Classical Euclidean general relativity from ``left-handed area = right-handed area"

TL;DR

This paper presents a classical continuum theory derived from a topological BF theory with a constraint linking left and right handed areas, showing that Euclidean GR is a specific sector within this broader framework.

Contribution

It introduces a classical theory with a constraint on areas in SO(4) BF theory, demonstrating Euclidean GR as a particular sector and analyzing the theory's sectors and quantization implications.

Findings

Euclidean GR is a sector of the constrained BF theory.

The theory's other sectors are characterized and distinguished.

Path integral quantization suggests non-GR sectors dominate semiclassically.

Abstract

A classical continuum theory corresponding to Barrett and Crane's model of Euclidean quantum gravity is presented. The fields in this classical theory are those of SO(4) BF theory, a simple topological theory of an so(4) valued 2-form field, $B^{IJ}_{\m\n}$, and an so(4) connection. The left handed (self-dual) and right handed (anti-self-dual) components of $B$ define a left handed and a, generally distinct, right handed area for each spacetime 2-surface. The theory being presented is obtained by adding to the BF action a Lagrange multiplier term that enforces the constraint that the left handed and the right handed areas be equal. It is shown that Euclidean general relativity (GR) forms a sector of the resulting theory. The remaining three sectors of the theory are also characterized and it is shown that, except in special cases, GR canonical initial data is sufficient to specify the GR sector as well as a specific solution within this sector. Finally, the path integral quantization of the theory is discussed at a formal level and a hueristic argument is given suggesting that in the semiclassical limit the path integral is dominated by solutions in one of the non-GR sectors, which would mean that the theory quantized in this way is not a quantization of GR.