Path integral and discontinuous paths

[Xiaomin Chu]

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.

 

[mfb]

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]

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 Feynman 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]

Well, in the (nonrigorous) development of path integrals in textbooks such as Feynman 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

 

[Xiaomin Chu]

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?

 

[Xiaomin Chu]

Well, in the (nonrigorous) development of path integrals in textbooks such as Feynman 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.

 

[rumborak]

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]

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

 

[bhobba]

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

 

[Xiaomin Chu]

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]

The good thing is that physicists' hand-waving mathematics can sometimes be made rigorous, giving rise to entirely new developments in pure math. A prime example is the advent of non-relativistic quantum mechanics. Dirac came up with the first complete understanding of the theory in 1926. Heisenberg had formulated the theory in terms of matrix mechanics (reinventing matrix calculus without having been aware of its existence; this was explained by Born immediately after he had seen Heisenbergs famous Helgoland work) and Schrödinger in terms of wave mechanics. Then Schrödinger also proved that both theories were in fact equivalent and independently Dirac gave the now established "representation free" formulation, which is still the clearest way to present the theory to begin with. Already Schrödinger's work was triggered by Hilbert's work on integral and partial differential equations (Schrödinger was using the methods from the famous textbooks by Courant and Hilbert on mathematical methods in physics).

Dirac used what's now called the "Dirac $\delta$ distribution" (although it was in fact invented already around 1910 by Sommerfeld in a work on electrodynamics). In the way it was treated it was mathematically totally weird, but it lead to very sensible results that could be checked by more rigorous means anyway. This gave a hint to mathematicians that there may be something into this, and this lead to the discovery of functional analysis. The first rigorous formulation in terms of Hilbert space theory was given by von Neumann. Dirac's much more convenient non-rigorous scheme was made rigorous by the discovery of the "rigged Hilbert-space formalism".

There's no rigorous treatment of relativistic quantum field theory yet, despite some efforts in terms of "axiomatic QFT". I don't know what to make out of this: Either it's not possible to make it rigorous or it's flawed in itself. The latter possibility is more exciting since this would indicate that QFTs are always only "effective theories" with a validity up to a certain energy-momentum scale, which is the modern understanding of QFTs anyway. Then one can hope for a more comprehensive theory (sometimes dubbed "Theory of Everything", including a solution of the puzzle of quantum gravity) which can be formulated rigorously with the QFTs as some approximative limit of the new theory.