I am have been dreaming of reformulating the Classical and Quantum Electrodynamics since mid 80-ties, but without any support (funding) it’s been practicaly in vain, so I am not dreaming of it anymore. I am tired and retired. Everything written below – my reasonings, new results and toy models – is already in the past – for curious and interested in this subject people.
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Reformulation: Why it is necessary?
It is necessary for better understanding the corresponding physics and for having better equations since currently the equations are such that their solutions need modifications. This fact reflects lack of physics understanding while constructing these equations.
Why it has not been done before?
In fact, many have tried, but none prevailed. And currently it is renormalizators (practitioners) who are teaching the subject, not theory developers, so they do everything to convince students to accept “bare particle” physics. In Classical Electrodynamics (CED) some teach that 
Similarly in QED – although the equation set is different from that of CED, the renormalization is still a crucial part of calculations. And in addition, soft mode contributions (absent in the first Born approximation) are obligatory for obtaining physically meaningful results. If one is obliged to sum up some of its contributions to all orders, it indicates a bad initial approximation used for the perturbation theory.
Many theory developers (founding fathers) were looking for better theory formulations. It happened to be an extremely difficult problem, mainly due to prejudices implicitly involved in theoretical constructions.
Paul Dirac, a rare physicist who was not thinking of fame and money at all, never gave up. His motto – a theory must be mathematically and physically sensible [6], and for the sake of that we must search for better Hamiltonians, better formulations, better description than the current one, is my motto too.
If you have read my blogs (this one, http://fishers-in-the-snow.blogspot.fr/ , http://vladimir-anski.livejournal.com/) and articles, (more here) you may have an idea what I mean by reformulation. If you like, my program can roughly be understood as both fulfilling the counter-term subtractions exactly:
and including some of this “good” (renormalized, to be exact) Lagrangian terms into a new initial approximation, i.e., figuratively speaking, I mean representing:
![\mathcal{L}_{good}= \left[{\mathcal{L}}_0+\mathcal{L}_{soft}\right]+\left[\mathcal{L}_{good}-{\mathcal{L}}_0-\mathcal{L}_{soft}\right]=\tilde{\mathcal{L}}_0+\tilde{\mathcal{L}}_{int}^R.\qquad (2)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BL%7D_%7Bgood%7D%3D+%5Cleft%5B%7B%5Cmathcal%7BL%7D%7D_0%2B%5Cmathcal%7BL%7D_%7Bsoft%7D%5Cright%5D%2B%5Cleft%5B%5Cmathcal%7BL%7D_%7Bgood%7D-%7B%5Cmathcal%7BL%7D%7D_0-%5Cmathcal%7BL%7D_%7Bsoft%7D%5Cright%5D%3D%5Ctilde%7B%5Cmathcal%7BL%7D%7D_0%2B%5Ctilde%7B%5Cmathcal%7BL%7D%7D_%7Bint%7D%5ER.%5Cqquad+%282%29&bg=ffffff&fg=333333&s=0&c=20201002)
The new “free” Lagrangian 
so that no renormalization will be needed, and the soft diagram contributions will be taken into account automatically in the first Born approximation by construction, like in [9], [11]. The resulting perturbation theory series will resemble a usual Taylor series with no necessity to cheat and modify its terms. This is an unexplored possibility of the theory formulation and it is what I would like to do.
What I need?
In order to pursue my research, I need funds. I believe that we can achieve a better description if we abandon some prejudices and employ some physical reasoning instead of doing by a blind analogy. I have already outlined possible directions in my articles [7-11]. But currently I am working for a private company, fulfilling subcontract studies, and it takes all my time and efforts. This activity is far from my dream, though. I have to abandon it in order to concentrate myself on my own subject. I’ve got to break free!
Academia does not support this “reformulation approach” any more. I can only count on private funding. If you or your friends or friends of your friends are rich people, then create a fund for supporting my research, run it and we will make it possible.
I do not need a crazy amount like a Milner prise, no! A regular salary of a theorist will suffice. And remember, this subject is not a piece of cake, but an exhausting job.
P.S. Et voilà, I became unemployed (27 April 2016). Sponsors, hurry up, I am getting older!
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[1] Sidney Coleman, Classical Electron Theory from a Modern Standpoint, http://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf
[2] Gilbert N. Plass, Classical Electrodynamic Equations of Motion with Radiative Reaction, Rev. Mod. Phys. V. 33, 37 (1961), http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.33.37 or https://drive.google.com/file/d/0B4Db4rFq72mLcUN6bEhweTgyWkE/edit?usp=sharing
[3] V. L. Ginzburg, Theoretical Physics and Astrophysics, Pergamon Press (1979), http://www.amazon.com/Theoretical-Physics-Astrophysics-Monographs-Philosophy/dp/0080230679 , https://drive.google.com/file/d/0B4Db4rFq72mLWGhCTXVJLUU1WVk/edit?usp=sharing
[4] Feynman Lectures on Physics, Volume II, Chapter 28.
[5] L. Landau, E. Lifshitz, The Classical Theory of Fields, § 75, p. 205.
[6] Jagdish Mehra (editor), The Physicist’s Conception of Nature, (1973), https://drive.google.com/file/d/0B4Db4rFq72mLWnIyM1FSOGcxaDA/edit?usp=sharing
[7] Reformulation instead of renormalization, http://arxiv.org/abs/0811.4416
[8] Atom as a “Dressed” Nucleus, http://arxiv.org/abs/0806.2635
[9] A toy model of Renormalization and Reformulation, http://arxiv.org/abs/1110.3702
[10] Unknown Physics of Short Distances, https://www.academia.edu/370847/On_Probing_Small_Distances_in_Quantum_World
[11] On integrating out short-distance physics, Physics Journal, V. 1. N. 3, pp. 331-342 (2015)
. A direct summations of its Taylor series is useless because it diverges from the exact function value at any finite 
. It is so not only for fast “growing” coefficient cases, but also for regular (converging) Taylor series truncated at some finite order, when we try to extrapolate the truncated series to finite (large) values of 
. “Why not to expand such a slow function in powers of slowly changing functions?“, thought I, for example: 
, where 
at small 
for finite 
, 
, and 
with adjustable coefficient 
. The corresponding figures are the following:
at
(note, the axis 
.
is approximated now much better than with its truncated asymptotic series in powers of 
works well too:
.
of the anharmonic oscillator in QM (anharmonicity 
) has also a divergent series: 
, which can be transformed into a series in powers of 
. It gives a good extrapolation of 
% within 
, Fig. 6), unlike the original series:
(a convergent one) whose convergence I managed to improve with partially summing up some of its terms into a finite function 
, so the resulting series 
became even better convergent (Chapters 3 and 4). It is somewhat similar to soft contributions summation in QED, if you like.
-like potential 
and calculates scattering amplitudes (reflection/transmission amplitudes). In particular, he obtains the “low-energy” formula: 
. It is just a regular calculation. Everything is physically reasonable and no renormalization is necessary so far. In particular, when the potential coefficient 
tends to infinity, the transmission amplitude vanishes and the incident wave is completely reflected. It is comprehensible in case of a positive 
. This potential gives “undesirable” results. Replacing 
with a “regularized” version of this kind ![V_1 (x)=c_1\cdot[\delta(x+a)-\delta(x-a)]/2a](https://s0.wp.com/latex.php?latex=V_1+%28x%29%3Dc_1%5Ccdot%5B%5Cdelta%28x%2Ba%29-%5Cdelta%28x-a%29%5D%2F2a&bg=ffffff&fg=333333&s=0&c=20201002)
, the author obtains a “regularized” amplitude 
. Again, so far so good. When 
, the transmission amplitude tends to zero too. It is qualitatively comprehensible because each 
grows in its absolute value as in case of 
in the problem considered just above. 
is a highly reflecting potential. 
from the region 
, like an infinite barrier of a finite width.
. As well, it is possible that in experiment one observes a non zero transmission amplitude and renormalization of 
. The author denotes 
and calls it a “phenomenological parameter” to compare with experiment via the famous formula 
is “not observable”, I wonder why and for what reason one then proposes such a 
depending on the absolute coordinates is conveniently presented as a product of a plane wave 
describing the atomic center of mass and a wave function of the relative or internal collective motion of constituents 
where ![\mathbf{R}_{\rm{A}}= \left[ M_{\rm{Nucl}}\mathbf{r}_{\rm{Nucl}}+m_{\rm{e}}(\mathbf{r}_{\rm{e}_{1}}+\mathbf{r}_{\rm{e}_{2}})\right ]/(M_{\rm{Nucl}}+2m_{\rm{e}})](https://s0.wp.com/latex.php?latex=%5Cmathbf%7BR%7D_%7B%5Crm%7BA%7D%7D%3D+%5Cleft%5B+M_%7B%5Crm%7BNucl%7D%7D%5Cmathbf%7Br%7D_%7B%5Crm%7BNucl%7D%7D%2Bm_%7B%5Crm%7Be%7D%7D%28%5Cmathbf%7Br%7D_%7B%5Crm%7Be%7D_%7B1%7D%7D%2B%5Cmathbf%7Br%7D_%7B%5Crm%7Be%7D_%7B2%7D%7D%29%5Cright+%5D%2F%28M_%7B%5Crm%7BNucl%7D%7D%2B2m_%7B%5Crm%7Be%7D%7D%29&bg=ffffff&fg=333333&s=0&c=20201002)
and 
are the electron coordinates relative to the nucleus 
(see Fig.1).
and 
are not separated (the interacting constituents are in mixed states). What can be separated in 
are normal (independent) modes of the collective motion (or “quasi-particles”). Normally it is their properties (proper frequencies, for example) who are observed.
), the wave function of internal motion, for our numerical estimations and qualitative analysis, can be quite accurately approximated with a product of two hydrogen-like wave functions 
where 
is a wave function of 
ion (
) and 
is a wave function of Hydrogen in a highly excited state (
).
is expressed via “collective” coordinates thanks to the relationship 
. I take a non-relativistic proton with 
as a projectile and I will consider such transferred momentum values 
that are insufficient to excite the inner electron levels by “hitting” the nucleus. Below I will precise these conditions. Thus, for the outer electron the proton is sufficiently fast to be reasonably treated by the perturbation theory in the first Born approximation, and for the inner electron the proton scattering is such that cannot cause its transitions. This two-electron system will model a target with soft and hard excitations.
. It is so because, roughly speaking, the atomic electrons are light compared to the heavy projectile and they cannot cause large-angle scattering for a kinematic reason. I can consider scattering angles superior to those determined with the direct projectile-electron interactions (
) or, even better, I may exclude the direct projectile-electron interactions 
in order not to involve 
into calculations any more. Then no “screening” due to atomic electrons exists for the projectile nor atomic excitations due to direct projectile-electron interaction at any scattering angle.
describes quantum mechanical smearing of the nucleus charge (a “positive charge cloud”) due to nucleus coupling to the first atomic electron. This form-factor may be close to unity (smearing may not be “visible” because of its small size 
). Form-factor 
describes quantum mechanical smearing of the nucleus charge (another “positive charge cloud”) due to nucleus coupling to the second atomic electron. In our conditions 
is much larger. In our problem setup the projectile “probes” these positive charge clouds and do not interact directly with the electrons.
tending to unity, as if this short-distance physics were absent. We can pick up such a proton energy 
, such a scattering angle 
, and such an excited state 
, that $
.
for the inner electron state [1]. (I remind that 
.) 
and that of exciting any “internal” state is described with the factor ![\left[1-|f1_0^0|^2\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B1-%7Cf1_0%5E0%7C%5E2%5Cright%5D&bg=ffffff&fg=333333&s=0&c=20201002)
):
stands 
for the 
and factor 5 originates from the expression 
. So, 
for 
. Fig. 2 shows just such a case (the red line) together with the other form-factor – for a third excited state of the other electron (the blue line) for demonstrating a strong impact of 
.
and 
at 
. 
form-factor 
becomes very close to unity (only elastic channel is open for the inner electron state) whereas form-factor 
may be still very small if 
. The latter form-factor describes a large and soft “positive charge cloud” in the elastic channel, and for inelastic scattering (
) it describes the soft target excitations energetically accessible when hitting the heavy nucleus.
at 
.
because its energy is way insufficient (its de Broglie wavelength is too large for that). The incident electron should be rather relativistic to be able to probe such short-distance details [1].
involved in 
and 
. The presence of the first electron would not be necessary in such a theory: the latter would work fine and without difficulties – it would reproduce low-energy target excitations.
) is not a “scale” in our theory in a Wilsonian sense.
in powers of “small coupling parameter” 
in the exponential (3): 
. For the first electron (i.e., for the hard excitations) the term 
may be small (see Fig. 3) whereas for the second one 
is rather large and diverges in the soft limit 
. In QED the hard and soft photon modes are treated perturbatively because the corresponding electron-field interaction is factually written in the so called “mixed variables” [5] and the corresponding series are similar to expansions of our inelastic form-factors 
in powers of 
excitations would not contribute (with some reservations). Roughly speaking, they would be integrated out (taken into account) automatically, like in my “two-electron” target model given above, reducing naturally to a unity factor.
and 
are bare particle mass and charge and the term 
is how bare particles interact. Of course, bare particles have spin and other quantum numbers.
where bare particles live. Thus we insightful nailed the bare particle properties and their interaction laws correctly despite their hiding from us.
was derived [1] and it remains an unsolved problem even today (there are still publications on this subject). We must not confuse it with the electromagnetic mass defect, though, i.e., with a calculable interaction energy.
, but if we take into account the electron magnetic moment field energy too, we will obtain another radius, closer to the Compton length 
. Still, in nature there is no electron of the classical or any other radius. And normally this part of the field energy is entirely discarded and what is left is an interaction energy of charges. Thus, when we calculate a field energy, the electromagnetic mass is just of no use.
enters the usual “mechanical” equation of a charge when we decide to insert the charge proper field into the charge equation of motion in the frame of a self-action ansatz. The latter is done for the sake of taking into account a weak radiation reaction force, which must provide the total energy conservation. The motivation – energy conservation – is understandable, but in a field approach with 
, we insert the entire filed 
, not just a radiation field 
, in the mechanical equation. We do it by (a wrong) analogy with an external force 
. So, before this our intervention we have a good “mechanical” equation (I use a non relativistic form)
with 
makes it impossible for a charge to change its state of a uniform motion 
. This is a self-induction force, an extremely strong one. It’s an understandable “physical effect”, but first, it is not observed as infinite, and second, the self-induction force is not a radiation reaction force in any way, so our approach to describing the radiation influence via self-action is blatantly wrong. Albeit of an anticipated sign (and even when made finite and small), it does not help conserve the total energy. Microsoft Windows would say:
that “absorbs” 
is introduced in physics by physicists, introduced exclusively with the purpose to subtract the harmful electromagnetic mass. This introduction is not convincing to me. A negative mass makes the particle move to the left when the force pulls it to the right. We never observed such a silly behaviour (like that of a stupid goat) and we never wrote the corresponding equations. We cannot pretend that (1) describes such a wrong particle in an external field, but adding its self-induction makes the equation right, as it does Kerson Huang. It’s all the way around: in order to make the wrong equation (2) closer to the original one (1), we just discard the electromagnetic mass whatever value it takes. Kerson Huang should have written honestly “One notices that the self‐mass ought to be omitted“.
and 
. But it is not true either: equation (1) contains the physical mass 
and in addition, if the external force in (1) contains the omnipresent gravity force, say, 
for simplicity, the latter does not acquire any addendum when we add that self-induction force. In reality, we fight our own invention 
, but they do not do it. They stick to the self-action patched with the “bare mass mechanism” and they hope that the jerk “remainder” of self-action will correctly describe the radiation reaction. Let us see.
, they are left with the jerk 
called a “radiation reaction” force:
and integrate it by parts to “show” that on average it is a radiation power.) If they insist on using quasi-periodic solutions in their proof, these solutions do not belong to Eq. (4). (A “jerky” equation like (4) does not even have any physical Lagrangian to be directly derived from!)
(or alike):
as if we solved (4) by iterations (perturbation method). However, in the true iterative procedure we obtain a given function of time 
on the right-hand side rather than a term 
and 
. For example, in an oscillator equation
is a known external (periodic and resonance (!) in case of harmonic oscillator) driving force whereas the replacement term 
is unknown damping force (kind of a friction):
(a red line in Fig. 2)
(a blue line in Fig. 2). Solution to a damped oscillator equation is non linear in 
, non linear in a quite certain manner. It is not a self-action, but an interaction with something else. This difference in equations is qualitative (conceptual) and it is quantitatively important in case of a strong radiation reaction force and/or when 
(I used in this example 
with 
, and 
). I conclude therefore that a damped oscillator equation (7) is not a perturbative version of (6), but is another guesswork result tried and left finally in practice because of its physically more reasonable (although still approximate) behaviour. Similarly, equation (5) is not a perturbative version of (4), but another (imperceptible) equation replacement [3], [4]. Of course, there is no and may not be any proof that perturbative series for (4) converge to solutions of (5). The term 
. Nowadays too few researchers see these steps as a severe lack of basic understanding of what is going on. On the contrary, the mainstream ideology consists in dealing with the same wrong self-action mechanism patched with the same discarding prescription (“renormalization”), etc., but accompanied also with anthems to these “guiding principles” and to their inventors. I do not buy it. I understand the people’s desire to look smart – they grasped principles of Nature, but they look silly to me instead.
, and together with Maxwell equations it works fine. Then why not to obtain (5) directly from (1) and from another physical concept different from a wrong self-action idea patched with several forced replacements of equations? Why do we present our silly way as the right and unique? Relativistic and gauge invariance (equation properties) must be preserved, nobody argues, but making them “guiding principles” only leads to catastrophes, so (5) it is not a triumph of “principles”, but a lucky result of our difficult guesswork done against the misguiding principles. Principles do no think for us researchers. Thinking is our duty. Factually we need in (1) a small force like that in (5), but our derivation gives (2). What we then do is a lumbering justification of replacements of automatically obtained bad functions with creatively constructed better ones. Although equations like (5) work satisfactorily in some range of forces, the lack of mechanical equation with exact radiation reaction force in CED shows that we have not reached our goal and those principles have let us down.
to the gauge one 
) is not small and should be taken into account exactly (joined with 
, hence, removed from the “perturbation”):
(3) and its series in powers of 
and 
in a relationship 
(à la renorm-group) is not physics, 
Reformulation instead of Renormalizations 