  We discuss relations between several relativistic spin observables and derive
a Lorentz-invariant characteristic of a reduced spin density matrix.A
relativistic position operator that satisfies all the properties of its
nonrelativistic analog does not exist. Instead we propose two
causality-preserving positive operator-valued measures (POVMs) that are based
on projections onto one-particle and antiparticle spaces, and on the normalized
energy density. They predict identical expectation values for position. The
variances differ by less than a quarter of the squared de Broglie wavelength
and coincide in the nonrelativistic limit. Since the resulting statistical
moment operators are not canonical conjugates of momentum, the Heisenberg
uncertainty relations need not hold. Indeed, the energy density POVM leads to a
lower uncertainty. We reformulate the standard equations of the spin dynamics
by explicitly considering the charge-independent acceleration, allowing a
consistent treatment of backreaction and inclusion of a weak gravitational
field.
