Coming from a mathematical background, I'm trying to get a handle on the path integral formulation of quantum mechanics.
According to Feynman, if you want to figure out the probability amplitude for a particle moving from one point to another, you 1) figure out the contribution from every possible path it could take, then 2) "sum up" all the contributions.
Usually when you want to "sum up" an infinite number of things, you do so by putting a measure on the space of things, from which a notion of integration arises. However the function space of paths is not just infinite, it's extremely infinite.
If the path-space has a notion of dimension, it would be infinite-dimensional (eg, viewed as a submanifold of $C([0,t] , {\mathbb R}^n))$. For any reasonable notion of distance, every ball will fail to be compact. It's hard to see how one could reasonably define a measure over a space like this - Lebesgue-like measures are certainly out.
The books I've seen basically forgo defining what anything is, and instead present a method to do calculations involving "zig-zag" paths and renormalization. Apparently this gives the right answer experimentally, but it seem extremely contrived (what if you approximate the paths a different way, how do you know you will get the same answer?). Is there a more rigorous way to define Feynman path integrals in terms of function spaces and measures?
Answer
Path integral is indeed very problematic on its own. But there are ways to almost capturing it rigorously.
Wiener process
One way is to start with Abstract Wiener space that can be built out of the Hamiltonian and carries a canonical Wiener measure. This is the usual measure describing properties of the random walk. Now to arrive at path integral one has to accept the existence of "infinite-dimensional Wick rotation" and analytically continue Wiener measure to the complex plane (and every time this is done a probabilist dies somewhere).
This is the usual connection between statistical physics (which is a nice, well-defined real theory) at inverse temperature $\beta$ in (N+1,0) space-time dimensions and evolution of the quantum system in (N, 1) dimensions for time $t = -i \hbar \beta$ that is used all over the physics but almost never justified. Although in some cases it was actually possible to prove that Wightman QFT theory is indeed a Wick rotation of some Euclidean QFT (note that quantum mechanics is also a special case of QFT in (0, 1) space-time dimensions).
Intermezzo
This is a good place to point out that while path integral is problematic in QM, whole lot of different issues enter with more space dimensions. One has to deal with operator valued distributions and there is no good way to multiply them (which is what physicist absolutely need to do). There are various axiomatic approaches to get a handle on this and they in fact do look very nice. Except that it's very hard to actually find a theory that satisfies these axioms. In particular, none of our present day theories of Standard model have been rigorously defined.
Anyway, to make the Wick rotation a bit clearer, recall that Schrödinger equation is a kind of diffusion equation but for an introduction of complex numbers. And then just come back to the beginning and note that diffusion equation is macroscopic equation that captures the mean behavior of the random walk. (But this is not to say that path integral in any way depends on the Schrödingerian, non-relativistic physics)
Others
There were other approaches to define the path-integral rigorously. They propose some set of axioms that path-integral has to obey and continue from there. To my knowledge (but I'd like to be wrong), all of these approaches are too constraining (they don't describe most of physically interesting situations). But if you'd like I can dig up some references.
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