We have studied a relativistically covariant Stern-Gerlach (SG) experimentfor a relativistic spin and a Dirac spin. We have obtained the relativisticspin in an arbitrary frame by using the classical spin dipole tensor, whichgives the covariant spin dipole interactions, and the relation between a spinand a spin magnetic dipole moment. The relativistic spin is shown to haveproblems to become a proper spin operator for a massive relativistic particlebecause of two reasons. First, the relativistic spin three-vector operatorscannot satisfy the spin algebra. Second, the SG experiment for the relativisticparticle provides a paradox between two observers in the particle rest frameand the laboratory frame, in which the particle is moving. We have shown thatthe paradox in the SG experiment is resolved by the Dirac spin, which iscovariantly defined by a Lorentz transformation in the Dirac spinorrepresentation. 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 theinformation of the momentum of the particle. This shows that the reduced spin density matrix for the Dirac particle can bewell-defined by integrating out the momentum degrees of freedom.
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