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arXiv:0802.0154

The Relativistic Stern-Gerlach Force

Ch. Tschalaer

Published Feb 1, 2008• 1 citation
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Abstract

The relativistic Lagrangian for a spinning particle in an electromagnetic field is derived from the known Lagrangian in the particle's rest frame. The resulting relativistic Stern-Gerlach and Thomas precession forces on the particle are then derived from the Lagrangian in the laboratory frame. In particular, the longitudinal component of this combined Stern-Gerlach-Thomas force does not contain a term proportional to gamma-squared as was claimed in a previous derivation [1].

DOI
10.48550/arxiv.0802.0154

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The relativistic Lagrangian for a spinning particle in an electromagnetic field is derived from the known Lagrangian in the particle's rest frame. The resulting relativistic Stern-Gerlach and Thomas precession forces on the particle are then derived from the Lagrangian in the laboratory frame. In particular, the longitudinal component of this combined Stern-Gerlach-Thomas force does not contain a term proportional to gamma-squared as was claimed in a previous derivation [1].
We have studied a relativistically covariant Stern-Gerlach (SG) experiment for a relativistic spin and a Dirac spin. We have obtained the relativistic spin in an arbitrary frame by using the classical spin dipole tensor, which gives the covariant spin dipole interactions, and the relation between a spin and a spin magnetic dipole moment. The relativistic spin is shown to have problems to become a proper spin operator for a massive relativistic particle because of two reasons. First, the relativistic spin three-vector operators cannot satisfy the spin algebra. Second, the SG experiment for the relativistic particle provides a paradox between two observers in the particle rest frame and the laboratory frame, in which the particle is moving. We have shown that the paradox in the SG experiment is resolved by the Dirac spin, which is covariantly defined by a Lorentz transformation in the Dirac spinor representation. The Dirac spin three-vector operators satisfy the spin algebra. It is shown that the SG experiment for the Dirac spin in the inertial frame, where there is only magnetic field, can determine the spin without the information of the momentum of the particle. This shows that the reduced spin density matrix for the Dirac particle can be well-defined by integrating out the momentum degrees of freedom.
The spin is the prime example of a qubit. Encoding and decoding information in the spin qubit is operationally well defined through the Stern-Gerlach setup in the nonrelativistic (i.e., low velocity) limit. However, an operational definition of the spin in the relativistic regime is missing. The origin of this difficulty lies in the fact that, on the one hand, the spin gets entangled with the momentum in Lorentz-boosted reference frames, and on the other hand, for a particle moving in a superposition of velocities, it is impossible to "jump" to its rest frame, where spin is unambiguously defined. Here, we find a quantum reference frame transformation corresponding to a "superposition of Lorentz boosts," allowing us to transform to the rest frame of a particle that is in a superposition of relativistic momenta with respect to the laboratory frame. This enables us to first move to the particle's rest frame, define the spin measurements there (via the Stern-Gerlach experimental procedure), and then move back to the laboratory frame. In this way, we find a set of "relativistic Stern-Gerlach measurements" in the laboratory frame, and a set of observables satisfying the spin su(2) algebra. This operational procedure offers a concrete way of testing the relativistic features of the spin, and opens up the possibility of devising quantum information protocols for spin in the special-relativistic regime.
Spinor wave functions; classical Lagrangian and Hamiltonian of a charged particle in an electromagnetic field, and its quantum Hamiltonian; gauge invariance; spin magnetic moment in a uniform magnetic field; magnetic resonance; the Stern--Gerlach experiment; neutron interferometry and rotations of spinor wave functions; treatment of a particle in a uniform magnetic field with and without the inclusion of spin degrees of freedom; Ahronov--Bohm effect for a charged spinless particle confined in a cylindrical shell.