Topological Quantum Numbers of Relativistic Two-Particle Mixtures

TL;DR

This paper explores the topological properties of relativistic two-particle quantum mixtures, revealing how their variables induce mappings to exchange groups and identifying topological quantum numbers related to the Hamiltonian.

Contribution

It introduces a topological framework for analyzing relativistic two-particle mixtures and identifies winding numbers as topological invariants of the system.

Findings

Identification of topological quantum numbers for two-particle systems

Mapping of space-time to exchange groups SU(2) and SU(1,1)

Existence of winding numbers as topological characteristics

Abstract

The relativistic two-particle quantum mixtures are studied from the topological point of view. The mixture field variables can be transformed in such a way that a kinematical decoupling of both particle degrees of freedom takes place with a residual coupling of purely algebraic nature ("exchange coupling"). Both separated sets of particle variables induce a certain map of space-time onto the corresponding "exchange groups", i.e. SU(2) and SU(1,1), so that for the compact case (SU(2)) there arises a pair of winding numbers, either odd or even, which are a topological characteristic of the two-particle Hamiltonian.