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A classical continuum theory corresponding to Barrett and Crane's model ofEuclidean quantum gravity is presented. The fields in this classical theory arethose of SO(4) BF theory, a simple topological theory of an so(4) valued 2-formfield, $B^{IJ}_{\m\n}$, 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 theorybeing presented is obtained by adding to the BF action a Lagrange multiplierterm that enforces the constraint that the left handed and the right handedareas be equal. It is shown that Euclidean general relativity (GR) forms asector of the resulting theory. The remaining three sectors of the theory arealso characterized and it is shown that, except in special cases, GR canonicalinitial data is sufficient to specify the GR sector as well as a specificsolution within this sector.  Finally, the path integral quantization of the theory is discussed at aformal level and a hueristic argument is given suggesting that in thesemiclassical limit the path integral is dominated by solutions in one of thenon-GR sectors, which would mean that the theory quantized in this way is not aquantization of GR.Comment: 19 pages latex. No figure

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