Collective Theory at Finite-$N$: Reduction of the Emergent Hilbert Space

TL;DR

This paper develops an algorithm to analyze finite-$N$ Hilbert spaces in emergent theories with $S_N$ symmetry, specifically for $N$ bosons in $d$ dimensions, revealing a finite, reduced gauge-invariant operator basis.

Contribution

It introduces a complete linear algebra-based method to identify independent invariants and relations, enabling the reduction of the emergent Hilbert space at finite $N$.

Findings

Successfully reproduces primaries and secondary invariants

Verifies the independence of gauge-invariant operators

Demonstrates a finite, emergent Fock space structure

Abstract

Continuing the formulation of finite $N$ Hilbert spaces in emergent theories we study in this work $S_{N}$ symmetric collective models. For the case of $N$ bosons in $d$ dimensions, which map to matrix models with commuting matrices, we describe a complete algorithm and give a detailed case study reproducing the expected primaries and determining secondary invariants at each bidegree (a Hironaka decomposition). The method is based on null spaces (of the full collective theory) which are seen to yield all the independent trace relations, reducing the construction to linear algebra. As a stringent check, of our algorithm, we have verified that the system of invariants generates a subset of gauge invariant operators with no redundancies. This results in a reduction of the Hilbert space, in particular the gauge invariant secondary invariants realize an emergent Fock space with finite-$N$ occupation-numbers.