Consistent discretization and canonical classical and quantum Regge calculus

TL;DR

The paper tackles obtaining a canonical, constraint-free formulation of Regge calculus for general relativity by applying the consistent discretization program to a discrete Regge action. It recasts the action into a time-sliced Lagrangian $L(n,n+1)$ and shows that, after eliminating Lagrange multipliers, the evolution becomes a canonical transformation generated by $-L(n,n+1)$ with reduced variables $(\ell_1,\ell_2,\ell_3; P_{\ell_1},P_{\ell_2},P_{\ell_3})$. In the Lorentzian sector, spikes are naturally suppressed and a well-defined path-integral measure arises from a unitary evolution, enabling topology-change discussions. Although demonstrated in 3D for concreteness, the construction extends to arbitrary dimensions and offers a practical computational avenue for classical and quantum gravity.

Abstract

We apply the ``consistent discretization'' technique to the Regge action for (Euclidean and Lorentzian) general relativity in arbitrary number of dimensions. The result is a well defined canonical theory that is free of constraints and where the dynamics is implemented as a canonical transformation. This provides a framework for the discussion of topology change in canonical quantum gravity. In the Lorentzian case, the framework appears to be naturally free of the ``spikes'' that plague traditional formulations. It also provides a well defined recipe for determining the measure of the path integral.

Figures (2)

• Figure 1: The simplicial decomposition considered. The figures on the right show prisms number 1 and 2 respectively, the other prisms are obtained by reflection and periodicity. The hinge length variables $\ell_i$ are assigned to the hinges in the following way: $A'A\mapsto\ell_1$, $A'B\mapsto\ell_2$, $A'B'\mapsto\ell_3$, $A'E\mapsto\ell_4$, $A'D\mapsto\ell_5$, $A'D'\mapsto\ell_6$$A'E'\mapsto\ell_7$.
• Figure 2: The framework can handle topology change.