  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.
