I have read this nice post and I have found it really interesting. The reason is the kind of approach of Lubos Motl, being physicist-like, on such somewhat old mathematical matter. The question of divergent series and their summation is as old as at least Euler and there is a wonderful book written by a great British mathematician, G. H. Hardy, that treats this problem here. Hardy is well-known for several discoveries in mathematics and one of this is Ramanujan. He had a long time collaboration with John Littlewood.
Hardy’s book is really shocking for people that do not know divergent series. In mathematics several well coded resummation techniques exist for these series. With a proper choice of one of these techniques a meaning can be attached to them. A typical example can be
and this is true exactly in the same way is true that the sum of all integers is -1/12. Of course, this means that discoveries by string theorists are surely others and most important than this one that is just good and known mathematics.
I agree with Lubos that these techniques are not routinely taught to physics students and come out as a surprise also to most mathematics students. I am convinced that Hardy’s book can be used for a very good series of lectures, for a short time, to make people acquainted with this deep matter that can have unexpected uses.
I think that mathematicians have something to teach us that is really profound: Do not throw anything out of the window. It could turn back in an unexpected way.
Update: I have three beautiful links about this matter that is very well explained leaving readers with a puzzle:
Dear Marco, thanks. I began to read the Hardy book,
and it is pretty fascinating. I especially like the sum f(x) = 1 – 1! x + 2! x^2 – 3! x^3 … because the result he announces looks like some S-dual instanton expansion of a kind. 😉
Do people actually know these Hardy’s formulae when they talk about divergent series in QFT? The usual wisdom that I’ve been saying is that the perturbative expansion is non-perturbatively undetermined and the minimal inaccuracy of the summing is equal to the first nonperturbative corrections.
But don’t the asymptotic expansions actually know more about the nonperturbative completion? It can’t be quite unique because there are theories that perturbatively agree but nonperturbatively don’t. But….
I would like to give an answer to your question that is surely an interesting mathematical problem too. The fact that what appears to be nonsense at a first sight proves to be really useful mathematics is something that always shocked me. But what is the link of asymptotic expansions and non-perturbative solutions is something I would like to know and to write a paper about.
“Divergent series are the invention of the devil, and it is shameful to base on them any demonstration
whatsoever.”—N. H. Abel.
I agree to Abel. If we speak about infinity, we can obtain a lot of paradoxes. It’s very difficult to believe we can obtaine some progress in physics starting from this subject.
As you may know, Abel quotations is rather old with respect to Hardy’s book (that cites it). I suggest you to read the book as this mathematics has several applications in physics and is sound as any other mathematical area.
It is difficult for me to think that a Fields medalist as Witten just swallows a mathematical abuse without complaining…
I’ve had the Hardy book since around 1998. Most of the interesting stuff is in the first 2 or 3 chapters. It makes it clear that any single divergent series can be summed in different ways to give different results. To obtain mathematical consistency, one must assume a method or methods of summation.
When I see divergent series show up in the foundations of physics it signifies to me that there are approximations being made and that we are not dealing with the fundamental theory. The fundamental theory might have additional degrees of freedom or whatever, in such a way that the methods of summation are implied when these are integrated over (or whatever).
The interaction with Ramanujan is fascinating and should be the subject of more blog posts (hint to Luboš).
I can understand your dislike about divergent series but their value cannot be overestimate. It is my view that some unexpected mathematical phenomena underlie a deeper reality or mathematical theory yet to be understood. This should be true also for divergent series. On a similar ground should be put also asymptotic series that anyhow fail to converge but give extremely precise results when properly used. They represent the main tool of a physicist.
Dear Carl, that’s very interesting. You must own a very different book.
The book by Hardy that I can read makes it, on the contrary, very clear that every divergent series that can be summed up by a legitimate method, leading to a finite result, always leads to the same result and that the method-independent result is important and can be verified independently, too.
That’s why Hardy wrote the book, after all. I am afraid that you got the whole point of the book upside down.
For example, you can read page 1 (one), last paragraph:
“Newton and Leibniz… had little temptation to use divergent series (although Leibniz played with them occasionally). The temptation became wider as analysis widened, and it was soon found that they were useful, and that operations performed on them uncritically often led to important results which could be verified independently. We give a few examples in the next section: in Chapter II we shall give others, of greater importance, from the work of the classical analysts.”
Could you please provide us with a glimpse of evidence that the book says something bad about summing divergent series – or even says that “anything goes” as a result? Until you will do so, I will consider all your words about the summation of divergent series to be vicious lies and libels.
Best wishes, Lubos
Let me offer one more quotation from the book to show that Carl’s conclusion is fundamentally wrong:
Page 6, second paragraph:
“It generally seems that there is only one sum which it is ‘reasonable’ to assign to a divergent series: thus all ‘natural’ calculations with the series (1.1.1) seem to point to the conclusion that its sum should be taken to be 1/2. We can devise arguments leading to a different value, but it always seems as if, when we use them, we are somehow ‘not playing the game’.”
In section 1.6, Hardy mentions some “extensions” of a series of numbers – such as 1+1+1+1 – in different ways that can lead to different results (zeta(0)=-1/2 or geometric series, which is exactly at the divergent point). But it’s important, and he clarifies it in the book, that in real applications, we are getting these series not only as the information about the numbers “1,1,1,1” but including the functional form that determines whether zeta(-1) or the geometric series (or something else) is legitimate in the given case.
Apparently Luboš has difficulty understanding the English phrase “We can devise arguments leading to a different value”. As a native speaker, let me assure the reader that Hardy is verifying that the sums of divergent series are not defined by the series themselves, but also require some sort of subsidiary assumptions. To assume otherwise is to play games with mathematics.
[…] divergentes en la física La semana pasada, The Gauge Connection hizo mención de un post de Lubos Motl donde comenta algunos usos de series divergentes en la […]
In this paper: http://www.math.mcgill.ca/bcais/Papers/Expos/div.pdf
the autor gives a proof that 1+2+3+4+…=-1/12 and he claims this result to be correct. If the proof according Abel and Cesaro summability is right, we have to conclude that all the differential calculus is wrong also from the point of view of the ‘non standard analysis’. Can these concepts help our understanding of physical phenomena? Until now our belief has been 1+2+3+4+…= infinity. If we replace infinity with -1/12 we have to write down a new physics.
Dear Carl, be sure that I have no problems of the kind you can ever imagine or understand with your extremely modest brain.
On the contrary, you have a problem to understand that what you call “subsidiary assumptions” are not just some nonsense that a good mathematician can ignore.
They’re the essential knowledge that determines which values of the a priori divergent sum and which calculation methods are legitimate or natural and which are not. You’ve decided to save the electricity in your brain and deny and ignore all of this math knowledge, which is why your words about all these issues are forever guaranteed to be pure noise of no value whatsoever.
Luboš, when a mathematician tells you that a glass is half empty he doesn’t somehow mean that it is a full glass. Hardy quite explicitly told you that the assumption that divergent series can be summed, without subsidiary assumptions, requires “game playing”. That should be enough of a clue for you.
If it is possible to make a divergent sum be different things (and it is), then you cannot assume that the sum is what is convenient for you and your physics; unless you wish to also explicitly include those assumptions. A clean theory is one where calculations give predictable results this is not possible in divergent series unless one makes additional assumptions.
A traditional and honest way of making those additional assumptions is to assume an energy scale where our present knowledge breaks down. The fact that it is possible to sweep this lack of understanding under the rug, by playing with divergent series, is not surprising (one can do similar things with extensions of analytic functions and the like).
Mathematically, the fact that there are alternatives to assuming an energy limit to our knowledge of elementary particles is not at all indicative that the there are no surprises at higher energy scales. This is how mathematics operates. If we understood the higher energy behavior the divergent series would no longer be divergent. This is fully compatible with alternative ways of summing them.
Summing up, my complaint about the way you throw around divergent series is not about the mathematics, it’s about the philosophy. You’re using it to avoid the admission that there is knowledge that we do not yet possess. This is against progress in physics because it is in denial of our present state of ignorance.
A related subject is the elimination of irrelevant interaction terms from the Lagrangian. While they may be eliminated in renormalization theory, it is possible that the degrees of freedom they provide may allow the unrenormalized theory to be written in a simpler way. That is, they could appear in the fundamental theory, but not in the effective theory.
The same thing applies to Lorentz invariance. To match experimental results, it must appear in the effective theory, but that is not proof that it must also appear in the fundamental theory. Luboš would reject such a theory without bothering to analyze it.
Once again, Carl, you think that these methods are about “philosophy” because you’re just a dumb, narrow-minded, severely limited, lazy crackpot who never wants to learn anything and who is happier if he promotes “his present state of ignorance” as the ideal state of affairs.
These things are maths, and they’re important maths in physics.
Of course that I would reject a theory of particle physics that breaks Lorentz invariance by generic terms. It’s called falsification and it is the most fundamental tool of all of science.
Please, could you avoid to offend Carl further? I think you should have enough sound arguments to respond on a scientific ground that is what this blog is aimed for.
I should probably note that I’m not at all offended by Luboš and have great respect for him due to his discovery of tripled Pauli statistics. As far as I’m concerned, a person only has to make one amazing discovery in order to overwhelm all possible obnoxious personal qualities.
In order to do physics, one must sacrifice great amounts of time. Psychologically, one does this by convincing oneself that one is not wasting one’s time. And that leads one to believing things that may not be true. Hence disagreements over the interpretation of facts can be emotional.
We all know Luboš and his way to approach scientific matter in disagreements. There is no problem about all this as it can happen to some people to evade the realm of rational arguing and entering into the emotional one.
But what you are saying is more serious. You are claiming that a person like Philipp Lenard should be forgiven thanks to his contributions to physics? I can find a lot of other examples like this in all fields of human endeavors .
There is a curious fact about this Nazi physicist. His research helped Einstein, an Hebrew physicist, to find a correct understanding of the photoelectric effect. This is a typical paradox of science. But should this be enough for him to be considered an acceptable man?
I think you are opening a kind of Pandora box here. I cannot share your view about this question and a great discovery cannot hide forever the bad man or woman the discoverer was. Rather, this fact should be used to remember her at the future generations for the kind of bad person she was.
I’m not saying that “Philipp Lenard should be forgiven”, but instead, that I forgive him. I had the flu a week ago, and for this sort of thing I keep books around that are certain to improve my mood. And so I read Goebbel’s diary of the last few months of his life, in 1945. It was fascinating, and it did cheer me up, what with the defeat of evil. Thinking of Luboš, I burst out laughing at what Goebbels said about Czechs.
While responsible for Nazi propaganda, Goebbels kept a clear understanding of the military situation as it slowly closed in on them. He was hopeful for a miracle but I did not think more so than the usual person would be in similar situations. I concluded that he was sane.
It’s amazing what the human mind can do in terms of what it is possible to believe, in physics as well as politics. I’ve ordered a biography of his beautiful, rich and fascinating wife, Magda, (sort of a 1933 Germany Paris Hilton) who refused to leave Hitler’s bunker and instead killed their 6 children and themselves. This is far more extreme, I think, than the incident in Guyana where a small city drank poison; these people were not nearly so self selected (for example, Magda Goebbels had, when younger, considered conversion to Judaism), and they were not made to practice suicide before committing it.
By contrast, our arguments about Lorentz invariance are quite polite.
I should add that the more I think about the Nazis, the less I understand them. I can’t help but suspect that their story is somehow a universal comment on the frailties of human reasoning and logic.