  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.
  So 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.
