Edge modes of gravity. Part III. Corner simplicity constraints
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 systemati...
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doaj-65143e1f611c4ca4b88d5b62793204e92021-01-24T12:06:33ZengSpringerOpenJournal of High Energy Physics1029-84792021-01-012021116410.1007/JHEP01(2021)100Edge modes of gravity. Part III. Corner simplicity constraintsLaurent Freidel0Marc Geiller1Daniele Pranzetti2Perimeter Institute for Theoretical PhysicsÉcole Normale Superieure (ENS) de LyonPerimeter Institute for Theoretical PhysicsAbstract 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 Poincaré 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 Poincaré 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 Poincaré 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 sl 2 ℂ $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) $$ subalgebra of Poincaré, and the components of the tangential corner metric satisfying an sl 2 ℝ $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$ 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.https://doi.org/10.1007/JHEP01(2021)100Classical Theories of GravityModels of Quantum GravitySpace-Time SymmetriesGauge Symmetry |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Laurent Freidel Marc Geiller Daniele Pranzetti |
spellingShingle |
Laurent Freidel Marc Geiller Daniele Pranzetti Edge modes of gravity. Part III. Corner simplicity constraints Journal of High Energy Physics Classical Theories of Gravity Models of Quantum Gravity Space-Time Symmetries Gauge Symmetry |
author_facet |
Laurent Freidel Marc Geiller Daniele Pranzetti |
author_sort |
Laurent Freidel |
title |
Edge modes of gravity. Part III. Corner simplicity constraints |
title_short |
Edge modes of gravity. Part III. Corner simplicity constraints |
title_full |
Edge modes of gravity. Part III. Corner simplicity constraints |
title_fullStr |
Edge modes of gravity. Part III. Corner simplicity constraints |
title_full_unstemmed |
Edge modes of gravity. Part III. Corner simplicity constraints |
title_sort |
edge modes of gravity. part iii. corner simplicity constraints |
publisher |
SpringerOpen |
series |
Journal of High Energy Physics |
issn |
1029-8479 |
publishDate |
2021-01-01 |
description |
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 Poincaré 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 Poincaré 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 Poincaré 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 sl 2 ℂ $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) $$ subalgebra of Poincaré, and the components of the tangential corner metric satisfying an sl 2 ℝ $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$ 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. |
topic |
Classical Theories of Gravity Models of Quantum Gravity Space-Time Symmetries Gauge Symmetry |
url |
https://doi.org/10.1007/JHEP01(2021)100 |
work_keys_str_mv |
AT laurentfreidel edgemodesofgravitypartiiicornersimplicityconstraints AT marcgeiller edgemodesofgravitypartiiicornersimplicityconstraints AT danielepranzetti edgemodesofgravitypartiiicornersimplicityconstraints |
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