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Quick Path Integral Question

  1. Oct 14, 2004 #1
    In path integrals, how does one deal with non-differentiable paths? Obviously non-differentiable paths are allowed, but with Feymann's formulation, one has to calculate the action for a path, and then sum over all possible paths. How is the action defined (if it is defined at all) for a non-differentiable path?

    Also, is it possible to construct path integral vigorously by constructing a measure on the space of possible paths?
  2. jcsd
  3. Oct 14, 2004 #2


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    Since you are integrating, differentiability of the path doesn't matter. You can compute the length of the perimeter of a square, although the path becomes nondifferentiable at the corners.
  4. Oct 15, 2004 #3


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    The nondifferentiabality of the paths is not so much a problem (in fact in general one expects that). However the measure is a problem in QFT. No one knows really what exactly *that* is, and is more or less poorly defined mathematically (but curiously, you *can* do a few things with it).

    People seem to have more or less given up on that problem for the general case, as its exceedingly hard.
  5. Oct 15, 2004 #4
    Thanks for your replies!

    To selfAdjoint,
    Maybe you mistakably thought that I was talking about the ordinary path integral? Or did I miss something?

    To Haelfix,
    Path integrals are new to me. May you explain why the non-differentiability of paths does not matter? I can imagine that there are many continuous yet everywhere non-differentiable paths (Brownian motion's deterministic counterparts?) that are allowed. I am not sure that they are of measure zero (if the measure is defined at all)?
  6. Oct 19, 2004 #5
    They dont have measure zero. Their contributions cancel out, because the lagrangian involves derivatives, and the action diverges. Its easiest to show that in the case of a Euclidean path integral, where the integrand is [itex]e^{-\infty}=0[/itex] for a non differentiable path.
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