Reformulation instead of Renormalizations

A popular explanation of renormalization

Vladimir Kalitvianski — Sun, 06 Jan 2013 14:14:46 +0000

I show where the error is made. Everyone can follow it.

Many think that renormalization belongs to relativistic quantum non linear field theories, and it is true, but it is not all the truth. The truth is that renormalization arises every time when we modify undesirably coefficients of our equations by introducing somewhat erroneous “interaction”, so we return to the old (good) values and call it renormalization. Both modifications of coefficients show our shameful errors in modeling and this can be demonstrated quite easily with help of a simple and exactly solvable equation system.

Let us consider a couple of very familiar differential equations with phenomenological coefficients (two Newton equations):

One can see that the particle acceleration excites the oscillator, if the particle is in an external force. In this respect it is analogous to the electromagnetic wave radiation due to charge acceleration in Electrodynamics.

The oscillator equation system can be equivalently rewritten via the external force:

It shows that the external force application point, i.e., our particle, is a part of the oscillator, and this reveals how Nature works (remember P. Dirac’s: “One wants to understand how Nature works” in his talk “Does Renormalization Make Sense?” at a conference on perturbative QCD, AIP Conf. Proc. V. 74, pp. 129-130 (1981)).

Systems (1) and (2) look like they do not respect an “energy conservation law”: the oscillator energy can change, but the particle equation does not contain any “radiation reaction” term. Our task is to complete the mechanical equation with a “radiation reaction” term, like in Classical Electrodynamics. It is namely here where we make an error. Indeed, let me tell you without delay that the right “radiation reaction” term for our particle is the following:

If we inject it in system (2), we will obtain a correct equation system:

Here we are, nothing else is needed for “reestablishing” the energy conservation law. System (4) can be derived from a physical Lagrangian in a regular way (see formula (22) here). We can safely give (4) to engineers and programmers to perform numerical calculations. ~~Period.~~ But it is not what we actually do in theoretical physics.

Instead, we, roughly speaking, insert (3) in (1) with help of our wrong ansatz on how “interaction” should be written. Let us see what then happens:

Although it is not visible in (5) at first glance, the oscillator equation gets spoiled – even the free oscillator frequency changes. Consistency with experiment gets broken. Why? The explanation is simple: while developing the right equation system, we have to keep the right-hand side of oscillator equation a known function of time, like in (2), rather than keep its “form” (1) (I call it “preserving the spirit, not the form”). Otherwise it will be expressed via unknown variable which is coupled now to , and this modifies the coefficient at the oscillator acceleration when in the oscillator equation is replaced with the right-hand side of the mechanical equation. In other words, if we proceed from (1), then we will make an elementary mathematical error because we not only add the right radiation reaction term, but also modify coefficients in the oscillator equation, contrary to our goal. As a result, both equations from (5) have wrong exact solutions. If we insist on this way, it is just our mistake (blindness, stubbornness) and no “bare” particles are responsible for undesirable modifications of equation coefficients.

In CED and QED they advance such an “interaction Lagrangian” (self-action) that spoils both the “mechanical” and the “wave” equations because it preserves the equation “form”, not the “spirit”. In our toy model we too can explicitly spoil both equations and obtain:

with advancing a similar “interaction Lagrangian”:

Here in (6) – masses with “self-energy corrections”. Thus, it is the “interaction Lagrangian” (7) to (1) who is bad, not the original constants in (1), whichever smart arguments are invoked for proposing (7).

Moreover, there is a physical Lagrangian for the correct equation system (4). Therefore, we simply have not found it yet, so we are the main responsible for modifying the equation coefficients in our passage from (1) to (6), not some “bare particle interactions”.

In QFT they perform a second modification of coefficients, now in perturbative solutions of (6) to obtain perturbative solutions of (4), roughly speaking. Such a second modification is called “renormalization” and it boils down to deliberately discarding the wrong and unnecessary “corrections” to the original coefficients in (6):

In other words, renormalization is our “repair” of spoiled by us coefficients of the original physical equations, whatever these equations are – classical of quantum. Although it helps sometimes, it is not a calculation in the true sense, but a “working rule” at best. A computer cannot do numerically such solution (curve) modifications. The latter only can be done in analytical expressions by hand. Such a renormalization can be implemented as a subtraction of some terms from (7), and it underlines again the initial wrongness of (7). It only may work by chance – if the remainder (3) is guessed right in the end, as in our toy model.

P. Dirac, R. Feynman, W. Pauli, J. Schwinger, S. Tomonaga, and many others were against such a “zigzag” way of doing physics: introducing something wrong and then subtracting it. However nowadays this prescription is given a serious physical meaning, namely, they say that no discarding we do, but it is the original coefficients who “absorb” our wrong corrections because our original coefficients in (1) are “bare” and “running”! Of course, it is not true: nothing was bare/running in (1) and is such in (4), but this is how the blame is erroneously transfered from a bad interaction Lagrangian to good original equations and their constants. Both modifications of coefficients (self-action ansatz and renormalization) are presented as a great achievement today. It, however, does not reveal how Nature works, but how human Nature works. Briefly, this is nothing else but a self-fooling, let us recognize it. No grand unification is possible until we learn how to get to (4) directly from (1), without renormalization.

Most of our “theories” are non renormalizable just for this reason: stubbornly counting that renormalization will help us out, we, by analogy, propose wrong “interaction Lagrangians” that not only modify the original coefficients in equations, but also bring wrong “radiation reaction” terms. Remember the famous leading to runaway exact solutions in CED. We must stop keeping to this wrong way of doing physics and pretending that everything is alright.

My presentations at INLN

Vladimir Kalitvianski — Fri, 16 Mar 2012 20:09:19 +0000

On the 15-th March I gave two talks à l’Institue Non Linéaire de Nice (Sophia-Antipolis), next to Nice and Cannes, France. My interlocutors were Thierry Grandou (INLN, France) and Herbert Fried (Brown University, USA). Both of them were interested in learning my position and in my explanations, and I am very grateful to them for their invitation. It is a very rare case when people do not reject the very idea that the renormalizations can be removed from our framework by reformulation of our theories in better terms.

The slides without comments are here and here, and with comments (but smaller in size) are here and here.

IVONA – the best text-to-speech converter and the best voices

Vladimir Kalitvianski — Thu, 03 Nov 2011 16:28:19 +0000

Recently I found a very good TTS converter with natural voices and other features. It is IVONA. Try it and maybe one day it will come in handy! It has British and American English male and female voices, as well as some other languages. It can not only be used as a simple text reader, but also voice up your applications if you are a software developer.

Ultimate explanation of renormalizations

Vladimir Kalitvianski — Sat, 16 Jul 2011 14:56:45 +0000

Trying to communicate my results and ideas to people, I started to prepare a PowerPoint document. Any theoretical physics student can follow it. An article version is here: http://arxiv.org/abs/1110.3702.

There are so many different “expoundings” of renormalizations in the literature. I think mine is the only correct one. The others mislead and even fool you. For example, one geek considers the Archimedes effect as a mass renormalization and says that it may give a negative effective mass. What a shit! Don’t buy it! Whatever is the resulting force applied to a body , the body mass remains the same.

Clarification of my position

Vladimir Kalitvianski — Thu, 30 Dec 2010 11:41:34 +0000

Some readers think that I am “against” QED and QFT results because I am against renormalizations. I think I might be insufficiently clear in my critics of renormalizations and thus produced such a false impression.

No, on the contrary, the final results of QED are right and I use them as a valuable data. I am just for a short-cut to these results. A careful reader can easily infer my position from my posts. I am convinced that we (I mean the QED fathers and followers) work with a wrong QED Hamiltonian. Because of this, we are forced to “repair” the calculation results “on the go”. “Repairing” includes discarding unnecessary corrections to the fundamental constants and a selective summation of soft diagrams to all orders. So we only obtain the right inclusive cross sections in the end, not before!

The right Hamiltonian can give the same final results directly, in a routine perturbative way, without discarding any corrections and without summation of divergent diagrams to all orders. The right Hamiltonian, if you like, can be equally called an “exactly renormalized” Hamiltonian. It contains only physical characteristics and it must be constructed just in a more physical way – what is coupled permanently in nature should be implemented so in the new Hamiltonian rather than “coupled perturbatively“. A better initial approximation leads to a better perturbative series – the latter turns into finite and reasonably small corrections due to the new initial approximation being closer to the exact solution. That’s it!

Some readers want me to produce QED results with even more precision than the actual QED provides. They say it is the only way to attract attention to my approach. Frankly, they want too much from me (I mean, from one person). A theory development is a result of years of work of many professional researchers. And I do not even hold an academic position with sufficient research freedom to carry our these laborious calculations. So my results are modest in this respect. But I hope I outline the right direction quite unambiguously.

P.S. See this.

Zoom in Atom or Unknown Physics of Short Distances

Vladimir Kalitvianski — Thu, 02 Dec 2010 20:51:00 +0000

In about 1985, while considering a banal problem of scattering form atoms, I occasionally derived a positive charge (“small” or a “second”) atomic form-factor

describing effects of electrostatic interaction of a charged projectile with a point-like (structureless) atomic nucleus in atom [1]:

(1)

(2)

(3)

To my surprise, it was unknown and it was absent in textbooks despite easiness of its derivation and a transparent physical sense. Namely, for elastic scattering describes the positive charge distribution in atom (not within the nucleus) and for inelastic processes the form-factors describe the atom excitation amplitudes due to transmitting an essential momentum q to the atomic nucleus while scattering (exciting atom due to pushing its nucleus).

In Fig. 1 one can see cross-sections (cuts) of the atomic (Hydrogen) wave functions squared describing the relative electron-nucleus motion in the Hydrogen atom or a Hydrogen-like ion. (A nice applet to visualize and rotate with your mouse 3D and 2D images of Hydrogen configurations is here.) Strictly speaking, they are the relative distance probabilities although they are often erroneously called “negative charge clouds”:

Fig. 1. Probability density 2D plots (electron-nucleus relative distance probability).

I say “erroneously” because in absolute coordinates quite similar, “positive charge clouds”, exist at much shorter distances due to nucleus motion around the atomic center of inertia (CI). The nucleus does not stay at the atomic CI, but moves around it. So the charge density pictures “seen” by a fast projectile include the positive clouds too. This fact is contained in formulas (1-3), both charge clouds being expressed via the same atomic wave function ! If we make a zoom in, we will see a picture similar to Fig. 1 since

Fig. 2 represents qualitatively such a picture for a particular state of a Hydrogen (or a Hydrogen-like ion).

Fig. 2. Qualitative image of atomic charge density, 2D plot of for the state |3,2,2>.

Two dots in the middle of Fig. 2 are “positive charge clouds”. In other words, Fig. 1, without scale indicated in it, describes the negative charge density 2D plots at the “atomic distances” equally well as it does the positive charge density at much shorter distances. This is the true, elastic (non destructive) physics of short distances. Unexpected?

Another beautiful qualitative picture, this time for the state |4,3,1>:

Fig. 3. Qualitative image of atomic charge density, 2D plot of for the state |4,3,1>.

I do not want to discuss here the fact that each “sub-cloud” contains a fractional charge . I would like to underline that the nucleus bound in atom is not seen as a point-like in elastic processes. Its charge and probability are quantum-mechanically smeared. And the most surprising thing about it is the smear size dependence on the electron configuration. The farther electron “orbit”, the larger the positive cloud. It looks counter-intuitively first, but it is so! The atomic form-factors (2) and (3) are just the Fourier images of the cloud densities. The corresponding elastic cross section (1) can be written also via an effective projectile-atomic potential which is softer than the Coulomb “singularity” at short distances [1]. By the way, in a solid state the positive charge “clouds” are rather large and comparable with the lattice step. So in elastic picture the positive charge in a solid is distributed as in a “plum pudding” model! Smearing is always the case for bound states. But one should not confuse how the atomic electron “sees” the nucleus and how a fast projectile sees the atom (Fig. 2, 3). The atomic electron does not see the positive clouds pictured in Fig. 2 and Fig. 3 because the latter is a result of mutual (not independent) motion.

In scattering experiments, however, it is extremely difficult to observe these pictures because of difficulties in preparing the target atoms in certain |n,l,m> states and in selecting the true elastic events. Normally it is impossible to distinguish (resolve in energy) an elastically scattered projectile from inelastically scattered one (the energy loss difference is relatively small). So experimentally, when all projectiles scattered in a given solid angle dΩ are counted, one observes an inclusive cross section. It is the inclusive cross section that corresponds to the Rutherford formula! So the point-like “free” nucleus in the Rutherford cross section is an inclusive (rich) picture, a sum of an elastic and all inelastic atomic cross sections.

Similarly, the “free point-like electron” in QED is not really free, but is permanently coupled to the quantized electromagnetic field. So its charge is also smeared quantum-mechanically and its smear size is state-dependent. This explains what elastic (non destructive) physics occurs at short distances: there is no Coulomb singularity, as a matter of fact. But the sum of an elastic and all inelastic cross sections results in a Rutherford-like cross section as if the electron were point-like and free.

—————————————————-

(By the way, in this approximation the Positronium looks as a rather neutral system: its positive and negative clouds coincide so due to cancellation of positive and negative charge form-factors. Inelastic channels are open instead: 0' title='d\sigma_n ^{n^{\prime}}>0' class='latex' /> )

References:

[1] V. Kalitvianski, Atom as a “Dressed” Nucleus, CEJP, V. 7, N. 1, pp. 1-11 (2009), http://arxiv.org/abs/0806.2635 .

Problem of infinitely big corrections

Vladimir Kalitvianski — Fri, 22 May 2009 13:27:06 +0000

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 [1])). 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 [1]).

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 [2]). 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 [3] (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):

[1] “On Perturbation theory for the Sturm-Liouville Problem with Variable Coefficients”, http://arxiv.org/abs/0906.3504.

[2] “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

[3] “Reformulation instead of Renormalizations”, (an APPENDIX recently added ), http://arxiv.org/abs/0811.4416.

[4] “A Toy Model of Renormalization and Reformulation”, http://arxiv.org/abs/1110.3702

With time I am going to develop, improve them and add new examples to this blog.

I was 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 [3]. 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 [2] (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 [3] 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 [2] 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. vladimir.kalitvianski@wanadoo.fr

———————-

P.S. Funny video of coffee cup experiments at work. You may think it’s a telekinesis, but it’s not: