Dirac Particle in an Aharonov-Bohm Potential: The Structure of the First Order S-matrix

TL;DR

This paper analyzes the algebraic structure of the first order S-matrix for a Dirac particle in an Aharonov-Bohm field, revealing how it splits wave components according to spin and angular momentum conservation.

Contribution

It uncovers the algebraic properties of the interaction Hamiltonian that lead to the splitting of wave components by spin and angular momentum in the first order S-matrix.

Findings

Wave components split into spin eigenstates.

Partial waves of total angular momentum split into orbital angular momentum channels.

Conservation of total angular momentum is maintained.

Abstract

The structure of the interaction Hamiltonian in the first order $S-$matrix element of a Dirac particle in an Aharonov-Bohm (AB) field is analyzed and shown to have interesting algebraic properties. It is demonstrated that as a consequence of these properties, this interaction Hamiltonian splits both the incident and outgoing waves in the the first order $S-$matrix into their $\frac{\Sigma_3}{2}-$components (eigenstates of the third component of the spin). The matrix element can then be viewed as the sum of two transitions taking place in these two channels of the spin. At the level of partial waves, each partial wave of the conserved total angular momentum is split into two partial waves of the orbital angular momentum in a manner consistent with the conservation of the total angular momentum quantum number.