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arXiv · 2111.14471

Spinor Representation of the Hamiltonian Constraint in 3D LQG with a Non-zero Cosmological Constant

Valentin Bonzom·Maïté Dupuis·Qiaoyin Pan
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Abstract

We develop in a companion article the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation which is well-known at zero cosmological constant. In this article, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum Hamiltonian constraint acts in the $q$-deformed spin network basis as difference equations on physical states, which are thus the Wheeler-DeWitt equations in this framework. Moreover, we study how physical states transform under Pachner moves of the canonical surface. We find that those transformations are in fact $q$-deformations of the transition amplitudes of the flat case as found by Noui and Perez. Our quantum Hamiltonian constraints therefore build a Turaev-Viro model at real $q$.

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Valentin Bonzom, Maïté Dupuis, Qiaoyin Pan. 2021-11-29. Spinor Representation of the Hamiltonian Constraint in 3D LQG with a Non-zero Cosmological Constant. https://arxiv.org/abs/2111.14471

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