Corrections to the Emergent Canonical Commutation Relations Arising in the Statistical Mechanics of Matrix Models

TL;DR

This paper investigates how canonical commutation relations emerge in matrix models' statistical mechanics, identifying conditions under which corrections are small and highlighting the role of supersymmetry in these emergent relations.

Contribution

It derives Ward identities to analyze corrections to commutation relations and establishes that supersymmetry is essential for their emergence in complex Hilbert space matrix models.

Findings

Emergent canonical commutators require equal fermionic and bosonic degrees of freedom.

Corrections to commutation relations can be minimized under specific conditions.

Supersymmetry is crucial for the emergence of canonical commutation relations.

Abstract

We study the leading corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models, by deriving several related Ward identities, and give conditions for these corrections to be small. We show that emergent canonical commutators are possible only in matrix models in complex Hilbert space for which the numbers of fermionic and bosonic fundamental degrees of freedom are equal, suggesting that supersymmetry will play a crucial role. Our results simplify, and sharpen, those obtained earlier by Adler and Millard.