Talk:Functional integration

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Nothing in the conventional Lebesgue definition of the integral assumes finite-dimensionality. I think the definition given here needs work; it's far too vague. Probabilists integrate over infinite-dimensional spaces all the time, but that is not the same thing as the Feynman integrals that this article is about. Michael Hardy 03:39, 1 Nov 2003 (UTC)

I liked the previous statement better, because spaces of paths are infinite dimensional topological vector spaces. Also, doesn't the conventional definition of Lebesgue integration assume local compactness?

Rigorous definitions of functional integration for purposes of physics (as in the work of Irving Segal) encompass the integrals that probabilists do over infinite dimensional spaces and are motivated by them. In addition, one sometimes sees people talk about the Feynman-Ito integral, which to me is an indication that the Feynman integral and the functional integrals from probability theory are the same. However, I don't know enough about the Ito integral to make a judgement on this. -- Miguel

  • But can we define the integral via Euler-McLaurin sum formula for infinite dimensional spaces ?? --85.85.100.144 20:56, 9 May 2007 (UTC)

I am not confident that integration over infinite-dimensional spaces that is done in probability should be called "functional integration", as opposed to that phrase's being restricted to those Feynman-integrals that physicists do. Does anyone know of it's being called that in the probability literature? If my suspicion is well-founded, the initial sentence needs to be changed: not all cases of integration over infinite-dimensional spaces should be considered "functional integration" as that term is used here. Michael Hardy 22:03, 4 Feb 2004 (UTC)

I have nothing against reserving the term "functional integration" for the Feynman integral and other rigorous integrals from mathematical physics, and using the term "stochastic integration" to refer to what is done in probability theory. However, there are numerous points of contact between the two, and once this article is sufficiently developed, I would like to see a section on functional integrals and stochastic integration. Miguel 03:34, 2004 Feb 26 (UTC)

I was trying to write a new article for functional integration, but it is getting a little too long and I am having trouble keeping it understandable for a reader with only calculus 201. It needs better motivation, why Gaussians, an exaplanation of the problems with having complex exponential, a history of the subject ....
There is a copy of what I typed in at Functional integration. Comments, suggestions, ....  — XaosBits 04:34, 27 February 2006 (UTC)
I read it, and that's really well written. I know nothing about functional integration, and a few about classical integration, but I followed up to the paragraph about gaussian measure, wich is less easy/clear/simple. I can just ask you for continuing this work ! Thanks. Dangauthier 13:44, 18 April 2006 (UTC)
 : But..couldn't we use Montecarlo Integration for Functional spaces?..(Infinite-dimensional spaces) in wich you propose a set of "random points" (random functions) along the set of integration getting a discrete series instead of a continous integral.

A possible method would be (at least perturbatively), let be the integral:

\int \mathcal D [\phi]exp(-S_{0}-\lambda G[\phi])

where 'lambda' is an small parameter,(using natural units with \hbar =1 ) and G[\phi]=\int dx^{4}V(\phi) , the integral is a wick-rotated version of Functional integral.

If 'lambda' were 0 then the field follows a Gaussian Measure similar to a Wiener process, since Gaussian measure can be generalized to infinite-dimensional spaces i think that the 'measure problem' for Feynmann integrals would be solved.--Karl-H 10:21, 24 January 2007 (UTC)

  • Another possible method is let be the exponential functional integral:

\int \mathcal D [\phi]exp(-aS)

a is a positive real number then we could perform saddle point method as if a\rightarrow \infty then the evaluation of the Functional integral above would be reduced to the problem of calculate a divergent series for a=1,2,3... of the form:

\sum_{n=0}^{\infty} b(n)a^{-n} using some resummation method, and after that put a=1

The cartier paper can be seen here www.arxiv.org HOwever..does anyone knows how is used (since it's very technical for non-mathematicians) to calculate functional integral of the form:

\int_{\mathcal I }D[\phi]F[\phi]

where F is a Polynomial functional analogue to the finite Polynomial, for example a generalization to functionals of K(x) = x3 + 2x + 3 and I is a finite region of the Infinite dimensional space.

[edit] Mathematical help wanted

This page links to the disambiguation page Action. I'm not sure which page it's supposed to link to - I don't think it's Group action, which is the only vaguely mathematical page under Action. Could someone more knowledgeable fix the link to point directly to the appropriate page? Soo 22:42, 24 August 2006 (UTC)

I have changed the link to point to action (physics). Alternatively, one could link to action integral but this redirects to action (physics) at the moment. — Tobias Bergemann 07:13, 25 August 2006 (UTC)

[edit] A thing to be clarified??

  • Reading the first paragraph of the article 'Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of [b] partial differential equations [/b] and in Feynman's approach to the quantum mechanics of particles and fields', could someone provide any references of the fact of HOw the functional integrals are applied in the study of a general PDE?? —The preceding unsigned comment was added by Karl-H (talk • contribs) 20:38, 28 January 2007 (UTC).