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 , 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].

Discrete Approaches to Quantum Gravity in Four Dimensions Renate Loll http://www.livingreviews.org/lrr-1998-13 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de