Kostka Numbers Constrain Particle Exchange Statistics beyond Fermions and Bosons

• URL: http://arxiv.org/abs/2505.17361v2
• Date: Mon, 26 May 2025 04:48:56 GMT
• Title: Kostka Numbers Constrain Particle Exchange Statistics beyond Fermions and Bosons
• Authors: Chi-Chun Zhou, Yu-Zhu Chen, Shuai A. Chen, Yao Shen, Fu-Lin Zhang, Wu-Sheng Dai,
• Abstract summary: We establish the microstate uniqueness theorem (MUT)<n>route 1 and symmetric-group-based quantum-mechanical exchange symmetry are mathematically incompatible under indistinguishability.<n>The MUT relies solely on the indistinguishability principle, without invoking Lorentz symmetry or any field-theoretic assumptions.
• Score: 2.1294282254359613
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Existing theoretical explorations of intermediate statistics beyond bosons and fermions have followed three routes: (1) a statistical-mechanics route that modifies microstate counting rules; (2) a quantum-mechanics route that generalizes wavefunction exchange symmetry via group representations; and (3) a quantum-field-theory route that deforms the creation-annihilation algebra. While each route has advanced individually, a unified formulation remains elusive. Recently, consistency between routes (2) and (3) was demonstrated (Nature 637, 314 (2025)). Here, employing combinatorial arguments with Kostka numbers, we establish the microstate uniqueness theorem (MUT). It demonstrates that statistical-mechanics counting constraints (route 1) and symmetric-group-based quantum-mechanical exchange symmetry (a restricted subset of route 2, excluding braid-group generalizations) are mathematically incompatible under indistinguishability. Consequently, intermediate statistics based on higher-dimensional irreducible representations of the symmetric group or on modified microstate-counting rules are mathematically ruled out for indistinguishable particles. The MUT relies solely on the indistinguishability principle, without invoking Lorentz symmetry or any field-theoretic assumptions.

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