  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.
