Loop Approach to Lattice Gauge Theories

Abstract

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

32 pages, 9 figures, typos corrected, new references added (to be published in Nucl. Phys. B)

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