representations, so one needs to restrict to the class of appropriate “regular” representations by hand. This should be compared with the use of the Weyl algebra for canonical systems, which contains many nonregular representations. It is a well-known practical necessity for the Weyl algebra to restrict representations by hand to regular representations when one analyzes physical systems. We are able to prove the existence of gauge invariant ground states which are regular.

Whereas each true local kinematics algebra (corresponding to a finite sublattice) has a unique ground state w.r.t. the local time evolution, for the infinite lattice limit, we do not presently have such a uniqueness property. The ground states are the weak \*-limit points of a sequence of “partial ground states”. This nonuniqueness needs further investigation. For the finite lattice, in a toy model the spectral problem for the lattice Hamiltonian has been solved exactly, yielding an explicit formula for the unique ground state, cf. [@HRS].

Our paper is organized as follows. In Sect. \[PM\] we state the model for the finite lattice taken from [@KR; @KR1], and give a very natural representation for it. We then generalize this representation for the infinite lattice in Sect. \[GFA\], and construct 