsh:Nuclear and particle physics. Atomic energy. Radioactivity ,Continuum (set theory) ,010306 general physics ,Mathematical physics ,Physics ,Simplex ,symplectic ,gravitation: partition function ,010308 nuclear & particles physics ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,Subalgebra ,Space-Time Symmetries ,Partition function (mathematics) ,Classical Theories of Gravity, Models of Quantum Gravity, Space-Time Symmetries, Gauge Symmetry ,Symmetry (physics) ,Hamiltonian ,High Energy Physics - Theory (hep-th) ,Irreducible representation ,Gauge Symmetry ,Computer Science::Mathematical Software ,lcsh:QC770-798 ,simplex ,Classical Theories of Gravity ,path integral - Abstract
In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincar\'e and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints. This amounts to fixing the Heisenberg frame with a choice of position and spin operators. The simplicity constraints then further reduce the Poincar\'e symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a particle-like description of (quantum) geometry: The internal normal plays the role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the flux that of a relativistic position, and the frame that of a spin harmonic oscillator. Moreover, we show that the corner area element corresponds to the Poincar\'e spin Casimir. We achieve this central result by properly splitting, in the continuum, the corner simplicity constraints into first and second class parts. We construct the complete set of Dirac observables, which includes the generators of the local $\mathfrak{sl}(2,\mathbb{C})$ subalgebra of Poincar\'e, and the components of the tangential corner metric satisfying an $\mathfrak{sl}(2,\mathbb{R})$ algebra. We then present a preliminary analysis of the covariant and continuous irreducible representations of the infinite-dimensional corner algebra. Moreover, as an alternative path to quantization, we also introduce a regularization of the corner algebra and interpret this discrete setting in terms of an extended notion of twisted geometries., Comment: 71 pages, 1 figure
- Published
- 2021
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