A finite spin-foam-based theory of three and four dimensional quantum gravity

TL;DR

This paper develops a finite, consistent spin-foam-based quantum gravity theory in three and four dimensions, providing exact solutions to the quantum constraints of general relativity and advancing the understanding of quantum spacetime.

Contribution

It introduces the first well-defined, finite spin-foam model with infinite degrees of freedom that solves the quantum gravity constraints, extending Ooguri's $BF$ theory.

Findings

Constructs a well-defined spin-foam quantum gravity theory.

Identifies exact solutions to the Hamiltonian constraint.

Highlights potential for semi-classical analysis of quantum spacetime.

Abstract

Starting from Ooguri's construction for $BF$ theory in three (and four) dimensions, we show how to construct a well defined theory with an infinite number of degrees of freedom. The spin network states that are kept invariant by the evolution operators of the theory are exact solutions of the Hamiltonian constraint of quantum gravity proposed by Thiemann. The resulting theory is the first example of a well defined, finite, consistent, spin-foam based theory in a situation with an infinite number of degrees of freedom. Since it solves the quantum constraints of general relativity it is also a candidate for a theory of quantum gravity. It is likely, however, that the solutions constructed correspond to a spurious sector of solutions of the constraints. The richness of the resulting theory makes it an interesting example to be analyzed by forthcoming techniques that construct the semi-classical limit of spin network quantum gravity.