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

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 $\mathbf{k}={\mathbf{p}}_{\mathbf{f}}+{\mathbf{p}}_{\mathbf{i}}$, the momentum transfer $\mathbf{q}={\mathbf{p}}_{\mathbf{f}}-{\mathbf{p}}_{\mathbf{i}}$, and $\mathbf{l}=\mathbf{k}\times \mathbf{q}$. 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.

arXiv comment4 figures, 12 pages, content refined