On the Classical Limit of the Balanced State Sum

In [1], a new state sum model (the “balanced” model) was suggested as a quantum theory of gravity. In [2], a proposal was made to construct a geometrical interpretation of it, and to demonstrate that it gives general relativity in the classical limit. The purpose of this note is to make several advances in the interpretation of the balanced model. First, we outline a shortcoming of the definition of the model in [1] pointed out to us by John Barrett and John Baez [3], and explain how to correct it. Second, we show that the classical limit of our state sum reproduces the Einstein-Hilbert lagrangian whenever the term in the state sum to which it is applied has a geometrical interpretation. Next we outline a program to demonstrate that the classical limit of the state sum is in fact dominated by terms with geometrical meaning. This uses in an essential way the alteration we have made to the model in order to fix the shortcoming discussed in the first section. Finally, we make a brief discussion of the Minkowski signature version of the model. This note is not intended to be self contained, it can only be read after familiarizing oneself with [1] and [2].