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• W3099695570 abstract "The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in $${mathbb R}^{3}$$ . For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in $${mathbb R}^{3,1}$$ ) in terms of normals living all in $$mathrm {SU}(2)$$ or in $$mathrm {SB}(2,{mathbb C})$$ . The latter fits perfectly with the classical phase space developed for q-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant $$Lambda >0$$ . This allows the interpretation of q-deformed twisted geometries as actual discrete hyperbolic triangulations for 4d quantum gravity." @default.
• W3099695570 created "2020-11-23" @default.
• W3099695570 creator A5016986364 @default.
• W3099695570 creator A5042649802 @default.
• W3099695570 date "2017-06-15" @default.
• W3099695570 modified "2023-10-06" @default.
• W3099695570 title "The closure constraint for the hyperbolic tetrahedron as a Bianchi identity" @default.
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• W3099695570 doi "https://doi.org/10.1007/s10714-017-2255-2" @default.
• W3099695570 hasPublicationYear "2017" @default.
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