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The configuration space is

**the connections on M:**

*A*by choice here we use SO(3) connections----the smooth rotation-valued 1-forms on M.

The connections represent all the possible configurations of gravity, or curvature, on the manifold-----in other words the possible geometries. There is no fixed prior choice of metric.

The quantum state space of the theory is a linear space

**consisting of complex-valued functions on**

*L***. We are following the notation in the Rovelli-Upadhya LQG primer.**

*A*A class of "cylindrical functions" is defined, spanning

**, and using these functions an inner product is defined, so that we have a Hilbert space.**

*L*Labeled networks enter here as a way of arriving at a basis for the Hilbert space. The set of "cylindrical functions" is highly redundant. They are very simple to define but there is a lot of overlap and the set is not linearly independent. To get a linearly independent spanning set of functions we have to be more methodical and selective. So to begin this thread I am going to describe a simplified version of labeled networks.

Without loss of generality, the networks can be taken to be trivalent---three legs meeting at each node. A node where more than 3 legs meet can always be broken down into a kind of "traffic circle" of tee-joint, or trivalent, nodes.