Lorentz-covariance of Position Operator and its Eigenstates for a massive spin $1/2$ field

TL;DR

This paper derives a Lorentz-covariant position operator for a massive spin 1/2 field, which accurately describes particle and antiparticle states, avoiding Zitterbewegung and aligning with classical velocity and Newton-Wigner criteria.

Contribution

It introduces a Lorentz-covariant, representation-independent position operator for Dirac fields that preserves particle-antiparticle distinction and satisfies locality and covariance.

Findings

The position operator's eigenvalues correspond to Lorentz-covariant space-time coordinates.

It avoids Zitterbewegung by maintaining particle and antiparticle character.

The operator aligns with Newton-Wigner position and classical velocity for free particles.

Abstract

We present a derivation of a position operator for a massive field with spin $1/2$, expressed in a representation-independent form of the Poincar\'e group. Using the recently derived Lorentz-covariant field spin operator, we obtain a corresponding field position operator through the total angular momentum formula. Acting on the Dirac spinor representation, the eigenvalues of the field position operator correspond to the spatial components of the Lorentz-covariant space-time coordinate $4$-vector. We show that the field position operator preserves the particle and the antiparticle character of the states. Thus, the field position operator can serve as a one-particle position operator for both particles and antiparticles, thereby avoiding an unusual fast-oscillating term, known as the Zitterbewegung, associated with the Dirac position operator. We show that the field position operator yields the same velocity as a classical free particle. The eigenstates of the field position operator satisfy the Newton-Wigner locality criteria and transform in a Lorentz-covariant manner. The field position operator becomes particle position and antiparticle position operators when acting on the particle and the antiparticle subspaces, both of which are Hermitian. Additionally, we demonstrate that within the particle subspace of the Dirac spinor space, the field position operator is equivalent to the Newton-Wigner position operator.