June 19, 2002
It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. Additionally, the traditional lattice field theory approach consists in fixing the gauge in a Euclidean action, which does not appear appropriate for general relativity. We analyze discrete lattice general relativity and develop a canonical formalism that allows to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a ``dynamical gauge'' fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. Among other issues the problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories like BF theory. We discuss a simple cosmological application that exhibits the quantum elimination of the singularity at the big bang.
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A crucial step in the history of General Relativity was Einstein's adoption of the principle of general covariance which demands a coordinate independent formulation for our spacetime theories. General covariance helps us to disentangle a theory's substantive content from its merely representational artifacts. It is an indispensable tool for a modern understanding of spacetime theories. Motivated by quantum gravity, one may wish to extend these notions to quantum spacetime th...
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A crucial step in the history of General Relativity was Einstein's adoption of the principle of general covariance which demands a coordinate independent formulation for our spacetime theories. General covariance helps us to disentangle a theory's substantive content from its merely representational artifacts. It is an indispensable tool for a modern understanding of spacetime theories. Motivated by quantum gravity, one may wish to extend these notions to quantum spacetime th...
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