From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Path integrals and Feynman diagrams Date: 26 May 2000 10:08:06 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8gb2fd$kj7$1@crib.corepower.com>, Nathan Urban wrote: >In article <8g770c$et9$1@pravda.ucr.edu>, baez@galaxy.ucr.edu (John Baez) wrote: >> In article <8g0f14$c64$1@crib.corepower.com>, >> Nathan Urban wrote: >> > Z = pi sqrt(2m)/b (1 - 3a + 105/2 a^2 - 3465/2 a^3 + 675657/8 a^4 - ...) >> And here's one thing you can see very clearly now: >> this series NEVER CONVERGES except for a = 0. So you might make >> the mistake of concluding right now that this whole approach to >> perturbatively computing path integrals is a complete waste of >> time! >Tempting, isn't it? Then I wouldn't have to do all this work. Yeah: you can become a mathematician, so that when someone asks you to compute a path integral perturbatively you can turn up your nose and say "Sorry, I don't do those, they don't converge." But that would be unwise... and not just because being a mathematician entails still greater hardship than being a physicist! So luckily, you decided to put in a bunch of time checking out exactly *how* the power series diverges.... >Well, I spent most of Sunday doing (and cross-checking) a bunch of >numerical calucations, in C (for the practice writing numerical codes) >and Mathematica (because I have more confidence in its answers). I made >lots of nice little tables and plots. Great!!! I guarantee you, this was time well spent. The phenomena you discovered occur in almost *all* perturbative path integral calculations! They happen in quantum field theory all over the place... but usually it's a whole lot more of a pain to study them. >Here is the general qualitative behavior: > > Typically, the error between the "approximate" (perturbation series) > and "exact" (direct numerical integration) values starts out fairly > reasonable and will decrease as you add terms to the series, up to > a point. But then the error starts increasing again and quite rapidly > (how fast?) diverges to something unreasonable. If a is large enough, > sometimes even adding a second term to the first is enough to start > the monotonic increase in error, and even the first term need not be > that great of an approximation. Right. That's the bad news. The good news is that if a is very close to zero, you can add a lot of terms before the series start to go berserk, and it will get very close to the right answer before it goes bad. I bet you saw this phenomenon too. When this phenomenon occurs - the good news part, not the bad news part! - one says the power series is *asymptotic* to the function it's trying to approximate. I could make this more precise but I won't just now. The point is that while all convergent series are automatically asymptotic to the functions they converge to, not all asymptotic series are convergent to the functions they are asymptotic to. And in perturbative path integral calculations, the power series you get are OFTEN asymptotic to the right answer, but ALMOST NEVER convergent to it. >One question that comes to mind is: in practice, is it possible to >predict the number of terms at which the series starts to go sour? Yes! I believe a very crude rule of thumb is that they go bad roughly around the nth term where n ~ 1/a and a is the quantity whose powers appear in the power series. The "~" means that this estimate is only good up to an overall constant - clearly this crude rule of thumb can't be more accurate than that, since for any asymptotic power series we can always cook up a new one just by doing the substitution a -> ca. Now, I don't really remember the conditions under which this crude rule of thumb is valid, and I could even be seriously mistaken... maybe it's not 1/a but some other power of a that matters? You can probably guess the right rule of thumb using your nice little tables and plots. But I seem to remember people saying that since the fine structure constant alpha is about 1/137, we expect the power series for perturbative quantum electrodynamics to go sour after roughly the 137th term! Of course, even the most insanely accurate QED computations only go out to about the 7th term, so this problem is not regarded as one of great practical importance. But it's interesting in principle, and *most* quantum field theories have coupling constants rather *greater* than 1/137, so perturbation theory is *worse* for them. Anyway, you don't need to rely on my feeble memory if this issue becomes important to you, there is a big literature on it. The best place to start is probably the volume of Reed and Simon's Methods in Modern Mathematical Physics which treats perturbation theory. They have a bunch of general theorems about asymptotic series, and also a bunch of tables illustrating what happens for the anharmonic oscillator. Unlike you, they don't consider the partition function of the classical anharmonic oscillator; they consider the ground state energy of the quantum anharmonic oscillator. But the issues are very much the same. Now you've gotten me interested in this. If I were in my office right now, I'd reach over and grab the relevant volume of Reed and Simon and start typing in cool facts. Maybe I'll do that tomorrow. >How do you tell when extending your series is no longer doing you good? >Without knowing the "exact" value ahead of time (which would presumably >obviate the need to do the perturbative approximation in the first place), >I'm not sure how to tell. Perhaps you compare differences between >successive partial sums and use that as a guide to what's going on. I'm sure there is a whole battery of tricks that people have developed for this kind of thing. However, there's something even cooler that you can do - you can often extract the *exact* answer from the asymptotic series by using tricks called "summation methods". A "summation method" is a way of defining a sum for a certain class of divergent series! There was a big fat thread once here on s.p.r. about summation methods, so if you look up stuff under "Borel summation" and "Abel summation" and "Cesaro summation" you can learn some of these tricks. People have proved, for example, that applying "Borel summation" to the perturbation expansion for the ground state of the quantum anharmonic oscillator gives the exact answer. Now remember, the anharmonic oscillator is just the phi^4 quantum field theory in 0+1 spacetime dimensions. With more work, people have shown that Borel summation can allow you to get exact answers for phi^4 theory in 1+1 and 2+1 dimensions. But nobody knows if Borel summation can handle physically realistic field theories in 3+1 dimensions. Nobody knows the exact answers *exist*, so it's hard to show some method like this works.... >Another question is whether it's always true that the error always either >goes up or goes down and up with the number of terms, or whether it's >possible for the error to oscillate up and down. I don't know about that. But I bet people have studied it. Here's a nice question, hopefully rather thought-provoking, first raised by Freeman Dyson. Suppose the power series you're studying *did* converge for small a > 0. Then by a basic result on power series, it would converge in a small disc |a| < r in the complex plane. Can you use this to cook up a hand-waving physicist's proof by contradiction showing that this would be weird.... so that the power series probably DOESN'T converge for small a > 0? (Don't worry about rigor here... remember, a physicist's proof by contradiction goes like this: "Assume A. I bet that A implies B. But B is really weird! So A must be false.") Of course, in this particular case it's perfectly obvious that the series diverges for all nonzero a. But if you can cook up a hand- waving proof based on general physics principles, you can perhaps apply this argument to more complicated situations where it's not possible to see "by inspection" that the power series diverges. More later.... ============================================================================== From: lgy@newton.phys.washington.edu (Laurence Yaffe) Subject: Re: path integral - sum over all paths? Date: 25 May 2000 10:25:20 GMT Newsgroups: sci.physics.research >>>As some previous responses have indicated, it's actually very >>>important to sum over a large class of paths, including nowhere >>>differentiable paths, when you are doing a path integral. >> >>I can just see Feynman saying "That's not a path!". >>I mean, I know he understood what had to be done, >>but this is surely *not* what he had in mind at first. >One point Feynman would make is that most of the paths have the phase of the >integrand varying so rapidly that they integrate out to zero. A sine wave with >a very high spatial frequency for example. When integrating a lagrangian over a >path, a discontinuity, or a reasonable approximation to one, would have a phase >contribution from a derivative that has a rapidly changing phase. Contributions >from such paths would tend to cancel each other out. With a bit of hand waving, >Feynman would then say that the dominant contribution comes from an extremum >path. Thus, you can take any path you wish. Contributions from crazy paths >cancel each other out. >William Buchman That is too cavalier. The dominant contribution does not come from the extremal path (which is a set of measure zero in the path integral). Rather it comes from a *neighborhood* of the extremal path. And with probability one, all paths in that neighborhood are continuous but everywhere non-differentiable. Feynman would definitely not have disputed this. In fact, anyone thinking about this should feel obligated to immediately go (re)read Feynman & Hibbs, "Quantum mechanics and path integrals", and take a look at Fig. 7-1 whose caption is: "Typical paths of a quantum-mechanical particle are highly irregular on a fine scale, as shown in the sketch. Thus, although a mean velocity can be defined, no mean-square velicity exists at any point. In other words, the paths are nondifferentiable." This is actually a rather deep result. Consider the two assertions: i) Typical paths in a quamtum mechanical path integral are non-differentiable. ii) The quantum operators mreasuring position and momentum satisfy the canonical commutation relation i[p,x] = hbar. The first is a statement about path integrals, while the second is a statement about the usual operator forumation of quantum mechanics. The two are naively completely unrelated. But they are really both aspects of the same thing! As Feynman & Hibbs discuss, one can *derive* the operator formulation of quantum mechanics from the path integral. This involves showing that (i) implies (ii). Or one can derive the path integral from the operator formulation. This involves showing that (ii) implies (i). This also connects with a seemingly unphysical technicality of quantum field theory: iii) Composite local operators in quantum field theory have short distance singularities which must be regularized and renormalized. This is illustrated by the mean square momentum (or kinetic energy operator, up to an irrelevant overall factor) in 1-d quantum mechanics, whose path integral representation may be written as either: lim_{t -> t'} v(t) v(t') or v(t)^2 - (some infinite constant) where v(t) = lim_{eps -> 0} [x(t+eps)-x(t) / eps] is the velocity at some point along the path (and I'm ignoring factors of the particle's mass). In other words, the quantum mechanical operator v^2, which is perfectly well defined, is represented in a path integral not by the obvious square of the velocity of the path [which is infinite], but by the finite remainder after suitably subtracting the infinity. This is a precursor of the whole apparatus of renormalization in quantum field theory. ------------------------------------------------------------------------ Laurence G. Yaffe yaffe@phys.washington.edu Department of Physics University of Washington 1-206-543-3902 (fax: 1-206-543-9523)