Discrete Wheeler-DeWitt Equation

TL;DR

This paper introduces a discrete lattice formulation of the Wheeler-DeWitt equation for quantum gravity, approximating continuum solutions with piecewise linear spaces and exploring their properties in the strong coupling limit.

Contribution

It develops a novel discrete form of the Wheeler-DeWitt equation based on Regge calculus, offering a computational alternative to Euclidean lattice methods in quantum gravity.

Findings

Wavefunctional depends on geometric quantities like areas and volumes.

Solutions show peaks at integer multiples of a fundamental volume.

Variational methods suggest links between quantum gravity and Euclidean path integrals.

Abstract

We present a discrete form of the Wheeler-DeWitt equation for quantum gravitation, based on the lattice formulation due to Regge. In this setup the infinite-dimensional manifold of 3-geometries is replaced by a space of three-dimensional piecewise linear spaces, with the solutions to the lattice equations providing a suitable approximation to the continuum wave functional. The equations incorporate a set of constraints on the quantum wavefunctional, arising from the triangle inequalities and their higher dimensional analogs. The character of the solutions is discussed in the strong coupling (large $G$) limit, where it is shown that the wavefunctional only depends on geometric quantities, such as areas and volumes. An explicit form, determined from the discrete wave equation supplemented by suitable regularity conditions, shows peaks corresponding to integer multiples of a fundamental unit of volume. An application of the variational method using correlated product wavefunctions suggests a relationship between quantum gravity in $n+1$ dimensions, and averages computed in the Euclidean path integral formulation in $n$ dimensions. The proposed discrete equations could provide a useful, and complementary, computational alternative to the Euclidean lattice path integral approach to quantum gravity.