The Hamiltonian constraint in 3d Riemannian loop quantum gravity

Details The Hamiltonian constraint in 3d Riemannian loop quantum gravity The Hamiltonian constraint in 3d Riemannian loop quantum gravity

2011-01-18

arxiv.org/abs/1101.3524v3

Class.Quant.Grav.28:195006,2011

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

24 pages, added proof that the regulated constraints are classically first class and explicit relationship to flat connections

Scientific paper

10.1088/0264-9381/28/19/195006

We discretize the Hamiltonian scalar constraint of three-dimensional Riemannian gravity on a graph of the loop quantum gravity phase space. This Hamiltonian has a clear interpretation in terms of discrete geometries: it computes the extrinsic curvature from dihedral angles. The Wheeler-DeWitt equation takes the form of difference equations, which are actually recursion relations satisfied by Wigner symbols. On the boundary of a tetrahedron, the Hamiltonian generates the exact recursion relation on the 6j-symbol which comes from the Biedenharn-Elliott (pentagon) identity. This fills the gap between the canonical quantization and the symmetries of the Ponzano-Regge state-sum model for 3d gravity.

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