The Hilbert space costratification for the orbit type strata of SU(2)-lattice gauge theory

We construct the Hilbert space costratification of G = SU(2)-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work [F. Fürstenberg, G. Rudolph, and M. Schmidt, J. Geom. Phys. 119, 66–81 (2017)], where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture. In this picture, each element τ of the classical stratification corresponds to the zero locus of a finite subset {p_i} of the algebra R of G-invariant representative functions on GNℂ . Viewing the invariants as multiplication operators p̂_i on the Hilbert space H, the union of their images defines a subspace of H whose orthogonal complement H_τ is the element of the costratification corresponding to τ. To construct H_τ, one has to determine the images of the p̂_i explicitly. To accomplish this goal, we construct an orthonormal basis in H and determine the multiplication law for the basis elements; that is, we determine the structure constants of R in this basis. This part of our analysis applies to any compact Lie group G. For G = SU(2), the above procedure boils down to a problem in combinatorics of angular momentum theory. Using this theory, we obtain the union of the images of the operators p̂_i as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of Wigner’s 3nj symbols. The latter are further expressed in terms of 9j symbols. Using these techniques, we are also able to reduce the eigenvalue problem for the Hamiltonian of this theory to a problem in linear algebra.