Loop approach to lattice gauge theories

Abstract

We solve the Gauss law and the corresponding Mandelstam constraints in the loop Hilbert space H L 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 ( 2 d − 1 ) prepotential intertwining operators, is used to transcribe the gauge dynamics directly in H L without any redundant gauge and loop degrees of freedom. Using generalized Wigner–Eckart theorem and Biedenharn–Elliot identity in H L , we show that the above loop dynamics for pure SU(2) lattice gauge theory in arbitrary dimension, is given by real and symmetric 3 n j 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.