Reconstruction of Quantum Particle Statistics: Bosons, Fermions, and Transtatistics

TL;DR

This paper classifies quantum particle statistics using operational assumptions, revealing known bosonic and fermionic types and discovering new 'transtatistics' with unique symmetry and degeneracy properties.

Contribution

It provides a new classification framework for quantum statistics based on operational principles, introducing novel 'transtatistics' beyond traditional bosons and fermions.

Findings

Bosons and fermions are the basic quantum statistics with minimal symmetry.

Discovered families of 'transtatistics' with hidden symmetries and degeneracies.

Identified effects like spontaneous symmetry breaking in new statistics.

Abstract

Identical quantum particles exhibit only two types of statistics: bosonic and fermionic. Theoretically, this restriction is commonly established through the symmetrization postulate or (anti)commutation constraints imposed on the algebra of creation and annihilation operators. The physical motivation for these axioms remains poorly understood, leading to various generalizations by modifying the mathematical formalism in somewhat arbitrary ways. In this work, we take an opposing route and classify quantum particle statistics based on operationally well-motivated assumptions. Specifically, we consider that a) the standard (complex) unitary dynamics defines the set of single-particle transformations, and b) phase transformations act locally in the space of multi-particle systems. We develop a complete characterization, which includes bosons and fermions as basic statistics with minimal symmetry. Interestingly, we have discovered whole families of novel statistics (dubbed transtatistics) accompanied by hidden symmetries, generic degeneracy of ground states, and spontaneous symmetry breaking -- effects that are (typically) absent in ordinary statistics.