I'm interested in how and how much classically-forbidden paths contribute to a path integral? Is there any good reference on the issue? Any discussion in QM or QFT context would be appreciated.
EDIT: I have a more specific question than the above but initially it was a reference request so I decided to make it more general. Now it doesn't seem appropriate any more so let me reformulate the question:
First of all let me apologize for the bad terminology. By "classically forbidden" I actually meant "differentiable"(i.e. for QM differentiable in time direction, for QFT in both space and time direction) instead of "forbidden by classical dynamics".
My motivation comes from path integral of QED, if we only integrate the fermionic degrees of freedom under some smooth gauge field, we will get a quantized theory of many electrons with a classical gauge background, and the fully quantized theory will emerge after we also integrate over gauge fields. This seems to be a reasonable way of thinking, but some of my subsequent derivations seem to suggest some quantum effects will disappear such as photon-photon scattering, but something is still preserved like the many-body feature of QFT(I'm not very sure about calculation yet so I'd rather not show it here). It occured to me it might be because I'm only including smooth backgrounds.
This motivates me to ask, what exactly is the role of smooth and non-smooth paths in path integral? Do they result in different and isolated features of QFT so that it's ok to consider them separately, or do their effects just mix with each other so that we always have to consider them as a whole?
Last but not least, the comments and answers below remind me of another question, if the classical path(this time I mean path predicted by classical dynamics) always contributes 0 to the path integral for any value of $\hbar$, then what do we mean by saying the classical path will dominate in $\hbar\to0$ limit? After all a simple fact of math is that a sequence of 0's cannot give you a limit of 1.
Best Answer
Other people have already addressed quantum mechanics, so let me comment on the field theory case.
In all of the QFTs which have been rigorously constructed, in spacetime dimension 2 & 3, the Euclidean path integral is supported on a space of distributions. The set of continuous classical fields sits inside this space of distributions, but it has measure zero with respect to the path integral. I see no reason to expect the QFTs that describe real world physics to be any better behaved. The path integral measure has to be supported on distributions to give an OPE with short distance singularities.
So yes, just summing over classical fields will probably not give you a good approximation to the path integral.
The only reference I know on this stuff is Glimm & Jaffe. (There may be more accessible references somewhere in the literature. I just don't know them.)