Grassmann Matrix Quantum Mechanics

TL;DR

This paper investigates quantum mechanical models with Grassmann-valued rectangular matrices, revealing how low-energy sectors can be described by bosonic Hermitian matrix quantum mechanics on a complex Kahler manifold, with potential holographic implications.

Contribution

It introduces a class of Grassmann matrix quantum mechanics models and demonstrates their low-energy classical phase space as a compact Kahler manifold, linking Grassmann variables to emergent bosonic matrices.

Findings

Low-energy sector described by bosonic Hermitian matrix quantum mechanics.

Classical phase space is a compact Kahler manifold parameterized by complex matrices.

Models suggest possible holographic interpretations with finite-dimensional Hilbert spaces.

Abstract

We explore quantum mechanical theories whose fundamental degrees of freedom are rectangular matrices with Grassmann valued matrix elements. We study particular models where the low energy sector can be described in terms of a bosonic Hermitian matrix quantum mechanics. We describe the classical curved phase space that emerges in the low energy sector. The phase space lives on a compact Kahler manifold parameterized by a complex matrix, of the type discovered some time ago by Berezin. The emergence of a semiclassical bosonic matrix quantum mechanics at low energies requires that the original Grassmann matrices be in the long rectangular limit. We discuss possible holographic interpretations of such matrix models which, by construction, are endowed with a finite dimensional Hilbert space.