  The Dirac equation has resided among the greatest successes of modern physics
since its emergence as the first quantum mechanical theory fully compatible
with special relativity. This compatibility ensures that the expectation value
of the velocity is less than the vacuum speed of light. Here, we show that the
Dirac equation admits free-particle solutions where the peak amplitude of the
wavefunction can travel at any velocity, including those exceeding the vacuum
speed of light, despite having a subluminal velocity expectation value. The
solutions are constructed by superposing basis functions with correlations in
momentum space. These arbitrary velocity wavefunctions feature a near-constant
profile and may impact quantum mechanical processes that are sensitive to the
local value of the probability density as opposed to expectation values.
