Can we make a Bohmian electron reach the speed of light, at least for one instant?

Abstract

In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light—a very nonclassical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem, Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current \documentclass[12pt]{minimal}\begin{document}$\overline{\psi }\gamma ^\mu \psi$\end{document}ψ¯γμψ is a lightlike 4-vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of faster-than-light tunneling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function \documentclass[12pt]{minimal}\begin{document}$\psi :\mathbb {R}^4\rightarrow \mathbb {C}^4$\end{document}ψ:R4→C4 that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of \documentclass[12pt]{minimal}\begin{document}$\mathbb {C}^4$\end{document}C4 along a given compact subset of space-time. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions \documentclass[12pt]{minimal}\begin{document}$\psi :\mathbb {R}^4\rightarrow \mathbb {C}^4$\end{document}ψ:R4→C4, Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.