On Physics of asymptotic series and resummation

Question

At the end of my second answer to the post on Q&A, I mentioned possibilities of improving convergence of the asymptotic series with "re-expanding the searched function in powers of less growing functions " $f(x)$ than powers of $x$ (or $g$). These possibilities reduce the "impact" of the "uniqueness theorem" to the problem of approximating functions in practice.

Here I am again approximating the toy integral $E(x)$, but this time I decided to use a simple function $f(x)=x/(1+k\cdot x)$, which has a coefficient $k$ to adjust in order to nullify the term $\propto f(x)^3$ in the expression $E(x)\approx 1-f(x)+a(k)\cdot f(x)^2-b(k)\cdot f(x)^3$.

Unfortunately (or fortunately?), the coefficient $b(k)$ cannot be nullified for certain series. However it can be minimized. For $E(x)$ such a $k$ is equal to 2, $b_{min}=2$ and $a(k=2)=0$ The corresponding curves are given in Fig. 4   (I extended the axis $x$ to $x=5$):

                                                                                   Fig. 4.

Precision of extrapolation of $E(x)$ into the region of finite $x$ is drawn in Fig. 5:

                                                                                    Fig. 5.

We can conclude that in order to successfully extrapolate the searched function $E$ to big $x$, we have to choose such a function $f(x)$, which provides sufficiently small coefficients or, better, smaller and smaller terms in the new expansion of $E(x)$ (banality).

This conclusion is also supported with the next two figures (Fig. 6, 7), where the searched function is $I(x)/\sqrt{\pi}$ and the extrapolating series are built in powers of a similar function $f(x)=0.75x/(1+k\cdot x)$. Here $b_{min}=30.14$, $k=4.375$, and $a=0$:

                                                                                     Fig. 6.

 

                                                                                          Fig. 7.

When the number of known terms is small, I think this constructive way of extrapolating functions may be rather practical. One bypasses the "uniqueness theorem" and arrives at a decent approximation.

Finally, the ground state energy $E_0(\lambda)$ of the anharmonic oscillator in QM (anharmonicity $\lambda x^4$) has also a divergent series: $E_0(\lambda)\approx 0.5(1+1.5\lambda-5.25\lambda^2+41.625\lambda^3)$, which can be transformed into a series in powers of $f(\lambda)=1.5\lambda/(1+3.5\lambda)$. It gives a good extrapolation of $E_0(\lambda)$ (error $\le\pm 1.5$% within $0\le\lambda\le 1$, Fig. 8), unlike the original series (the exact curve was taken from here http://arxiv.org/pdf/quant-ph/0305128.pdf):

                                        Fig. 8. The ground state energy $E_0(\lambda)$ of 1D anharmonic oscillator.

Thus, my idea was not too stupid as it allowed to extrapolate the asymptotic (divergent) series in the region of big with a reasonable accuracy.

The moral is the following: if you want to have a “convergent” series, then build it yourself and enjoy.

Comments

What you do here is more or less what is generally called variational perturbation theory, just with different choices in detail. See also http://www.physics.wustl.edu/sbrandt/talks/slideschicago.pdf and http://users.physik.fu-berlin.de/~kleinert/b8/psfiles/19.pdf

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Even without reading these papers I know I am not always original. I mean, I admit that somebody else did similar things too.

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I added the references to give you an incentive to upgrade your knowledge. It makes a very poor impression if you write something in bliss ignorance of related work in the literature. Knowing (and studying) what others do with similar techniques is always a great plus for one's own research.

Nobody achieves high goals without standing on the shoulders of others who added to one's own strength. You complained repeatedly about not having the strength to push your ideas forward. But it may well be that the main reason is that you don't link what you do to what others have done. 

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Thank you, Arnold, for the references and your kindness. The idea of re-expansion of series in powers of "less growing functions" occurred to me in the beginning of my career (1981-1983), when I learned Pade approximants, Borel summation, etc. I tested this idea and abandoned it because it seemed to me so evident, trivial, and maybe less effective than other ways of approximating. It concerned any series, not only "perturbation theory". Later on I learned from others other ways of improvement of perturbative series and proposed my own ones. I never returned to this subject because it was not an actuality to me any more. It is true, I do not know works of many other researchers, but you could flatter me for creativity instead of blaming me for illiteracy.

Answer 1

The last sentence of your post says: ''if you want to have a convergent series, then build it yourself". This is poor advice as it just amounts to ''forget about all other people have done and discover for yourself a good way to do it''. This is the approach of the amateur.

But PO is a site for professionals and those who aspire to become it. Professionals know that they are likely to find far better solutions if they first consult the literature, and consult it again after they found their first successful approach. For example, concerning variational perturbation theory, it can be pushed to give highly accurate and uniformly converging solutions for the energy levels of the quartic anharmonic oscillator, while your trial and error approach only gives a heuristic without any guarantee about the errors involved.

When may years ago I started my career in mathematics (still at high school) I had very limited access to the literature and discovered lots of things for myself, only to find them later in books written 100, 80, or 50 years ago. (At that time, literature search extended only into the past, and I was essentially limited to books, Each book I could understand had references to other books, typically at least 10 years older....) When I entered university and had access to the full literature, it became worse, as I got better in searching for the right literature. With time, the ''new'' results I kept rediscovering were of more and more recent origin, and I counted it as success when the distance to the present got smaller - it meant that I got closer to the state of the art. Until I started to find results which were really new and publishable....

but you could flatter me for creativity instead of blaming me for illiteracy.

Spending time reading the relevant literature (and finding out what is relevant) always pays in the long run - especially when one has high aspirations. Creativity is by far not enough when your ultimate goal is to reform QED. 

In science one never gets credit for a late rediscovery, only for progress beyond the current state of the art. I measure your work at the goals they claim to lead to. Unfortunately, this means mostly having to blame it for not being adequate. Repeatedly I had advised you to remove all poorly justified claims as it is these that degrade your work. But you repeat them in every new preprint you write, and so you need to be blamed again and again.

Comments

The last sentence of your post says: ''if you want to have a convergent series, then build it yourself". This is poor advice as it just amounts to ''forget about all other people have done and discover for yourself a good way to do it''. This is the approach of the amateur.

Saying this I wanted to underline that one is not left with a useless divergent series and the "uniqueness" theorem. This was one of concerns of this question.

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I wanted to underline that one is not left with a useless divergent series and the "uniqueness" theorem

But what you actually underlined by how you chose your words was that one should do it oneself (and therefore forget about the literature). Not that the question you just referred to  mentioned various resummation techniques in its first paragraph, which turn many divergent sums into convergent ones. 

And your post didn't address at all the question of uniqueness. You just gave one answer, which could have been the right or the wrong converged version, since it can be proved that there is no unique answer without making further assumptions. 

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Arnold, you always find something to blame me. What a cruel person you are!

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It is not meant personally; in fact I treat you generously, and my critique is only about what and how you write and claim. 

But you want to have a scientific discussion (after all, this is what PO is for), and I state what needs to be stated from the scientific perspective.

You are creative, and you do work that is original on the basis of what you know about physics. Your creativity would be much better applied if you would also bother about learning the state of the art - others are creative as well, and some of them did very appreciable work in the same direction as you.

You could learn from our discussions, improve your writing style, and take the literature better into account - this would already make a lot of difference.

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Arnold, maybe it is enough to put me down on PO? I indeed want a scientific discussion, but you discuss my drawbacks and I do not see the end of it.

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What do you perceive as a put down? That I say you can improve, and point out the things that deserve improvement? That I expose the discrepancy between your claims and what you actually achieve? That I give references to show that what you do is not so novel but is related to others who also try to understand asymptotic series and QED? If this hurts you, what is the point of having your work discussed? 

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You do not discuss my work, Arnold. At best, you list what is absent in my work.

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Your exaggerated claims and your lack of references are part of your work. What you wrote in the post of this thread is correct and useful as a heuristics, so what is there to discuss? I am not writing a referee's report, so there is no point summarizing what you did. The discussion can therefore only be about how your work fits into the state of the art, and which impact it might have. This is what I did.

Your work is related to stuff that merits discussion, as it expands on the scope of your methods. And one can get error bounds in that way, hence a higher quality of knowledge, so this is worth mentioning. That's what discussions are about - extending the vision of the reader. (I care for all readers here, not just for you.)

But instead of looking into these papers and saying, perhaps, 'yes, interesting, but in don't understand this ...', which would have made the discussion more valuable, you complained and saw it as me blaming you for illiteracy. You brought in the personal aspect, saying I should flatter you for creativity. So I conceded that you are creative, but addressed that creativity must be applied in the right context to be useful for the community.

If I am only allowed to praise you, don't ask me for further comments on anything!

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I did not say you should flatter me for creativity. I said that you could, but you did the opposite. And I said this just because you write nothing about the subject, but about me. According to you, I am not professional, I am against relativistic and gauge invariance, I am against learning, I write a low-level stuff, I advance exaggerated claims, and my works are worth nothing because it is a half-baked rather than a completed result. All this is rather discouraging for me and for an occasional reader who might be interested in the subject.

As to my claims, I show where and how we physicists make a physical and mathematical error while coupling good equations. This is a strict and general result. Without this error, the new equations become coupled correctly, relativistic, non relativistic, whatever. So my claim is the following - do not commit such an error and you will succeed. You, Arnold, completely missed this important point.

I do not need any flattery, but as you concentrated on putting me down and crossed the red line, what I am saying to you now is the following: you, Arnold, only allowed to please me.

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@VladimirKalitvianski: Dear Vladimir, please look at what you are doing in your blog and here on PO - endlessly blaming the excellent, hard-discovered and highly predictive work on QED of many scientists as being wrong, just because it doesn't fit into your dream of what QED should be. Clearly you think that blaming established work for its perceived errors and gaps paves the way to better work in the future. 

But when someone else with a slightly different perspective applies this principle to your work you get upset. How strange!

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I am a strange man. But not as strange as P. Dirac.