# Path integral and discontinuous paths

Somebody asked about this but that thread was closed very soon. In physics, discontinuous paths breaks locality so they must be 0; but mathematically, they causes some problems. Discontinuous functions must not be differentiable, so it's impossible to calculate the action over that path. However this does not say that one will be zero, it will be infinity. This really causes a lot of problems while evaluating path integrals. Can anyone explain more?
Another problem: path integral formula allows to take some ugly functions like Weierstrass function into the integration. It's not differentiable but there seems no reason to exclude these functions.

## Answers and Replies

mfb
Mentor
Where do you see discontinuous functions in path integrals? And if there is something that cannot be calculated, how can it have a result?

stevendaryl
Staff Emeritus
Where do you see discontinuous functions in path integrals? And if there is something that cannot be calculated, how can it have a result?

Well, in the (nonrigorous) development of path integrals in textbooks such as Feynmann and Hibbs, integrating over all possible paths includes discontinuous paths, as well. The usual argument is that the discontinuous paths don't contribute much to the integral, since the amplitudes due to the discontinuous paths tend to cancel.

bhobba
Mentor
Well, in the (nonrigorous) development of path integrals in textbooks such as Feynmann and Hibbs, integrating over all possible paths includes discontinuous paths, as well.

When one breaks the path in little bits and takes the limit the only paths you can do that to are differentially continuous.

Thanks
Bill

When one breaks the path in little bits and takes the limit the only paths you can do that to are differentially continuous.

Thanks
Bill

There are always some functions that remain discontinuous when being broken to pieces. In fact there are " much more" this kind of functions than continuous functions, "almost every" function has this property. Does that mean path integral will not even converge?

Well, in the (nonrigorous) development of path integrals in textbooks such as Feynmann and Hibbs, integrating over all possible paths includes discontinuous paths, as well. The usual argument is that the discontinuous paths don't contribute much to the integral, since the amplitudes due to the discontinuous paths tend to cancel.

Can you tell more mathematical details about this? I'm not satisfied by Srednicki's introduction, I want strict mathematics.

I'm not all too familiar with the matter at hand, but integrating over discontinuous paths seems like a very weird practice, since (it would seem to me) you would have to account for the gap somehow. Also, you could take the practice to the extreme, shrinking each line segment to a dot, in which case your "path" would just be a smattering of dots across space. That clearly can't be a valid practice.

bhobba
Mentor
There are always some functions that remain discontinuous when being broken to pieces. In fact there are " much more" this kind of functions than continuous functions, "almost every" function has this property. Does that mean path integral will not even converge?

The convergence of path integrals is a difficult issue requiring Hida distributions from white noise theory:
http://mathlab.math.scu.edu.tw/mp/pdf/S20N41.pdf

In the semi heuristic presentation of the usual physics texts only paths that satisfy the usual physical requirements of continuous differentiability are considered and its assumed the only ones that appear in the path integral. Its exactly the same thing that's done when the path of a classical particle is considered.

Unfortunately for those of a rigorous mathematical bent physics can be a bit sloppy. For example it took mathematicians quite a while to develop Rigged Hilbert Spaces so Dirac's version of QM that physicists actually use, rather than Von Neumann's mathematically correct version, was mathematically valid.

Thanks
Bill

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bhobba
Mentor
I'm not all too familiar with the matter at hand, but integrating over discontinuous paths seems like a very weird practice, since (it would seem to me) you would have to account for the gap somehow. Also, you could take the practice to the extreme, shrinking each line segment to a dot, in which case your "path" would just be a smattering of dots across space. That clearly can't be a valid practice.

Its exactly the same thing that's done in physics all the time where paths are continuously differentiable. The path integral simply wouldn't work with discontinuous paths.

Thanks
Bill

The convergence of path integrals is a difficult issue requiring Hida distributions from white noise theory:
http://mathlab.math.scu.edu.tw/mp/pdf/S20N41.pdf

In the semi heuristic presentation of the usual physics texts only paths that satisfy the usual physical requirements of continuous differentiability are considered and its assumed the only ones that appear in the path integral. Its exactly the same thing that's done when the path of a classical particle is considered.

Unfortunately for those of a rigorous mathematical bent physics can be a bit sloppy. For example it took mathematicians quite a while to develop Rigged Hilbert Spaces so Dirac's version of QM that physicists actually use, rather than Von Neumann's mathematically correct version, was mathematically valid.

Thanks
Bill

Sometimes physics seems incorrect mathematically, that's because the current mathematics doesn't include the objects physicists use. I believe physics includes two sides, one is experiments, that goes from "top" to "bottom", the other is mathematics, that goes from "bottom" to "top". When they meet somewhere, they form a complete theory. It is not allowed to make assumptions just based on every day experiences, or Dirac, Pauli, Heisenberg wouldn't think of quantum physics!

vanhees71