Theoretical High Energy Physics Progress Report May 1 1979 April 30 1980 Columbia Univ New York PDF Download

The following results have been obtained: (a) A generalization of the Villian-type action to non-Abelian gauge groups has been developed. (This exploits the solution to the diffusion equation on the group manifold.) (b) The equivalence between the Villian U(1) lattice gauge theory and a noncompact Abelian gauge theory with monopoles has been generalized to relate a lattice gauge theory with gauge group G to a second gauge theory with group G bar and non-Abelian monopoles. Here G bar is the covering group of G and common terms in the character expansions of the two theories must agree. (c) A definition of a random lattice in n dimensions has been developed. The resulting lattice has greater rotational symmetry than the usual rectangular lattice and is a promising candidate for both numerical and analytical investigation. So far analytical results have been obtained for (1) various average properties (i.e. coordination number, link length, etc.) of the lattice, (2) the long distance, weak coupling limit of certain Green's functions, and (3) the strong coupling limit of the expectation value of a Wilson loop. The latter, (4), shows properties of a roughened loop suggesting that the random lattice may have a strong coupling limit that is more physical than in the usual rectilinear case.