Talk:Functional integration
From Academic Kids
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
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)
