Canonical transformations and loop formulation of SU(N) lattice gauge theories

We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop and string flux operators along with their canonically conjugate loop and string electric fields. The canonical relations between the initial SU(N) link operators and the final SU(N) loop and string operators, consistent with SU(N) gauge transformations, are explicitly constructed over the entire lattice. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space H-p. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the (1/g(2)) magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the (g(2)) electric field terms describe all their interactions. In the weak coupling (g(2) -> 0) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbor interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group SO(4, 2) in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.