What's the role of classically forbidden paths in path integral?

Question

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.

Answer 1

I think I know what you are thinking. Here's how it goes: in classical mechanics, we have a Lagrangian describing the system. Our principle of least action says that the system will follow a path that extremizes $S = \int L dt$. This amounts to taking the E-L equations on it. The resulting path is called the classical path and we say that it is the only path that the system follows.

In QM, the Feynman path integral says, let us talk about the amplitude for a particle to go from $a$ to $b$. Let us call this $K(b,a)$. Heuristically this is given by $K(b,a) = \sum_{all paths} \phi[x(t)]$, where the contribution of each path that goes from $a$ to $b$ is $\phi[x(t)] = \text{const} e^{i/\hbar S[x(t)]}$, where $S[x(t)]$ is the classical action of that path.

So to answer your question, how much do classically forbidden paths contribute (and I'll add, to the transition amplitude)? It's just $e^{i/\hbar S[x(t)]}$.

Now how do we recover classical mechanics? Send $\hbar \to 0$, and we see that the most contribution to the sum comes from the smallest value of $S$, as $e^{i/\hbar S[x(t)]}$ oscillates the least. (This is, or is like, the saddle point approximation). The path that gives the most contribution is the classical path, and it is that path that the particle takes classically.

Of course that wasn't very rigorous, but I would recommend 'Quantum mechanics and path integrals' by the guy who came up with this himself, Richard P. Feynman.

Cheers.

Answer 2

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.)

Answer 3

Unfortunately, after the edit to the question, this answer only refers to the final "last but not least" part of it.

In certain experiments there is no classical path that leads to the outcome. In such cases classically-forbidden paths contribute 100% to the path integral. See Feynman's QED, fig. 27. The picture is on the internet.

The reverse "might" be said of (most) macroscopic objects (in particular not QM-experimental set-ups). There the classical path would in itself describe the total path integral. (But they would't contribute. See below.)

But I take it that whenever there are non-classical paths, you could always take out the exact classical paths and still end up with the same path integral, because the classical paths would have measure $0$.

So that would mean that, if there are (a measurable amount of) non-classical paths, they contribute 100%.

But there are always (a measurable amount of) non-classical paths, therefore:

Classically forbidden paths contribute 100% to a path integral.

(Plenty of handwaving here, e.g., you might want to show that all non-classical paths together don't destructively interfere. But they don't, because then there wouldn't be any amplitude left.)

Added after the edit to the question:

I gather this picture, also taken from Feynman's QED, should clear up the "last but not least" part. (And in the limit, amplitudes are amplitude densities.) To me, "dominate" appears to be the wrong word. I think it is the other way around: The nearby paths determine the classical path (in situations where it is OK to use CM).

Answer 4

QM deals with waves, and for waves every point of space takes part in creating the resulting wave. The same is valid for a path-integral formulation. It is difficult to present a general weight of these or those paths. It depends.