SU(2) Symmetry and Conservation of Helicity for a Dirac Particle in a Static Magnetic Field at First Order

TL;DR

This paper analyzes the conservation of helicity and spin dynamics of a Dirac particle in static magnetic fields at first order, providing a symmetric description and a general transition matrix formula.

Contribution

It introduces an alternative symmetric framework for helicity conservation and derives a general transition matrix formula independent of specific magnetic field configurations.

Findings

Helicity conservation corresponds to invariance of spin component along total momentum.

Spin flip occurs along the momentum transfer vector, analogous to a bouncing ball.

Derived a general, field-independent transition matrix formula.

Abstract

We investigate the spin dynamics and the conservation of helicity in the first order $S-$matrix of a Dirac particle in any static magnetic field. We express the dynamical quantities using a coordinate system defined by the three mutually orthogonal vectors; the total momentum $\mathbf{k}=\mathbf{p_f}+\mathbf{p_i}$, the momentum transfer $\mathbf{q}=\mathbf{p_f-p_i}$, and $\mathbf{l}=\mathbf{k\times q}$. We show that this leads to an alternative symmetric description of the conservation of helicity in a static magnetic field at first order. In particular, we show that helicity conservation in the transition can be viewed as the invariance of the component of the spin along $\mathbf{k}$, and the flipping of its component along $\mathbf{q}$, just as what happens to the momentum vector of a ball bouncing off a wall. We also derive a "plug and play" formula for the transition matrix element where the only reference to the specific field configuration, and the incident and outgoing momenta is through the kinematical factors multiplying a general matrix element that is independent of the specific vector potential present.