Considering the sum over histories approach to quantum mechanics. This considers all histories consistent with certain starting configurations and ending configurations.
How "smooth" do these histories have to be? Are we only considering, for instance, smooth paths of an electron?
Or when considering quantum field theory, how smoothly must the fields vary between starting and ending states? Can they just change randomly? And just the histories that vary more smoothly contribute more to the amplitude?
When taking a slice of time of one of the histories of a field, will the field itself on this spacial slice be a smooth function or would it also be completely random noise for most histories? In which case how do differentials work with such noise?