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
, 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
for holomorphic
holonomies on the lattice was used in [146,
99].