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

Abstract

We construct the Hilbert space costratification of $G=\mathrm{SU}(2)$-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work where we have implemented the classical gauge orbit strata on quantum level within a suitable holomorphic picture. In this picture, each element $\tau$ of the classical stratification corresponds to the zero locus of a finite subset $\{p_i\}$ of the algebra $\mathcal R$ of $G$-invariant representative functions on the complexification of $G^N$. Viewing the invariants as multiplication operators $\hat p_i$ on the Hilbert space $\mathcal H$, the union of their images defines a subspace of $\mathcal H$ whose orthogonal complement $\mathcal H_\tau$ is the element of the costratification corresponding to $\tau$. To construct $\mathcal H_\tau$, one has to determine the images of the $\hat p_i$ explicitly. To accomplish that goal, we construct an orthonormal basis in $\mathcal H$ and determine the multiplication law for the basis elements, that is, we determine the structure constants of $\mathcal R$ in this basis. This part of our analysis applies to any compact Lie group $G$. For $G = \mathrm{SU}(2)$, 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 $\hat p_i$ 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.