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In Bohmian mechanics, a version of quantum mechanics that ascribes worldlines to electrons, we can meaningfully ask about an electron's instantaneousspeed relative to a given inertial frame. Interestingly, according to therelativistic version of Bohmian mechanics using the Dirac equation, a massiveparticle's speed is less than or equal to the speed of light, but notnecessarily less. That is, there are situations in which the particle actuallyreaches the speed of light---a very non-classical behavior. That leads us tothe question of whether such situations can be arranged experimentally. Weprove a theorem, Theorem 5, implying that for generic initial wave functionsthe probability that the particle ever reaches the speed of light, even if atonly one point in time, is zero. We conclude that the answer to the question isno. Since a trajectory reaches the speed of light whenever the quantumprobability current psi-bar gamma^mu psi is a lightlike 4-vector, our analysisconcerns the current vector field of a generic wave function and may thus be ofinterest also independently of Bohmian mechanics. The fact that the current isnever spacelike has been used to argue against the possibility offaster-than-light tunnelling through a barrier, a somewhat similar question.Theorem 5, as well as a more general version provided by Theorem 6, are alsointeresting in their own right. They concern a certain property of a functionpsi: R^4 --> C^4 that is crucial to the question of reaching the speed oflight, namely being transverse to a certain submanifold of C^4 along a givencompact subset of space-time. While it follows from the known transversalitytheorem of differential topology that this property is generic among smoothfunctions psi: R^4 --> C^4, Theorem 5 asserts that it is also generic amongsmooth solutions of the Dirac equation.

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