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

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.