Discrete WheelerDeWitt equation
Abstract
We present a discrete form of the WheelerDeWitt equation for quantum gravitation, based on the lattice formulation due to Regge. In this setup, the infinitedimensional manifold of 3geometries is replaced by a space of threedimensional 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 wave functional, arising from the triangle inequalities and their higherdimensional analogs. The character of the solutions is discussed in the strongcoupling (largeG) limit, where it is shown that the wave functional 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 wave functions 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.
 Publication:

Physical Review D
 Pub Date:
 November 2011
 DOI:
 10.1103/PhysRevD.84.104033
 arXiv:
 arXiv:1109.2530
 Bibcode:
 2011PhRvD..84j4033H
 Keywords:

 04.60.m;
 04.60.Gw;
 04.60.Nc;
 98.80.Qc;
 Quantum gravity;
 Covariant and sumoverhistories quantization;
 Lattice and discrete methods;
 Quantum cosmology;
 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology
 EPrint:
 45 pages, 8 figures