Quasicharacters, recoupling calculus and costratifications of lattice quantum gauge theory

TL;DR

This paper develops a mathematical framework using quasicharacters and recoupling calculus to analyze invariant functions in lattice quantum gauge theory, enabling explicit calculations of operators and spectral problems.

Contribution

It introduces quasicharacters as a basis for invariant functions, linking their structure to recoupling coefficients and binary trees, and applies this to quantum lattice gauge theory.

Findings

Explicit basis of invariant functions (quasicharacters) constructed.

Multiplication laws expressed via recoupling coefficients and 9j symbols.

Sample calculations for SU(2) and SU(3) gauge groups.

Abstract

We study the algebra of 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. We construct a basis of invariant representative functions referred to as quasicharacters. The form of the quasicharacters depends on the choice of a reduction scheme. We determine the multiplication law of quasicharacters and express their structure constants in terms of recoupling coefficients. Via this link, the choice of the reduction scheme acquires an interpretation in terms of binary trees. We show explicitly that the structure constants decompose into products over primitive elements of 9j symbol type. For SU(2), everything boils down to the 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 occuring 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 may be useful in the study of virtually all quantum models with polynomial constraints related to some symmetry.