  Does the Heisenberg uncertainty principle (HUP) apply along the time
dimension in the same way it applies along the three space dimensions?
Relativity says it should; current practice says no. With recent advances in
measurement at the attosecond scale it is now possible to decide this question
experimentally. The most direct test is to measure the time-of-arrival of a
quantum particle: if the HUP applies in time, then the dispersion in the
time-of-arrival will be measurably increased. We develop an appropriate metric
of time-of-arrival in the standard case; extend this to include the case where
there is uncertainty in time; then compare. There is -- as expected --
increased uncertainty in the time-of-arrival if the HUP applies along the time
axis. The results are fully constrained by Lorentz covariance, therefore
uniquely defined, therefore falsifiable. And therefore we have an experimental
question on our hands. Any definite resolution would have significant
implications with respect to the role of time in quantum mechanics and
relativity. A positive result would also have significant practical
applications in the areas of quantum communication, attosecond physics (e.g.
protein folding), and quantum computing.
