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 [146Jump To The Next Citation Point99Jump To The Next Citation Point].
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