### 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 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 that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of 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 , Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.

Received 15 August 2010
Accepted 04 November 2010
Published online 28 December 2010

Acknowledgments:
Roderich Tumulka thanks Frank Loose (Tübingen) and Feng Luo (Rutgers) for helpful discussions. This research was supported by grant RFP1-06-27 from The Foundational Questions Institute (fqxi.org).

Article outline:

I. INTRODUCTION
II. MOTIVATION AND RELEVANCE
III. BOHMIAN ELECTRONS
IV. SPINORS LEADING TO SPEED *c*
V. DISCUSSION OF THE QUESTION IN THE TITLE
VI. GENERICITY THEOREM
VII. PROOFS
VIII. VARIATIONS OF THE QUESTION IN THE TITLE
IX. COMPARISON WITH ORTHODOX QUANTUM MECHANICS

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