Can We Make a Bohmian Electron Reach the Speed of Light, at Least for One Instant?

TL;DR

This paper investigates whether a Bohmian electron can reach the speed of light, proving that for typical initial states, the probability of this happening even momentarily is zero, thus reinforcing relativistic constraints in Bohmian mechanics.

Contribution

The paper proves that in relativistic Bohmian mechanics, the probability of an electron reaching the speed of light at any instant is zero for generic wave functions, extending transversality results to solutions of the Dirac equation.

Findings

Probability of reaching the speed of light is zero for generic initial states.

The current vector field of a generic wave function is never spacelike.

Theorems 5 and 6 establish generic properties of solutions to the Dirac equation.

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 non-classical 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 psi-bar gamma^mu psi 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 tunnelling 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 psi: R^4 --> C^4 that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of C^4 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 psi: R^4 --> C^4, Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.