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 ${\textbf {1/2}}$, expressed in a representation-independent form of the Poincaré 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 $\textbf{4}$-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.

Appendix A: Dirac Hamiltonians

In this appendix, we derive the Dirac Hamiltonians for a particle and an antiparticle by using the parity-space inversion-operation. The parity operation transforms the left-handed spinor $\psi _L(p,\lambda )$ to the right-handed spinor $\psi _R(p,\lambda )$ and vice versa. Hence, the parity operation for the spinors $\psi _{\pm \epsilon }(p,\lambda )$ in (9) can be represented by $\pm \gamma ^0$, as shown in (10). Interestingly, the parity operation is also represented by the square of the inverse standard Lorentz boost, denoted as $M^{-2}(\mathcal {L}_{{\textbf {p}}})$, which becomes:

$$\begin{aligned} M^{-2}(\mathcal {L}_{{\textbf {p}}})= e^{-\gamma ^5 \varvec{\sigma }\cdot {{\textbf {f}}}}=\frac{E_{{\textbf {p}}}}{m}-\frac{\gamma ^5 \varvec{\Sigma }\cdot {{\textbf {p}}}}{m}, \end{aligned}$$

(A1)

referring to $M(\mathcal {L}_{{\textbf {p}}})$ in (24).

By equating the two representations of the parity as

$$\begin{aligned} M^{-2}(\mathcal {L}_{{\textbf {p}}}) \psi _{\pm \epsilon }(p,\lambda ) =\pm \gamma ^0 \psi _{\pm \epsilon }(p,\lambda ), \end{aligned}$$

(A2)

we can derive

$$\begin{aligned} \pm m \gamma ^0 \psi _{\pm \epsilon }(p,\lambda ) =( E_{{\textbf {p}}} -\gamma ^5 \varvec{\Sigma }\cdot {{\textbf {p}}}) \psi _{\pm \epsilon }(p,\lambda ). \end{aligned}$$

(A3)

These equations yield the Hamiltonians for a particle and an antiparticle, expressed as:

$$\begin{aligned} H_{P/AP} \psi _{\pm \epsilon }(p,\lambda ) = \gamma ^0 \varvec{\gamma }\cdot {{\textbf {p}}} \pm m \gamma ^0 \end{aligned}$$

(A4)

by using $\gamma ^5 \Sigma ^k = \gamma ^0 \gamma ^k$.