Fermions, Bosons, Anyons, Boltzmanions and Lie-Hopf Algebras

TL;DR

This paper develops an algebraic framework for quantum statistics using Lie-Hopf groups, revealing new connections between algebraic structures and particle statistics, including fermions, bosons, anyons, and Boltzmann particles.

Contribution

It introduces a novel algebraic formalization of quantum statistics via Lie-Hopf groups, challenging traditional group-statistics relations and incorporating thermal bath effects.

Findings

Different statistics are formalized as coherent states of Lie-Hopf groups.

Boltzmann statistics relates to h(1) in presence of a thermal bath.

Fermions are described using su(2) in the fundamental representation.

Abstract

Usual quantum statistics is written in Fock space but it is not an algebraic theory. We show that at a deeper level it can be algebraically formalized defining the different statistics as (multi-mode) coherent states of the appropriate (but different from the usual ones) Lie-Hopf groups. The traditional connection between groups and statistics, established in vacuum, is indeed subverted by the interaction with the thermal bath. We show indeed that h(1), related in quantum field theory to bosons, must be used to define in presence of a bath the Boltzmann statistics while, to build the Bose statistics, we have to take into account su(1,1). Astonishing to describe fermions we are forced to use not the superalgebra h(1|1) but su(2) in the fundamental representation. Higher representations of su(2) allow also to give a possible definition of anyon statistics with generalized Pauli principle. Physical implications are discussed; the results is more general then the usual on the discrete spectrum, but everything collapses to standard theory when the continuum limit is performed.