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.

47 pages. arXiv admin note: text overlap with arXiv:1702.01047

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Cited by in corpus (1)

• Quasicharacters, recoupling calculus and costratifications of lattice quantum gauge theory