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 ˆH. It was shown by Loll [146Jump To The Next Citation Point] that solutions exist in the Renteln–Smolin formulation, where ∑ Hˆ(n) = 𝜖ijkTr (ˆU (n,⊓⊔ ˆ)τk)ˆpi(n, a)ˆpj(n,b) a,b aˆb. They are given by multiple, non-intersecting Polyakov loops (the lattice is assumed to have compact topology 3 T). 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 1 × 1 × 1-lattice in [146Jump To The Next Citation Point].

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 k (U ⊓⊔), where a single lattice plaquette ⊓⊔ 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. [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.

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