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

journal contribution

posted on 2023-05-20, 16:55 authored by Fuchs, E, Peter JarvisPeter Jarvis, Rudolph, G, Schmidt, MWe 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*} of the algebra R of_{i}*G*-invariant representative functions on*G**N*ℂ . Viewing the invariants as multiplication operators*p̂*on the Hilbert space H, the union of their images defines a subspace of H whose orthogonal complement H_{i}_{τ}is the element of the costratification corresponding to*τ*. To construct H_{τ}, one has to determine the images of the*p̂*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_{i}*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*as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of Wigner’s 3_{i}*nj*symbols. The latter are further expressed in terms of 9*j*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.## History

## Publication title

Journal of Mathematical Physics## Volume

59## Issue

8## Article number

083505## Number

083505## Pagination

1-32## ISSN

0022-2488## Department/School

School of Natural Sciences## Publisher

Amer Inst Physics## Place of publication

Circulation & Fulfillment Div, 2 Huntington Quadrangle, Ste 1 N O 1, Melville, USA, Ny, 11747-4501## Rights statement

© 2018 Author(s). The published version of the article and the accepted manuscript may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Fuchs, E., Jarvis, P. D. Rudolph, G. Schmidt, M, 2018 The Hilbert space costratification for the orbit type strata of SU(2)-lattice gauge theory, Journal of mathematical physics, 59(8), and may be found at https://doi.org/10.1063/1.5031115## Repository Status

- Open