A Hamiltonian Lattice Theory for Homogeneous Curved Spacetimes in 2+1 Dimensions

Details A Hamiltonian Lattice Theory for Homogeneous Curved Spacetimes in 2+1 Dimensions A Hamiltonian Lattice Theory for Homogeneous Curved Spacetimes in 2+1 Dimensions

1996-01-19

arxiv.org/abs/gr-qc/9601028v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

TeX file, 19 pages, please contact the author for figures

Scientific paper

We propose an exact Hamiltonian lattice theory for (2+1)-dimensional spacetimes with homogeneous curvature. By gauging away the lattice we find a generalization of the ``polygon representation'' of (2+1)-dimensional gravity. We compute the holonomies of the Lorentz connection ${\bf A}_i = \omega_i^a {\bf L}_a + e_i^a {\bf K}_a$ and find that the cycle conditions are satisfied only in the limit $\Lambda \to 0$. This implies that, unlike in (2+1)-dimensional Einstein gravity, the connection ${\bf A}$ is not flat. If one modifies the theory by taking the cycle conditions as constraints, then one finds that the constraints algebra is first-class only if the Poisson bracket structure is deformed. This suggests that a finite theory of quantum gravity would require either a modified action including higher-order curvature terms, or a deformation of the commutator structure of the metric observables.

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