Quasicharacters, recoupling calculus, and Hamiltonian lattice quantum gauge theory

Abstract

We study the algebra R of G-invariant representative functions over the N-fold Cartesian product of copies of a compact Lie group G modulo the action of conjugation by the diagonal subgroup. Using the representation theory of G on the Hilbert space H=L2(GN)G, we construct a subset of G-invariant representative functions, which, by standard theorems, span H and thus generate R. The elements of this basis will be referred to as quasicharacters. For N = 1, they coincide with the ordinary irreducible group characters of G. The form of the quasicharacters depends on the choice of a certain unitary G-representation isomorphism, or reduction scheme, for every isomorphism class of irreps of G. We determine the multiplication law of R in terms of the quasicharacters with structure constants. Next, we use the one-to-one correspondence between complete bracketing schemes for the reduction of multiple tensor products of G-representations and rooted binary trees. This provides a link to the recoupling theory for G-representations. Using these tools, we prove that the structure constants of the algebra R are given by a certain type of recoupling coefficients of G-representations. For these recouplings, we derive a reduction law in terms of a product over primitive elements of 9j symbol type. The latter may be further expressed in terms of sums over products of Clebsch–Gordan coefficients of G. For G = SU(2), everything boils down to combinatorics of angular momentum theory. In the final part, we show that the above calculus enables us to calculate the matrix elements of bi-invariant operators occurring in quantum lattice gauge theory. In particular, both the quantum Hamiltonian and the orbit type relations may be dealt with in this way, thus reducing both the construction of the costratification and the study of the spectral problem to numerical problems in linear algebra. We spell out the spectral problem for G = SU(2), and we present sample calculations of matrix elements of orbit type relations for the gauge groups SU(2) and SU(3). The methods developed in this paper may be useful in the study of virtually any quantum model with polynomial constraints related to some symmetry.