Three-geometry and reformulation of the Wheeler-DeWitt equation

TL;DR

This paper presents a reformulation of the Wheeler-DeWitt equation emphasizing gauge-invariant three-geometry elements, aiming to clarify quantum gravity constraints and address factor ordering ambiguities within loop quantum gravity.

Contribution

It introduces a new gauge-invariant reformulation of the Wheeler-DeWitt equation and proposes polynomial quantum constraints to help resolve factor ordering issues in quantum gravity.

Findings

Reformulation highlights the role of gauge-invariant three-geometry elements.

A polynomial quantum constraint is proposed to address factor ordering ambiguity.

Both volume and Chern-Simons functional are shown to be significant in the reformulation.

Abstract

A reformulation of the Wheeler-DeWitt equation which highlights the role of gauge-invariant three-geometry elements is presented. It is noted that the classical super-Hamiltonian of four-dimensional gravity as simplified by Ashtekar through the use of gauge potential and densitized triad variables can furthermore be succinctly expressed as a vanishing Poisson bracket involving three-geometry elements. This is discussed in the general setting of the Barbero extension of the theory with arbitrary non-vanishing value of the Immirzi parameter, and when a cosmological constant is also present. A proposed quantum constraint of density weight two which is polynomial in the basic conjugate variables is also demonstrated to correspond to a precise simple ordering of the operators, and may thus help to resolve the factor ordering ambiguity in the extrapolation from classical to quantum gravity. Alternative expression of a density weight one quantum constraint which may be more useful in the spin network context is also discussed, but this constraint is non-polynomial and is not motivated by factor ordering. The article also highlights the fact that while the volume operator has become a preeminient object in the current manifestation of loop quantum gravity, the volume element and the Chern-Simons functional can be of equal significance, and need not be mutually exclusive. Both these fundamental objects appear explicitly in the reformulation of the Wheeler-DeWitt constraint.