### 2.13 Solutions to the Wheeler–DeWitt equation

Part of solving the canonical quantum theory is to determine the states annihilated by the Hamiltonian
constraint . It was shown by Loll [146] that solutions exist in the Renteln–Smolin formulation, where
. They are given by multiple, non-intersecting Polyakov
loops (the lattice is assumed to have compact topology ). 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 -lattice
in [146].
The search for solutions was continued by Ezawa [99] (see also [100] for an extensive review), who used
a symmetrized form of the Hamiltonian. His solutions depend on multiple plaquette loops , where a
single lattice plaquette is traversed by the loop 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. [102], 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.