University of Pennsylvania
In a recent paper we asked "should the wave function extend in time in the same way it extends in space?". This hypothesis is motivated by the way relativity treats time & space as interchangeable; is not ruled out by existing observational or experimental evidence; and is falsifiable with current technology.
The most direct test is to measure the time-of-arrival of a quantum particle: if the wave function is extended in time, then the dispersion in the time-of-arrival will be significantly & measurably increased.
However we first need to have a well-defined measure of the time-of-arrival in standard quantum mechanics. Unfortunately the field does not yet appear to have settled on an acceptable measure. Perhaps the most popular existing measure proposed by Kijowski & others violates unitarity by an arbitrarily large amount when presented with a wave function crafted with sufficient malice.
The Kijowski metric was built using assumptions that make more sense classically than they do quantum mechanically. We therefore take a fully quantum mechanical approach: using path integrals and a simple definition of detection to get a reasonable definition of time-of-arrival: consistent with Kijowski in the appropriate limit but satisfying unitarity in general.
With this we get an unambiguous & reasonable result for time-of-arrival in standard quantum mechanics. And we show that when the wave function is extended in time the dispersion in time-of-arrival is appropriately increased.
We therefore conclude that the hypothesis of the wave function being extended in time is falsifiable using time-of-arrival measurements.