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

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Abstract
A classical continuum theory corresponding to Barrett and Crane's model of Euclidean simplicial 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)-algebra-valued 2-form field, , 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 presented is obtained by adding to the BF action a Lagrange multiplier term which enforces the constraint that the left- and the right-handed areas should be be equal. The solution space of the theory has six branches, two of which reproduce Euclidean general relativity (GR). The other four branches are also characterized and it is shown that the GR branches are stable in the sense that non-degenerate initial data of a solution in one of the GR branches does not, except in special cases, admit an alternative development in another branch. Finally, the path-integral quantization of the theory is discussed at a formal level and a heuristic 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.
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