Physicists like to tell people that they’re interested in computing integrals of the form
where is the classical action, which appears here divided by , and is a measure on some space of classical fields.
Mathematicians, of course, know that this is nonsense. There is no translation invariant even-remotely-Lebesgue-y measure on a space of functions. At best, when isn’t too far from Gaussian, you have some measure which behaves likes like the product of and .
What I’m wondering is this: The world is quantum mechanical, not classical, so let’s assume that the “true” object here is the measure . How do we recover the classical action from it? Normally, you’d take the log of the Radon-Nikodym derivative with respect to , but we can’t do that here.
Nor, for that matter, should we really expect to recover a unique action from a measure. We’re supposed to recover the classical physics by taking the limit , but there are plenty of different actions that give the same classical physics. So what’s going on here? Do we really only just care about the support of the measure as ? How do we recover a symplectic structure?
I assume this is well known in the right circles, but I’m not sure where one might look in the literature.