Unitary Irreducible Representations of a Lie Algebra for Matrix Chain Models

TL;DR

This paper develops a decomposition-based method to construct all unitary irreducible representations of a Lie algebra associated with open matrix chain models, linking them to physical states and extending to a generalized Virasoro algebra.

Contribution

It introduces a novel decomposition of the Lie algebra for matrix chain models and constructs all corresponding unitary irreducible representations, including a new quotient algebra extension.

Findings

All multiple meson states are obtained through this construction.

States with finite quantum numbers are characterized as multiple meson states.

The quotient algebra's representation theory may describe thermodynamic limit physics.

Abstract

There is a decomposition of a Lie algebra for open matrix chains akin to the triangular decomposition. We use this decomposition to construct unitary irreducible representations. All multiple meson states can be retrieved this way. Moreover, they are the only states with a finite number of non-zero quantum numbers with respect to a certain set of maximally commuting linearly independent quantum observables. Any other state is a tensor product of a multiple meson state and a state coming from a representation of a quotient algebra that extends and generalizes the Virasoro algebra. We expect the representation theory of this quotient algebra to describe physical systems at the thermodynamic limit.