Spin and localization of relativistic fermions and uncertainty relations

Understand

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.