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American Journal of Physics

Simple approach to relativistic spin dynamics

Krzysztof Rębilas

Published Sep 26, 2011• 7 citations
Publisher

Abstract

Spin dynamics is considered from the point of view of relativistic classical mechanics. It is shown that the assumptions of the Bargmann-Michel-Telegdi theory may be used to obtain an intuitive and convenient equation of motion. The advantage of this approach is that the presence of Thomas precession is automatically and manifestly demonstrated. An easy and unambiguous way to achieve the original Bargmann-Michel-Telegdi equation is also proposed.

DOI
10.1119/1.3619868

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Spin dynamics is considered from the point of view of relativistic classical mechanics. It is shown that the assumptions of the Bargmann-Michel-Telegdi theory may be used to obtain an intuitive and convenient equation of motion. The advantage of this approach is that the presence of Thomas precession is automatically and manifestly demonstrated. An easy and unambiguous way to achieve the original Bargmann-Michel-Telegdi equation is also proposed.
This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.
A great effort has been devoted to formulate a classical relativistic theory of spin compatible with quantum relativistic wave equations. The main difficulty in order to connect classical and quantum theories rests in finding a parameter which plays the role of proper time at a purely quantum level. We present a partial review of several proposals of classical and quantum spin theories from the pioneer works of Thomas and Frenkel, revisited in the classical BMT work, to the semiclassical model of Barut and Zanghi [Phys. Rev. Lett. 52, 2009 (1984)]. We show that the last model can be obtained from a semiclassical limit of the Feynman proper time parametrization of the Dirac equation. At the quantum level we derive spin precession equations in the Heisenberg picture. Analogies and differences with respect to classical theories are discussed in detail. 1