# A Question about Path Integrals and Measures

Physicists like to tell people that they’re interested in computing integrals of the form
$\int_F df e^{-S(f)/\hbar} O(f)$

where$S$ is the classical action, which appears here divided by $\hbar$, and $df$ is a measure on some space $F$ of classical fields.

Mathematicians, of course, know that this is nonsense. There is no translation invariant even-remotely-Lebesgue-y measure $df$ on a space of functions. At best, when $S(f)$ isn’t too far from Gaussian, you have some measure $d\mu_\hbar(f)$ which behaves likes like the product of $df$ and $e^{-S(f)/\hbar}$.

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 $d\mu_\hbar(f)$. How do we recover the classical action from it? Normally, you’d take the log of the Radon-Nikodym derivative with respect to $df$, 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 $\hbar \to 0$, 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 $\hbar \to 0$? 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.

## 17 thoughts on “A Question about Path Integrals and Measures”

1. thomas1111 says:

Hi, first I’d like to congratulate you about this nice blog, don’t worry all of you if not so many people comment your posts at the moment, I’m sure you’ll have regulars very soon (plain curiosity: how many clicks a day do you get a the moment?)

Now as for this post in particular and your question of recovering a unique action from the quantum measure, all I know is that it obviously drastically depends on the configuration space your quantum fields are living on. In the case of ordinary QM, i.e. finitely many dofs, if your configuration space is a smooth compact connected Riemannian manifold then it’s all been worked out in details, look in particular at section 3 of this nice survey paper by Colin de Verdière (who is a mathematician) http://www-fourier.ujf-grenoble.fr/~ycolver/festival.pdf

I have no clue for other cases (non-compact config space, or infinitely many dofs), which presumably were the core of your question… Also, one should say that the non-degeneracy conditions mentionned in the aforementionned paper are quite restrictive in terms of the kind of classical dynamics one can treat.

2. A.J. Tolland says:

Hi Thomas,

Thanks for the reference.

As for blog statistics: I hadn’t thought to look until you asked, but it looks like we’re getting a few hundred clicks a day.

3. We spiked up to around 750 when Not Even Wrong linked to us. We’re currently in the 300-400 range, though with what seems to be an upward trend (which is slightly shocking to me. Who are these people?).

4. Geoff says:

Is that 300-400 distinct visitors, or a smaller number with multiple clicks per visit?

5. A.J. Tolland says:

That’s number of clicks. I assume a lot of it is bots and reloads, but even still, there’s more people reading this than I expected.

6. Same thing happened when NEW linked to me, and it dropped off further after a while. In the long-term you can build a regular readership, and even mine it for help on occasion. I’ve found a co-author through my weblog, for instance. Still, don’t pay too much attention to the specific numbers on the stats page. What you want to watch is “Referrers” and “Incoming Links” so you can build little mutual-linking relationships and help share readerships.

7. While I don’t know the general answer to the above question, I do think this touches upon a couple of really deep issues.

I tend to suspect that quite generally we should think of the kinetic term of the action as qualitatively different from the rest. The kinetic term is really part of the measure.

We know this is exactly true for the charged point particle:

there the action is kinetic + gauge coupling and the rigorous form of the path integral is the integral of the gauge coupling part using the Wiener measure on paths.

By the way: does anyone know of a rigorous result of this kind which would apply to the 2-dimensional Polyakov action (the string): can the purely kinetic part of of the Polyakov action (i.e. no B-field or things like that around) be regarded, rigorously, as something like a Wiener measure on paths in loop space, or the like?

Not that it would really help answer your above question, but we did discuss aspects of this phenomenon “kinetic action = part of measure” on more abstract grounds over at the n-Cafe a bit.

I am quite fond of a probably unreadable entry I once wrote (here) where I discuss what I believe is pretty much the proof that the “obvious” (once one thinks about it…) push-forward/colimit operation on the category corresponding to a latticezed line does reproduce the (correspondingly latticized) path integral for the charged particle propagating on that line, with the kinetic term dropping out automatically from the purely abstract-nonsense colimit operation.

Well, anyway. Whatever I really do there, ever since I did that I have the strong suspicion that it is no coincidence that

a) the kinetic part of the action wants to be part of the measure (e.g. Wiener measure)

a) the kinetic part of the action is usually Gaussian

b) the Gaussian part is usually the only one we can make rigorous sense of

8. A.J. Tolland says:

Hi Urs,

I’m intrigued by the idea that the kinetic part of the action wants to be part of the measure….but I’m even happier if all of the interactions are “kinetic” in origin, i.e. arise via minimal coupling as in Yang-Mills.

I have no clue if the Polyakov action can be regarded as a measure on paths in loop space… It wouldn’t surprise me, but you might have to ask someone who does SLE.

9. Aaron Bergman says:

The whole exponential of the action should be considered part of the measure. I think the general expectation is that the way to make sense of a path integral is to somehow discretize the space being integrated over and then define a one parameter family of actions depending on the “spacing” such that the joint limit of the action and the “spacing” converges to…something. This limit would somehow be the “measure”.

I can’t say I understand the original question so much. I can’t imagine recovering a classical theory in general (what about dualities?). Perhaps you want something like a one-parameter family of measures that localize in some sense?

10. I’m even happier if all of the interactions are “kinetic” in origin, i.e. arise via minimal coupling as in Yang-Mills.

Right. Everything I said is really, at the moment, to be read as concerning only “worldvolume theories” in low dimensions – as opposed to “target space theories” in dimension four and higher.

So, I am really thinking of Polyakov-like actions here, with various couplings in various dimensions (though you can easily scare me by passing to d greater than two). But not of Yang-Mills and the like. These I understand even less.

By the way, this is another related issue which seems to be relevant:

it is not entirely clear to me that we should carelessly be speaking about one and the same “quantum field theory” for both worldvolumes and for target spaces.

For instance: I am convinced that the right way to conveive the quantum theory of

the particle. the string, the membrane, etc (the “n-particle” in general)

(in a loose sense, not necessarily the objects that appear instance in string theory) is representations of cobordism categories.

There should be some crank which nobody can as yet turn at the moment, which reads in such a functor on cobordisms and spits out its “second quantized” QFT on target space.

It is not a priori clear to me that second quantizing a cobordism representation results itself again in a cobordism representation!

Hope you see what I mean. I am saying: maybe we need to distinguish between the quantum theory of “fundamental objects” (namely: charged n-particles, I think) and the second quantized (effective) theory they induce.

For me this means that it might be that for integration measures of “fundamental n-particles” (particle, string, membrane, etc) some things are true (like: the kinetic action is really part of a higher dimensional Wiener measure) which are not true for “target space theories” like Yang-Mills.

But I don’t know.

(In case anyone cares: I have commented on the issue fundamental quantum system versus non-fundamental quantum system here. And I mentioned an idea about how to conceive “second quantization of extended cobordisms reps” here. All input is welcome.)

11. The whole exponential of the action should be considered part of the measure.

Hm.

Let’s see: in the simple standard case of the particle propagating on a space X, with the action being

action = standard kinetic part + non-kinetic part

it so happens that the right way to do the path integral is to integrate

int (non-kinetic part) dWiener

I suppose with the Wiener measure dWiener given ,we may regard the product

(non-kinetic part) dWiener

as another measure. While technically true, it somehow misses the point that the kinetic part is the essential content of the measure.

12. Aaron Bergman says:

The kinetic part of the measure is the only thing that’s currently defined, but I don’t see anything physical about splitting the exponential into kinetic and interactions except that it allows us to do perturbation theory. Much as perturbation theory has proven to be extraordinarily fruitful, it’s not fundamental.

13. A.J. Tolland says:

Aaron,

Yes, there are some quantum systems that don’t have much of a classical limit. The question is whether, knowing that a system does have a well-defined $\hbar \to 0$ limit, how can we extract the action/symplectic structure from the path integral measure? Basically, can we run quantization in reverse?

So yeah, we’re looking for a one parameter family of measures, with $\hbar$ the parameter in question. Maybe it’s actually this easy:

$\frac{d}{d\hbar} d\mu_\hbar(f) = \frac{S}{\hbar^2}du_\hbar(f)$

14. Sorry, just to see if I know how to do TeXed formulas, here a formula

$$\int \exp(-i S(f)) \; d\mu(f)$$

(too bad there is no comment preview…)

15. Oh, dear, I forgot the syntax. Anyway, here is a comment:

I don’t see anything physical about splitting the exponential into kinetic and interactions

Now, that’s a different question. How to interpret this situation physically. I am not entirely sure yet.

For the charged particle, everything is quite clear. The path integral is

int holonomy(path) dWiener(path)

which is nice, because it is clearly the natural expression to form given a holonomy map from paths to phases and a measure on the space of paths.

And here it does seem that the distinction between interaction and kinetic part makes good sense physically. They are of quite different nature.

What I am not sure about yet is what exactly happens to this situation as we pass from the charged particle to the charged string.

In principle (assuming the kinetic Polyakov action does correspond to a Wiener measure on paths in path space) the action for the string in a “B-field background” would have the same form

int holonomy(surface) dWiener(surface)

But now this splitting into interaction and kinetic part is, while correct, a little puzzling, since we know that there is actually a nice symmetry between the two. This I don’t fully understand yet, fundamentally.

16. A.J. Tolland says:

Urs,
The syntax is (dollar-sign)latex e^{-iS(f)}(dollar-sign).

17. A.J.,

ah, I have to follow the dollar sign by “latex”? Let me see:

$e^{-iS(f)}$