Quantum particles in non-commutative space-time: an identity crisis

TL;DR

This paper reveals that in quantum systems with non-commutative space-time, the concept of identical particles becomes ill-defined due to non-abelian momentum composition, challenging previous Fock space constructions and resolving exchange symmetry issues.

Contribution

It demonstrates the breakdown of traditional particle identity notions in non-commutative space-time and introduces a covariant braiding of momentum quantum numbers.

Findings

Previous Fock space models are based on incorrect assumptions.

Total momentum remains well-defined despite individual momenta being non-unique.

Braiding of momentum quantum numbers is covariant under relativistic transformations.

Abstract

We argue that the notion of identical particles is no longer well defined in quantum systems governed by non-commutative deformations of space-time symmetries. Such models are characterized by four-momentum space given by a non-abelian Lie group. Our analysis is based on the observation that, for states containing more than one particle, only the total momentum of the system is a well defined quantum number. Such total momentum is obtained from the non-abelian composition of the particles individual momenta which are no longer uniquely defined. The main upshot of our analysis is that all previous attempts to construct Fock spaces for these models rested on wrong assumptions and indeed have been unsuccessful. We also show how the natural braiding of momentum quantum numbers which characterizes the exchange of factors in the tensor product of states is covariant under relativistic transformations thus solving a long standing problem in the field.