The measure

2.11 The measure

There is a natural measure for the quantum theory, given by the product over all lattice edges of the Haar measures $dg$ . However, since the Ashtekar connections $A$ are complex-valued, the gauge group is the non-compact group $SO (3,ℂ ) = SL (2, ℂ)$ , and the gauge-invariant Wilson loop functions are not square-integrable. For the alternative formulation in terms of real $SU (2)$ -variables (see below), these problems are not present. An alternative heat kernel measure $dν$ for holomorphic $SL (2,ℂ )$ holonomies on the lattice was used in [146

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