Spin and localization of relativistic fermions and uncertainty relations

TL;DR

This paper explores relativistic spin and position measurements, proposing new POVMs that preserve causality and examining how uncertainty relations are modified in relativistic quantum mechanics.

Contribution

It introduces two causality-preserving POVMs for relativistic position and analyzes their properties, challenging the existence of a canonical relativistic position operator.

Findings

The proposed POVMs predict identical position expectation values.

Variances differ by less than a quarter of the squared de Broglie wavelength.

Heisenberg uncertainty relations may not hold in the relativistic context.

Abstract

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.