Part of solving the canonical looks matters most quantum theory is to determine the states annihilated by the Hamiltonian constraint tex2html_wrap_inline2373 . It was shown by Loll [146Jump To The Next Citation Point In The Article] that solutions exist in the Renteln-Smolin formulation, where tex2html_wrap_inline2375 . They are given by multiple, non-intersecting Polyakov loops (the lattice is assumed to have compact topology tex2html_wrap_inline2377). Such solutions are trivial in the sense that they correspond to quantum states ``without volume''. The difficulties one encounters when trying to find other solutions is illustrated by the explicit calculations for the tex2html_wrap_inline2379 -lattice in [146Jump To The Next Citation Point In The Article].
The search for solutions was continued by Ezawa  (see also  for an extensive review), who used a symmetrized form of the Hamiltonian. His solutions depend on multiple plaquette loops tex2html_wrap_inline2381, where a single lattice plaquette tex2html_wrap_inline2383 is traversed by the loop k times. The solutions are less trivial than those formed from Polyakov loops, since they involve kinks, but they are still annihilated by the volume operator.
A somewhat different strategy was followed by Fort et al , who constructed a Hamiltonian lattice regularization for the calculation of certain knot invariants. They defined lattice constraint operators in terms of their geometric action on lattice Wilson loop states, and reproduced some of the formal continuum solutions to the polynomial Hamiltonian constraint of complex Ashtekar gravity on simple loop geometries = h_o_t