Furthermore, if the action is quadratic then the theory is solvable. This means that interacting field theories can be defined and treated perturbatively. Since the discreteness of the group lattice renders the system ultraviolet finite, these theories are of interest as possibilities for finite quantum versions of gravity. Twisted group lattices offer a straightforward, competitive, yet mathematically similar approach to quantum hyperspaces. Roughly speaking, group lattices are more regular than random lattices but more random than regular lattices.
Enough structure is preserved so as to maintain solvability.
The structure is determined by a group G . Nearest neighbor sites are determined by a subset of G denoted by NN . A precise definition is as follows. Starting with G , one associates a site with each element in G . The origin is the identity element, e .