Loop-string-hadron approach to SU(3) lattice Yang-Mills theory, II: Operator representation for the trivalent vertex

TL;DR

This paper develops an explicit infinite-dimensional matrix representation for gauge-invariant operators at a trivalent SU$(3)$ lattice gauge theory vertex within the Loop-String-Hadron framework. Building on irreducible Schwinger bosons, the authors derive closed-form matrix elements for a broad set of operators acting on the naive LSH basis, enabling calculations to proceed directly in LSH variables rather than the Schwinger-boson formalism. A companion Mathematica notebook accompanies the analytic results, showcasing state normalization, Gram–Schmidt orthogonalization, and eigenstate construction for the seventh Casimir, with a practical emphasis on Hamiltonian-based QCD simulations. This work removes the need to revert to underlying bosonic variables for operator actions, improves computational efficiency, and lays the groundwork for Part III, which will extend the framework to full lattices and realistic QCD dynamics.

Abstract

This work is the second installment of a series on the Loop-String-Hadron (LSH) approach to SU(3) lattice Yang-Mills theory. Here, we present the infinite-dimensional matrix representation for arbitrary gauge-invariant operators at a trivalent vertex, which results in a standalone framework for computations that supersedes the underlying Schwinger-boson framework. To that end, we evaluate in closed form the result of applying any gauge-invariant operators on the LSH basis states introduced in Part I. Classical calculations in the LSH basis run significantly faster than equivalent calculations performed using Schwinger bosons. A companion code script is provided, which implements the derived formulas and aims to facilitate rapid progress towards Hamiltonian-based calculations of quantum chromodynamics.

Figures (1)

• Figure 1: A trivalent vertex with legs $i$ labeled as $i=1$, 2, or 3.