How to commute

Andreas Walter Aste,Won Sang Chung

doi:10.12988/astp.2016.512116

Abstract

A simple exposition of the rarely discussed fact that a set of free boson fields describing different, i.e. kinematically different particle types can be quantized with mutual anticommutation relations is given by the explicit construction of the Klein transformations changing anticommutation relations into commutation relations. The q-analog of the presented results is also treated. The analogous situation for two independent free fermion fields with mutual commutation or anticommutation relations is briefly investigated.

Highlights

All hitherto existing experimental evidence indicates that physical systems with one type of integer spin particles solely obey the laws of Bose-Einstein statistics, whereas systems with one type of half-odd integer spin particles respect Fermi-Dirac statistics
The natural way to arrive at Bose-Einstein or FermiDirac statistics is to describe the particles by the help of quantum fields which commute or anticommute for space-like separations, repectively
When one turns from the commutation relations for a given field to those between different fields in the sense that the fields cannot be mapped by space-time transformations onto each other, the situation becomes more complicated

Summary

Introduction

All hitherto existing experimental evidence indicates that physical systems with one type of integer spin particles solely obey the laws of Bose-Einstein statistics, whereas systems with one type of half-odd integer spin particles respect Fermi-Dirac statistics. One observes that ‘abnormal’ commutation relations in theories in which, e.g., two different integer spin fields anticommute, may arise, but such theories possess special symmetries which allow to link them to the case with regular commutation relations. In this paper, it is shown how this link can be constructed from simple algebraic considerations for systems with a finite number of degrees of freedom which can be generalized in a straightforward manner to the case of inifinitely many degrees of freedom. The abstract work on a quantum field theoretical level in connection with the spin-statistics theorem given much later by Huzihiro Araki [2] was the basis for a short discussion given by Ray Streater and Arthur Wightman in their famous book on PCT, spin and statistics, and all that [3]

Several degrees of freedom

How to q-commute

Changing the mutual commutation relations of two different Dirac fields