Talk:Functional integration

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Infinite dimensions

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 $K(x)=x^{3}+2x+3$ and I is a finite region of the Infinite dimensional space.

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)

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).

A functional integral is a method for computing the Green's function of a partial differential equation. The free particle functional integral is a formula for the Green's function of the diffusion equation in one dimension. More interesting applications are the study of the symmetries of solutions of partial differential equations by using their functional integral representation, such as in the work by DeWitt-Morette. —Preceding unsigned comment added by XaosBits (talk • contribs) 14:40, 28 June 2008 (UTC)

Functional integration and P.J. Daniell

In February 2010, User:Dr. Universe edited the second paragraph of this article to say that Functional integration was created "by P. J. Daniell in 1919 (in Annals of mathematics)". As far as I can tell from "But you have to remember P.J. Daniell of Sheffield" (linked from from Daniell's page), he did indeed have a role in the creation of functional integration and the relevant article is probably Daniell, P. J. (1919-07). "Integrals in An Infinite Number of Dimensions". The Annals of Mathematics. Second Series. 20 (4): 281–288. ISSN 0003-486X. {{cite journal}}: Check date values in: |date= (help)

For the moment I'll just tidy up the text and will come back only if I get the chance to look more at this. –Syncategoremata (talk) 20:11, 13 March 2010 (UTC)

"Inverse of functional derivative"!?

This entire section appears to be original research, and there are no citations to back any claims made. Indeed, there isn't very detailed claims made!

There are serious problems with transferring the intuition underlying basic calculus to the functional integral domain, and it appears this is just an example of such confusion.

I think this section should be removed, lest it confuse poor graduate students...

—Pqnelson (talk) 19:58, 3 February 2012 (UTC)

So I've decided to remove it. The reason is the following (basic sketch of argument: fundamental theorem of calculus fails to generalize appropriately/accordingly/at-all):

Suppose we have some functional integral $\int \exp(I[\varphi ]){\mathcal {D}}\varphi$, right? Well, the first problem is where do we put the functional derivative: inside or outside the integral?

Well, if we put it outside the integral, it's meaningless. We have an expression of the form

${\frac {\delta }{\delta \varphi }}$(a number)

Recall functional derivatives make sense if and only if acting on functionals!!! So we have to put the functional derivative inside the integral. But then it's observably incorrect to assert

$\int {\frac {\delta }{\delta \varphi }}\exp(I[\varphi ]){\mathcal {D}}\varphi =\exp(I[\varphi ])$

Why? Well, the routine calculations in QFT suggest this won't produce the desired result...

So unless someone can back up the assertion "Functional integration is the inverse of functional differentiation", I am going to remove it....

—Pqnelson (talk) 15:59, 14 February 2012 (UTC)

Definitions impenetrable for readers not familiar with subject

With regards to the "confusing" tag:

None of the notation given is explained, and will be unfamiliar or confusing to even those readers who are familiar with traditional calculus. For example:

• What does G[f] do? What kind of object is it?
• What is Df and how is it analogous to `dx` in a traditional integral equation?
• What is df(x)? How is it different from whatever Df is?
• What is the capital Pi notation doing in this equation? Does it refer to the arithmetic product, or to function application in this context? Why is it there?
• How is the series of df(x) defined? Why is there a series at all? What are the bounds of the series (i.e. what values may x take)?
• What, exactly, takes on values between negative infinity to positive infinity? It is not clear what those extents "connect" to, given that the notation and meaning of the equation is altered in unspecified ways from a "traditional" integral.

To somebody not already completely familiar with this topic, the definition given is likely to read as ambiguous hieroglyphics, which share vague but impenetrable similarities to an ordinary integral.

A better version of this article would explain what each object in the equation does, why it is there, and how it compares to the form and meaning of a traditional integral. Timrb (talk) 10:08, 15 January 2014 (UTC)