Loop Approach to Lattice Gauge Theories

We solve the Gauss law and the corresponding Mandelstam constraints in the loop Hilbert space HL using the prepotential formulation of (d+1) dimensional SU(2) lattice gauge theory. The resulting orthonormal and complete loop basis, explicitly constructed in terms of the d(2d − 1) prepotential intertwining operators, is used to transcribe the gauge dynamics directly in HL without any redundant gauge and loop degrees of freedom. Using generalized Wigner-Eckart theorem and Biedenharn -Elliot identity in HL, we show that the above loop dynamics for pure SU(2) lattice gauge theory in arbitrary dimension, is given by real and symmetric 3nj coefficients of the second kind (e.g., n=6, 10 for d=2, 3 respectively). The corresponding “ribbon diagrams” representing SU(2) loop dynamics are constructed. The prepotential techniques are trivially extended to include fundamental matter fields leading to a description in terms of loops and strings. The SU(N) gauge group is briefly discussed.