Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field

TLDR

The precession of a particle’s spin in a homogeneous electromagnetic field, a topic of recent experimental interest, has been studied for spin‑½ particles using the Dirac equation and related to classical spin dynamics, with earlier covariant formulations employing an antisymmetric tensor M by Frenkel and Kramers. The study aims to derive covariant classical equations of motion for arbitrary spin particles in a relativistic homogeneous electromagnetic field, expressed in terms of a polarization four‑vector s. The authors formulate the dynamics using a covariant axial four‑vector s to describe polarization, deriving the equations of motion directly in terms of this vector.

Abstract

The problem of the precession of the “spin” of a particle moving in a homogeneous electromagnetic field — a problem which has recently acquired considerable experimental interest — has already been investigated for spin ½ particles in some particular cases.1 In the literature the results were derived by explicit use of the Dirac equation, with the occasional inclusion of a Pauli term to account for an anomalous magnetic moment. On the other hand, following a remark of Bloch2 in connection with the nonrelativistic case, the expectation value of the vector operator representing the “spin” will necessarily follow the same time dependence as one would obtain from a classical equation of motion. To solve the problem for arbitrary spin in the relativistic case, it will thus suffice to produce a consistent set of covariant classical equations of motion. Such equations have been indicated a long time ago by Frenkel3 and are discussed by Kramers.4 These authors use an antisymmetric tensor M as the relativistic generalization of the intrinsic angular momentum observed in the rest-frame of the particle. A formulation in terms of the (axial) four-vector s which describes the polarization in a covariant fashion5 — though basically equivalent — is however much more convenient for our problem. We shall therefore derive first the equations of motion directly in terms of this four-vector s.