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arXiv · 2507.01219

From Symmetry to Structure: Gauge-Invariant Operators in Multi-Matrix Quantum Mechanics

Robert de Mello Koch·Minkyoo Kim·Hendrik J. R. Van Zyl
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Abstract

Recently the algebraic structure of gauge-invariant operators in multi-matrix quantum mechanics has been clarified: this space forms a module over a freely generated ring. The ring is generated by a set of primary invariants, while the module structure is determined by a finite set of secondary invariants. In this work, we show that the number of primary invariants can be computed by performing a complete gauge fixing, which identifies the number of independent physical degrees of freedom. We then compare this result to a complementary counting based on the restricted Schur polynomial basis. This comparison allows us to argue that the number of secondary invariants must exhibit exponential growth of the form $e^{cN^2}$ at large $N$, with $c$ a constant.

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Robert de Mello Koch, Minkyoo Kim, Hendrik J. R. Van Zyl. 2025-12-18. From Symmetry to Structure: Gauge-Invariant Operators in Multi-Matrix Quantum Mechanics. https://arxiv.org/abs/2507.01219

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