(T/E) GAUGE INVARIANT VARIATIONAL METHODS FOR HAMILTONIAN LATTICE GAUGE THEORIES

This paper develops variational methods for calculating the ground state and excited state spectrum of Hamiltonian lattice gauge theories defined in A0 = 0 gauge. The scheme introduced in this paper has the advantage of allowing one to convert more familiar tools such as mean-field, Hartree-Fock and real space renormalization group approximations, which are by their very nature gauge non-invariant methods, into fully gauge invariant techniques. We show that these methods apply in the same way to both abelian and non-abelian theories, and that they are at least powerful enough to correctly describe the physics of periodic quantum electrodynamics in 2+1 and 3+1 space-time dimensions. This paper formulates the problem for both abelian and non-abelian theories and shows how to reduce the Rayleigh-Ritz problem to that of computing the partition function of a classical spin system. We discuss the evaluation of the effective spin problem which one derives for PQED and then discuss ways of carrying out the evaluation of the partition function for the system equivalent to a non-abelian theory. The explicit form of the effective partition function for the non-abelian theory is derived, but because the evaluation of this function is considerably more complicated than the one derived in the abelian theory no explicit evaluation of this function is presented. However, by comparing the gauge-projected Hat-tree-Fock wavefunction for PQED with that of the pure SU(2) gauge theory, we are able to show that extremely interesting differences emerge between these theories even at this simple level. We close with a discussion of fermions and a discussion of how one can extend these ideas to allow the computation of the glue-ball and hadron spectrum.-2