2.9 Hamiltonian treatment. Introduction

Relatively little work has been done on discretized Hamiltonian formulations of gravity. This can in part be understood from the fact that the numerical methods available for lattice gauge field theories rely mostly on the Euclidean path-integral description. Unfortunately the relation between the Lagrangian and Hamiltonian quantizations for generally covariant theories without a fixed background is far from clear.

In the usual metric formulation, the complicated non-polynomial form of the Hamiltonian constraint has been a long-standing problem. In this framework, neither the functional form of the quantum representations nor the nature of the quantization problems suggest that a discrete approach might yield any advantages. This situation has improved with the introduction of new Hamiltonian gauge-theoretic variables by Ashtekar [2223].

By a Hamiltonian lattice approach one usually means a formulation in which the time variable is left continuous, and only the spatial 3-slices are discretized. In continuum gravity, the 3+1 decomposition leads to the (non-Lie) Dirac algebra of the three-dimensional diffeomorphism generators and the Hamiltonian constraint, associated with the deformation of three-surfaces imbedded in four-space. One usually requires this algebra to be realized in the quantum theory, without anomalous terms, for a set of self-adjoint quantum constraint operators, for some factor-ordering.

Since a discretization of space-time breaks the diffeomorphism invariance, there is no reason to expect the Dirac algebra to be preserved in any discrete approach, even classically. This raises the question of whether and in what form part of the diffeomorphism symmetry can still be realized at the discrete level. Using gauge-theoretic variables, one can maintain the exact local gauge invariance with respect to the internal degrees of freedom, but there is no analogous procedure for treating the coordinate invariance.

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