2.11 The measure2 Gauge-theoretic discretizations of gravity2.9 Hamiltonian treatment. Introduction

2.10 Hamiltonian lattice gravity

There is a gauge-theoretic Hamiltonian version of gravity defined on a cubic lattice, which in many aspects resembles the Lagrangian gauge formulations described earlier. It also is virtually the only discrete Hamiltonian formulation in which some progress has been achieved in the quantization (see also [155] for a recent review).

Renteln and Smolin [180] were the first to set up a continuous-time lattice discretization along the lines of Hamiltonian lattice gauge theory. Their basic configuration variables are the link holonomies U (l) of the spatial Ashtekar connection along the edges. The lattice analogues of the canonically conjugate pairs tex2html_wrap_inline2341 are the link variables tex2html_wrap_inline2343, with Poisson brackets

  eqnarray168

with the SU (2)-generators satisfying tex2html_wrap_inline2347 . Lattice links tex2html_wrap_inline2349 are labelled by a vertex n and a lattice direction tex2html_wrap_inline2353 . These relations go over to the usual continuum brackets in the limit as the lattice spacing a is taken to zero. In this scheme, they wrote down discrete analogues of the seven polynomial first-class constraints, and also attempted to interpret the action of the discretized diffeomorphism and Hamiltonian constraints in terms of their geometric action on lattice Wilson loop states.



2.11 The measure2 Gauge-theoretic discretizations of gravity2.9 Hamiltonian treatment. Introduction

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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