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Quantum Statistics Forbids Particle Exchange Statistics beyond Bosons and Fermions in 3D

Chi-Chun Zhou, Shuai A. Chen, Yu-Zhu Chen, Yao Shen, Fu-Lin Zhang, Wu-Sheng Dai

Abstract

Quantum matter in three spatial dimensions is observed to consist exclusively of bosons and fermions. Whether this empirical fact follows from basic consistency requirements of quantum theory itself or must be imposed as an additional principle has for 80 years remained a fundamental conceptual gap. Here we close this gap by establishing a no-go theorem that excludes any particle exchange statistics beyond bosons and fermions in three dimensions. We identify the consistency conditions linking the many-body Hilbert-space structure of quantum mechanics with the statistical microstate counting of indistinguishable particles. As a corollary, we demonstrate that higher-dimensional representations of the symmetric group cannot give rise to genuinely distinct particle exchange statistics in any spatial dimension.

Quantum Statistics Forbids Particle Exchange Statistics beyond Bosons and Fermions in 3D

Abstract

Quantum matter in three spatial dimensions is observed to consist exclusively of bosons and fermions. Whether this empirical fact follows from basic consistency requirements of quantum theory itself or must be imposed as an additional principle has for 80 years remained a fundamental conceptual gap. Here we close this gap by establishing a no-go theorem that excludes any particle exchange statistics beyond bosons and fermions in three dimensions. We identify the consistency conditions linking the many-body Hilbert-space structure of quantum mechanics with the statistical microstate counting of indistinguishable particles. As a corollary, we demonstrate that higher-dimensional representations of the symmetric group cannot give rise to genuinely distinct particle exchange statistics in any spatial dimension.

Paper Structure

This paper contains 72 sections, 98 equations, 1 figure, 8 tables.

Table of Contents

  1. The Motivation
  2. A Long-Standing Foundational Question
  3. Existing No-Go Theorems and Their Scope
  4. An Eighty-Year Investigation: Existing Intermediate Particle Exchange Statistics
  5. Statistical-mechanical generalizations.
  6. Quantum-mechanical approaches.
  7. Symmetric-function and combinatorial approaches.
  8. Dynamical and effective-model approaches.
  9. Summary
  10. Fundamental Premises of Our No-Go Theorem
  11. Quantum statistics of indistinguishable particles
  12. Many-body Hilbert Space in Quantum Mechanics
  13. Label Configuration Space and Permutation Symmetry
  14. Label Configuration Space
  15. Configuration Space of Identical Particles and Fundamental Group
  16. ...and 57 more sections

Figures (1)

  • Figure 1: Multi-particle spectra for systems obeying Bose, Fermi, Gentile, and parastatistics for (a) a general Hamiltonian as in Eq. \ref{['eq:htij']} and (b) a diagonal Hamiltonian. Gentile-1 and Gentile-2 refer to the two possible realization schemes. For a non-diagonal Hamiltonian, only bosonic and fermionic systems remain free. For a diagonal Hamiltonian, all statistics yield free many-body systems. However, only bosons and fermions lead to non-negative coefficients in the Schur-function expansion of the partition function. For parastatistics, to construct the many-body spectrum from the single-particle energy states, we simply assume a maximal occupation number to be $m$ which may lead to additional spectral branches.