Laplacians in polar matrix coordinates and radial fermionization in higher dimensions

TL;DR

This paper introduces matrix-valued polar coordinates to analyze the Laplacian in invariant states of multi-matrix quantum systems, revealing a fermionic equivalence in certain sectors and identifying inter-eigenvalue interactions.

Contribution

It develops a new polar coordinate framework for matrix Hamiltonians and demonstrates fermionization in the radial invariant sector of multi-matrix models.

Findings

Radially invariant sector maps to non-interacting 2+1D fermions.

Density description of the fermionic system is established.

Identification of a repulsive inter-eigenvalue potential for multiple matrices.

Abstract

We consider the quantum mechanical hamiltonian of two, space indexed, hermitean matrices. By introducing matrix valued polar coordinates, we obtain the form of the laplacian acting on invariant states. For potentials depending only on the eigenvalues of the radial matrix, we establish that the radially invariant sector is equivalent to a system of non interacting 2+1 dimensional fermions, and obtain its density description. For a larger number of matrices, the presence of a repulsive radial inter-eigenvalue potential is identified.