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 . However, since the Ashtekar connections are complex-valued, the gauge group is
the non-compact group , and the gauge-invariant Wilson loop functions are not
square-integrable. For the alternative formulation in terms of real -variables (see below), these
problems are not present. An alternative heat kernel measure for holomorphic holonomies
on the lattice was used in [146, 99].