On Stern-Gerlach forces allowed by special relativity and the special case of the classical spinning particle of Derbenev-Kondratenko

Abstract

This work is devoted to an examination of Stern-Gerlach forces consistent with special relativity and is motivated by recent interest in the relativistic Stern-Gerlach force acting on polarized protons in high-energy particle accelerators. The equations for the orbital and spin motion of a classical charged particle with arbitrary intrinsic magnetic dipole moment in an external electromagnetic field are considered and by imposing the constraints of special relativity and restricting to first order in spin (= first order $\hbar$) a well-defined class of spin-orbit systems is obtained. All these systems can be treated on an equal footing including such prominent cases as those considered by Frenkel and by Good. The Frenkel case is considered in great detail because I show that this system is identical with the one introduced by Derbenev and Kondratenko for studying spin motion in accelerators. In particular I prove that the spin-orbit system of Derbenev and Kondratenko is (nonmanifestly) Poincaré covariant and identify the transformation properties of this system under the Poincaré group. The Derbenev-Kondratenko Hamiltonian was originally proposed as a way to combine relativistic spin precession and the Lorentz force. The aforementioned findings now demonstrate that the Derbenev-Kondratenko Hamiltonian also provides a legitimate framework for handling the relativistic Stern-Gerlach force. Numerical examples based on the Frenkel and Good cases for the HERA proton ring and electromagnetic traps are provided.