Wigner rotations and an apparent paradox in relativistic quantum information

It is shown that a general model for particle detection in combination with a linear application of the Wigner rotations, which correspond to momentum-dependent changes of the particle spin under Lorentz transformations, to the state of a massive relativistic particle in a superposition of two counterpropagating momentum states leads to a paradox. The paradox entails the instantaneous transmission of information between two spatially separated parties. A solution to the paradox is given when the physical construction of the corresponding state is taken into account, suggesting that we cannot in general linearly apply the Wigner rotations to a quantum state without considering the appropriate physical interpretation.

Figure 1

Modulus squared of particle 2 wave functions in reference frame $S^{\left(1\right)}$: $|{\varphi }^{\prime }{\left(y\right)|}^2$ from Eq. (9) and $|{\varphi }^{\prime \prime }{\left(y\right)|}^2$ from Eq. (10) (continuous red curve) and $|{\psi }^{\prime }{\left(y\right)|}^2$ from Eq. (6) and $|{\psi }^{\prime \prime }{\left(y\right)|}^2$ from Eq. (7) (dashed black curve) for ${\gamma }_{\beta }=10$ and ${\gamma }_p=1.2$ in Eq. (5).

Figure 2

Modulus squared of the position wave function of a particle in a quantum state with a momentum wave function $\tilde{\psi }\left(p\right)\propto e^{-p^2/\left(2m^2{c}^2\right)}$ in the $y$ direction seen from a observer moving with velocity $0.995c\widehat{\mathbf{z}}$ when the spin state points in the $x$ direction (continuous red curve) and the $z$ direction (dashed black curve).