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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?

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For standard QM (i.e. "0-dimensional QFT"), the path integral is over the space of all continuous paths between the start and the end point, and the measure is rigorously a conditional Wiener measure. As a matter of fact, the differentiable functions have measure zero in this Wiener space, so in some sense the smooth paths do not contribute at all to the value of the path integral. (Note that by this logic one could also always say that the rationals do not contribute to any integral over the real line. Whether this is a useful thing to say is left for the reader to decide)

For full quantum field theory, no general rigorous formulation of the path integral is known, so the question is ill-defined. Where a rigorous formulation does exist (e.g. in two dimensions and in some cases in three, see e.g. the book by Glimm and Jaffe), the path integral is not taken over functions but over tempered distributions (the dual of the Schwartz functions), so it does not make sense to talk about whether or not they are smooth. The differential of a field is then of course also to be interpreted in the distributional sense.

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