Hilbert Space of Finite $N$ Multi-matrix Models

TL;DR

This paper explores the Hilbert space of gauge-invariant operators in large-N multi-matrix quantum mechanics, revealing a truncation in the spectrum due to finite-N trace identities that constrain degrees of freedom.

Contribution

It identifies a class of light single-trace operators that behave like free creation operators but saturate at high energy, showing how finite-N effects limit the spectrum.

Findings

Finite-N trace identities cause spectrum truncation.

Emergent degrees of freedom are fewer than semiclassical estimates.

High-energy states are constrained by nonperturbative effects.

Abstract

We study the Hilbert space structure of gauge-invariant operators emergent in large-$N$ multi-matrix quantum mechanics. Building on the framework of \cite{deMelloKoch:2025ngs}, we identify a class of light single-trace operators that behave like free creation operators at low energy but saturate beyond a critical excitation level, ceasing to generate new states. This $q$-reducibility is a direct consequence of finite N trace identities and leads to a dramatic truncation of the high-energy spectrum of the emergent theory. The resulting number of independent degrees of freedom is far smaller than na\"ive semiclassical expectations, providing a concrete mechanism for how nonperturbative constraints shape the ultraviolet behaviour of emergent theories.