In this web log I would like to share my findings on reformulation of problems with big (infinite or divergent) perturbative corrections and discuss them. (The blog is regularly updated so do not pay attention to the date – it is a starting date.)
I myself encountered big (divergent) analytical perturbative corrections in practice long ago; it was the beginning of my scientific career (1981-1982 years). It was a simple and exactly solvable Sturm-Liouville problem, with transcendental eigenvalue equations solvable exactly only numerically. Analytical solutions (series) were divergent. First I thought to develop a renormalization prescription to cope with the “bad” perturbative expansion, as I was taught to at the University, but soon I managed to reformulate the whole problem with choosing a better initial approximation by a better variable choice (variable change (see )). Since then I have been persuaded that we have to seek a physically/mathematically better initial approximation each time when the perturbative corrections in calculations are too big (in particular, infinite).
In fact, here may be at least two types of difficulties:
1) A particular physical and mathematical problem has exact, physically meaningful solutions, but perturbation theory (PT) corrections are divergent, like in the Sturm-Liouville problem considered in my articles. Then a better choice of the initial approximation may improve the PT series behaviour. No renormalizations are necessary here (although possible, see Appendix 5 in ).
2) A particular physical and mathematical problem has not any physically meaningful solutions and PT corrections are divergent, like in theories with self-actions. In this case no formal variable change can help – it is a radical reformulation of the theory (new physical equations) which is needed.
In about 1985, considering non-relativistic scattering of charged projectiles from atoms, I derived the positive charge atomic form-factors surprisingly unknown to the wide public (English publication is in ). These form-factors described correctly the physics of elastic, inelastic, and inclusive scattering to large angles. Briefly, according to my results, scattering from an atom with a very large momentum transfer is inelastic rather than the elastic, Rutherford. All textbooks describe it in a wrong way – they obtain an elastic cross section due to erroneously neglecting an essential (“coupling”) term.
This physics is quite analogous to that of QED with its soft radiation which accompanies any scattering in reality (also inelastic channel), but which is not obtained in the first Born approximation in the theory. QED does not obtain the soft radiation due to decoupling the quantized field from the charge in the initial approximation. Solution for a coupled system (charge + filed oscillators) is not known. In my “atomic” case the corresponding “coupled” solution is formally known and unambiguous, at least conceptually, and this helped me construct a better initial approximation in QED – by a physical ansatz, so that I obtain now the soft radiation automatically.
Let me underline here that the QFT Hamiltonians are guessed. And the “standard guess” includes a self-action term first appeared in H. Lorentz works.
The self-action idea was supposed to preserve the energy-momentum conservation laws in the point-like electron dynamics, but it failed – it led to infinite correction to the electron mass and “runaway” exact solutions after discarding the infinity (after mass “renormalization”). In other words, the self-action ansatz in a point-like charge model is just wrong. Many physicists have tried to resolve this problem – to advance new equations with new physics. They were M. Born, L. Infeld, P. Dirac, R. Feynman, and many many others. As I said, in this case no variable change can help – it is a reformulation of the theory (equations) which is needed and what has been sought by researchers.
I personally found that the energy-momentum conservation laws can be preserved in a different, more physical way, if one considers the electron and the electromagnetic filed as features of one compound system: intrinsically coupled charge and field. A physical and mathematical hint of this coupling is the following: as soon as the charge acceleration excites the field oscillators, the charge is a part of these oscillators. Then the external force work splits into two parts – acceleration of the center of inertia of the compound system and exciting its “internal” degrees of freedom (oscillators). So I propose to start from different theory formulation – without self-action, but with another coupling mechanism. This should be done non perturbatively – from the very beginning, just by constructing a better, more physical initial Hamiltonian. Here my understanding corresponds to that of P. Dirac’s who insisted in searching new physical ideas and new Hamiltonians (see, for example, The Inadequacies of Quantum Field Theory by P. Dirac. Reminiscences about a Great Physicist / Ed. B. Kursunoglu, E.P. Wigner. — Cambridge: Univ. Press, 1987. P. 194-198.) In the “mainstream” theories it is the renormalizations that fulfil this “dirty job” perturbatively – they discard unnecessary self-action contributions to the fundamental constants at each PT order. Renormalizations are in fact a transition to another, different result or to the perturbative solution of different, unknown equations. Recently I found a similar explicit statement by P. Dirac in his “The Requirements of Fundamental Physical Theory”, Europ. J. Phys. 1984. V. 5. P. 65-67 (Lindau Lecture of 1982). Being done perturbatively, such a transition is not quite visible. Usually everything is presented as the constant redefinitions in the frame of the same theory. As a result, it is not clear at all to what formulation without self-action the renormalized solutions correspond and if they are physical at all. A very simplified analysis of the renormalization “anatomy” in its “working” in an exactly solvable problem is presented in  (see also Transparent_Renormalization_1.pdf).
In this web-log, in order to demonstrate all this, I am going to present flawless and transparent examples rather than hand waving. References to available publications are the following (they are English translations and adaptations of my Russian publications):
 “On Perturbation theory for the Sturm-Liouville Problem with Variable Coefficients”, http://arxiv.org/abs/0906.3504.
 “Atom as a “Dressed” Nucleus”, http://arxiv.org/abs/0806.2635
(invited and published in CEJP, V. 7, N. 1, pp. 1-11, (2009), http://www.springerlink.com/content/h3414375681x8635/?p=309428ad758845479b8aeb522c6adfdd&pi=0), and
 “Reformulation instead of Renormalizations”, (an APPENDIX recently added ), http://arxiv.org/abs/0811.4416.
 “On integrating out short-distance physics”, http://arxiv.org/abs/1409.8326.
With time I am going to develop, improve them and add new examples to this blog.
I have been repeatedly told that my style of writing is too absolutist and imperfect anyway. I apologize for that. It is not my goal to offend anyone. I do not consider the people advocating self-action and renormalizations as stupid or evil. I consider them as “trapped” and innocent. My expositions, made simple on purpose, are written just to present the moment when and how we all got trapped in this trap. This subject turned out to be extremely tricky for researchers and the only known “resort” has been the “renormalization prescription” for a too long time. Fortunately now there is another physical and mathematical solution and I try to advance it in my works.
First of all it is, of course, a new physical insight that makes it possible to reformulate physical problems in the micro-physics. It “contradicts” to the very idea of “elementary” (in the true sense!) particles. That is why it has been hard for fundamental physicists to figure it out – the mainstream development in micro-physics is based on attempts to deal with “elementary”, independent, separated particles. This idea turned out to be blocking the right insight. On the other hand, the quasi-particle ideas and solutions are widely used in many-body problems. Agree, if some particles are in interaction, they can form compound (non elementary) systems. And some compound systems cannot be ever “disassembled”, unlike bricks in a wall. Some compound systems are “welded” by nature rather than made of “separable” bricks. In a compound system the observable variables are those of quasi-particles . So, the electron and the quantized electromagnetic field, always coupled together, form a compound system – I call it an electronium. The photons in it remain photons, the electron remains the electron; what is different is the way how they are coupled in the electronium. The electron is not free any more, but it moves in electronium around the electronium center of inertia, somewhat similarly to the nucleus motion in an atom  (the nuclei in atoms are not free).
Indeed, it is known that charge-field interaction cannot be “switched off”, even “adiabatically”. The notion of electronium implements this intrinsic property of the charge nature by construction. The photons are just excited states of the electronium – they are quasi-particles describing the “relative” or “internal” motion of this compound system [2, 3]. The electron (a charge) is a part of oscillators and is the external force application point. In the frame of such a compound system the energy-momentum conservation laws hold without the electron’s “self-action”. That is why no corrections to mass (=rest energy) and charge (=coupling constant between “particle” and “wave” subsystems) arise in my approach.
The true understanding of electronium is only possible in Quantum Mechanics. It is based on the notion of charge form-factor. The latter describes the charge “cloud” in a bound state. It is practically unknown, but true, that the positive (nucleus electric) charge in an atom is quantum mechanically smeared, just like the negative (electron) charge  in a smaller volume. It is also described with an atomic (positive charge or “second”) form-factor, so the positive charge in an atom is not “point-like”. The positive charge “cloud” in atoms is small, but finite. It gives a natural “cut-off” or regularization factor in atomic calculations just because of taking the electron-nucleus coupling exactly rather than perturbatively.
Similarly, the electron charge in electronium is quantum mechanically smeared. This gives correct physical and mathematical description of quantum electrodynamics: emission, absorption, scattering, bound states, and all that – without infinities since the electronium takes into account exactly the charge-field coupling – by construction. Thinking of electron as of a free point-like particle is not correct since the point-like free “elementary particle” appears as the inclusive, secondary picture, not a fundamental one (see  for details). The point-like electron “emerges” from this theory as the inclusive, classical or average picture.
Any mathematician knows that the “better” is the initial approximation in a Taylor series, the smaller are corrections to it. (“Better” here means closer to the exact function.) So the problem of “big” corrections is often the problem of “bad” choice of the initial approximation in an iterative procedure. It is the case 1.
In the theoretical physics it holds as well as in the mathematics – the problems are formulated as mathematical problems describing a given physical situation. Theorists choose the total Hamiltonians and the initial approximations following their ideas about physical reality. Unfortunately one can easily obtain the case 2 where the very formulation is non physical and the divergences just show it. I consider the point-like electron model, free electromagnetic field, and the “self-action” ansatz (by H. Lorentz) to be the worst ones although explainable historically. It failed as a physical model (corrections to mass, runaway solutions). Worse, it has given a bad example to follow – the mass renormalization and the perturbative “treatment” of the non-physical remainder. The notion of “infinite bare” mass and an “infinite mass counter-term” is the top of “bad” physics. As long as we follow the flawed approach, we will not advance in physical description of many phenomena. This is what we see nowadays.
Fortunately the theory can be reformulated in quite physical terms. The only sacrifice to do on this way is the idea of “elementariness” of electron in the sense of its being “free” of electromagnetic field and being just a “point-like” in reality.
My research is not finished yet – I am quite busy with other things at my job. I do not hold an academic position. On the contrary, I am on subcontract works implying no freedom and strict timing for each subcontract. As soon as I find a grant or a position (or at least a part time position) to be able to devote myself to the relativistic calculations, I will carry out the Lamb shift and anomalous magnetic moment calculations at higher orders. If you hold a post in science with sufficient responsibilities , you may take an initiative to make my researching possible. I cannot do everything on my own and the resistance of renormalizators is very high. If you are an extremely rich person, consider sponsoring my research via my PayPal account (all you need for that is my e-mail address).
Any constructive proposals/discussions/questions are welcome.
Vladimir Kalitvianski. firstname.lastname@example.org
P.S. Funny video of coffee cup experiments at work. You may think it’s a telekinesis, but it’s not: