Commuting Quantum Matrix Models

TL;DR

This paper investigates quantum systems of commuting matrices, revealing the necessity of curvature-dependent potentials for finite energy ground states and analyzing eigenvalue distributions, especially highlighting the critical role of p=4.

Contribution

It introduces the need for curvature-dependent potentials in commuting matrix quantum systems and analyzes eigenvalue distributions in the large matrix limit, emphasizing the special case p=4.

Findings

Curvature-dependent potential is essential for finite energy ground states.

Eigenvalues form a sharp spherical shell for p≥4.

p=4 is a critical dimension for eigenvalue distribution behavior.

Abstract

We study a quantum system of $p$ commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to avoid such curvature dependence in the Hamiltonian. We study the eigenvalue distribution for such systems in the large matrix size limit. A critical r\^ole is played by $p=4$. For $p\ge4$ the competition between eigenvalue repulsion and the attractive potential forces the eigenvalues to form a sharp spherical shell.