It is indeed, and how to understand that the quantum effects lead to a finite energy for the ground state of the hydrogen atom was found by W. Pauli in 1926,http://link.springer.com/article/10.1007%2FBF01450175  and, more fully, by V. Fock in 1935, http://link.springer.com/article/10.1007/BF01336904#page-1  and this (and more) is reviewed by M. Bander and C. Itzykson in http://inspirehep.net/record/48744 in work that was published fifty years ago. Mathematics and physics are amenable to rational discourse-esthetic judgements are not. The results of impersonal calculations are what count, not personal opinions about esthetics.

I am sorry, Stam,  but Fock had reported his work in Leningrad in 1935 (NOT IN 1927!) and it was published in 1936 in German. Bander and Itzykson's  2 papers are VERY ugly.  Much better treatment of this (hydrogen) problem could be found in http://www.amazon.com/Variations-Colloquium-Publications-American-Mathematical/dp/082184184X    and even more digestible treatment  could be found in the thesis by Laat "Regularization and quantization of the Kepler problem"(available readily in pdf format on Google).

Dear Stam and Arkady,

I am sorry I cannot read works in German. Other works, such as that of Laat, they don't reflect what I would like to discuss. My aim is comparatively modest, if we use Coulomb potential then we cannot explain why the electron does not radiate when it is orbiting the nucleus. Therefore, we need to modify the potential in a classical manner in order to resolve this problem, without quantisation methods, because Coulomb potential is a classical law. (Of course, the results, such as Bohr's results, must be retained). And as shown in my work, even though the work is very scratchy, Einstein's general relativity could be the answer.

Best regards,

Vu. 

Coulomb potential was already obtained using general relativity methods. E.g. look at "Spectrum generating algebras for the classical Kepler

problem" by A.Keane.J.Phys.A 35(2002) 8083-8108. 

Dear Arkady,

Yes, it's true. But, have you seen my work? I am talking about a modified potential derived from GR.

Kind regards,

Vu.

Of course it's possible to explain the absence of radiation in the Coulomb potential, when quantum effects are taken into account-no modification, beyond quantum mechanics is necessary. In particular, spacetime doesn't become either curved, nor does it fluctuate, which is what the relevance of general relativity would imply: the quantum dynamics of a single particle in the Coulomb potential, in flat spacetime, in the non-relativistic approximation,  is equivalent, mathematically, to the classical dynamics of a field on a compact manifold, in particular, the 3-sphere of finite radius, in momentum space:  that's what Fock found. It's the fact that this manifold is compact that implies that the ground state energy is finite (and that the spectrum is discrete). The radius of the 3-sphere in momentum space depends on Planck's constant and becomes infinite, in the limit when Planck's constant is taken to be zero, which recovers the expected result, that, in the classical limit, the ground state doesn't exist and the system is unstable due to radiation.   What Pauli had found was  that the quantum dynamics of a single, non-relativistic,  particle in a Coulomb potential was integrable, which ensured stability and the absence of radiating states.

Dear Stam,

I think I should state my stand more clearly. I am trying to express and interpret quantum phenomena in terms of classical dynamics within the framework of differential geometry and topology, without the use of quantisation methods. I could not obtain those works that you have referred to, but I guess they all involve quantisation methods in some way. I don't believe in sophisticated mathematical methods, because it doesn't matter how advancedly you represent them, Newton's laws are still Newton's laws. In order to advance in physics, we should try to find new physical laws, not to find new mathematical methods to represent the old laws.

Regards,

Vu.

One way to describe quantum phenomena in flat space(time) is through classical gravitational phenomena in higher dimensional spacetimes. But this isn't the case for the Coulomb potential discussed here, since, as Fock showed, the higher dimensional space doesn't fluctuate-which, in fact, is why the quantum fluctuations are so constrained, the quantum system is integrable. The technical issues as such  are not relevant, the answers matter, so any metaphysical issues about what particular method to use, should be best avoided. It doesn't matter. For an introduction to real applications, it's recommended to read http://arxiv.org/abs/1310.4319

Dear Vu, unfortunately, this problem is not as simple as you are willing to portray  http://www.amazon.com/Kepler-Problem-Regularization-Quantization-Perturbations/dp/303489421X/ref=sr_1_3?ie=UTF8&qid=1459434892&sr=8-3&keywords=kepler+problem     Stam is absolutely wrong when he is saying about radius of the 3 sphere depending on the Planck constant. Just read the book.

The Kepler problem is a classical problem, that of the hydrogen atom is a quantum problem and the two have very little to do with each other, so there's no point bringing them together. If one is interested in the former one shouldn't be writing about the latter, however. To understand the hydrogen atom it suffices to learn quantum mechanics. 

Stam, this is what everybody believes, including myself. However, situation is not as simple as ugly review by Bander and Itzykson is telling us. All depends in the end upon what one wants to figure out from the solution of the hydrogen atom problem. 

It doesn't matter what anyone says-what matters is understanding the calculations themselves. The hydrogen atom is an integrable system, in the non-relativistic approximation, for the particle describing the relative motion and when one can neglect the fluctuations of the electromagnetic field  and the reason can be understood thanks to Fock's work. What Fock did is what matters, from where one reads it is irrelevant.  If one reads German, one can read his paper, if one doesn't one can read any number of presentations, e.g. 

http://info.phys.unm.edu/~ideutsch/Classes/Phys531F11/Runge-Lenz%20Vector/weinberg.pdf

which one can find simply by searching for fock hydrogen atom

and which one uses  is a matter of taste, that doesn't make any sense to focus on. What does matter is to understand that, in this case,  the classical potential is unbounded from below, but that quantum effects change it completely-the quantum potential is completely different and the reasons are known for 70 years, so it's useful to learn them and not imagine that they aren't known. 

Stam, the paper you just sited I am well aware of. This is a simplistic paper. The Thesis I've cited is much better. As we speak, I am working on some topics related to these issues. Your understanding of these things, is  too vague at  the very least. PLEASE, do not be offended!!! The literature on the subject is HUGE. You can read , for example,  book by Wybourne http://www.amazon.com/Classical-Groups-Physicists-Brian-Wybourne/dp/0471965057/ref=sr_1_1?s=books&ie=UTF8&qid=1459448261&sr=1-1&keywords=wybourne   or 2 books I had cited already , or the book by Gilmore  http://www.amazon.com/Lie-Groups-Physics-Geometry-Introduction/dp/0521884004/ref=sr_1_3?s=books&ie=UTF8&qid=1459448337&sr=1-3&keywords=lie+groups+for+physicists and so on. Fortunately, there is still a room for some improvements. Since you are having no taste for serious mathematics, it is difficult for me to make guesses how you will take the paper I am working on  now. Still, I would be glad to let you know when the paper will be put on arxiv.org . It may take another 4-5 months before this will happen though.This is given that I am working on it day and night  for about 5 months already..

Dear Arkady and Stam,

I have never imagined the problems of physics are simple. Since we haven't made significant progress for so long on the foundations of physics, to the point now most people working in physics seem to believe that, or just simply to accept, as Stam has said, the classical problem and the quantum problem have very little to do with each other, and in particular, it is believed that the current formulation of quantum physics is the ultimate theory of all. To me, it sounds more like a religious belief than scientific. I don't believe in a big ad hoc theory either. What I am trying to do is to try to uncover bit by bit what behind all physical laws and to express them in terms of well-defined geometrical and topological entities, and then put them together to form a coherent theory. The simpler the better. However, I believe that we are entitled to our own beliefs and we should keep hold of them until they are proved otherwise.  

Vu you are surely entitled to your beliefs and I am NOT sharing beliefs of Stam that classical and quantum problems have very little to do with each other. Should you give yourself a trouble to look into references I gave to you, this would become pretty obvious What puzzles me is the easiness with which you ignore these references written by the best of the best people in the field. The choice is always yours, as they say...

If one use Coulomb gauge condition, the scalar potential coincides with electrostatic potential. Radiation in this description is related with vector potential only.

Dear Arkady,

I know, as you said, that the literature on the subject of representation of classical and quantum physics is huge, especially group theories, and I did read many of them, including those books that you referred to when I did research for my PhD dissertation more than 20 years ago. The title of my thesis is: Geometrical and Topological Methods in Classical and Quantum Physics. As far as I understand them, group theories alone cannot be used to describe a dynamical process which is an essential component of a physical theory. Group theories are used to classify objects, but not to be used to describe how an object manifests according to physical laws. Please let me know when you post your article.

Dear Mikhail,

Could you please give more details for you comment, in particular, whether the radiation that you mentioned is related to that of a hydrogen-like atom as I would like to ask here.

Regards,

Vu.

Vu, in references I've provided,  group theory is occupying only a part. Much-much more you will get by reading works by Jurgen Moser, Guillemin and Sternberg and so on which are contained in references. These are world -wide renown people.They are not group theory people

Dear Vu and Arkady,

First of all, I agree to Vu in his comment when he said that "we haven't made significant progress for so long on the foundations of physics".  I think this is because Modern Physics (started with Planck and Einstein works at the beginning of the past century) take the wrong path.

One interesting alternative to quantum (Schrodinger or Dirac) models for the Hydrogen atom, all of which apply the coulombian ELECTROSTATIC potential (as you can easily check in textbooks) is to take into account the (classical) Weber electrodynamics for representing the electron-nucleus atractive interaction, since this approach makes use of a potential which includes the motion of the particles. I strongly suggest you a reading of the book "Weber's planetary model of the atom", by A.K.T. Assis et al., as well as its main references.

All previous comments and responses of other people will not offer you an "electrodynamic alternative", since they follow the present paradigma of modern physics (Einstein relativity + quantum theory), varying only the mathematical depth.

Sincerely,

Prof. Fabio M. S. Lima

Dear Fabio, I believe, first one should select a problem and then talk about how to solver it.In the present case, what exactly you would like to accomplish by looking at Hydrogen atom can you provide an itemized list of your objections to what is available already. Then by looking at each item in the list it would be possible to be more focused.

Dear Arkady,

Do you agree in using Coulomb electrostatic potential to model a VERY electrodynamic system (such as the H atom)? The mean speed of the electron around the proton is about 1% of light velocity!

Your opinion, above, clearly reveals your "positivism", i.e., you would accept any physical theory which reveals a good agreement to experiments.  This is just the approach of the physicsts of the beginning of century XX, which I criticize above.

In fact, the theories (Relativity and Quantum Mechanics) created in that time were good in the sense of yielding results in agreement to experiment, BUT they both depart much from classical physics (you know, the physics that makes sense). Of course, the H-atom is a simple, very well-known two-body system, but one in which the use of an ELECTROSTATIC potential does not makes sense (except if you, as Dirac himself, don't know any dynamical alternative)!!!

With respect to problems in modern physics, we have a lot of them! For instance:

i) The large number (more than 100) of fundamental particles (should it not to be reduced to one?)

ii) Standard model predicts neutrinos with an (exactly) null mass, but 2015 Nobel prize was for experiments which prove that m>0  !!!

iii) What is the neutrino speed? Most recent experiments (years after that famous ones at CERN and Gran-Sasso) indicate a very short statistical distribution: $v/c = 1 \pm 1.1 \times 10^{-6}$ (LaTeX notation).  For a reference, search arXiv for 2015 MINOS experiments. Superluminal speeds clearly contradict Einstein relativity!

iv) Hensen experiments (see Nature magazine, Oct/Nov 2015) reveal, above all possible suspects, instantaneous action at a distance ('spooky' action-at-a-distance, in Einstein words).

v) Cosmology: the standard model gives account of only 4% of the mass of universe. Where are the resting 96%? Dark matter? Dark energy? This must be a joke!

vi) There is no quantum theory of gravity...  In other words, Einstein General Relativity and Quantum Mechanics are incompatible! At least one of them is wrong.

vii) The present model for the electron is a joke: a particle with null size (no internal structure), but one that got a finite mass (9.11 x 10^{-31} kg), a finite electric charge (1.6 x 10^{-19}C), and a finite angular momentum $S_z = \pm \hcut/2$. How could these finite physical quantities be attributed to a sizeless particle?

viii) A photon also does not have an internal structure, so it has also a null size. However, it has a null mass, though it has a non-null linear momentum p. Note the great effort of textbook writers when they have to explain this to undergraduate students (see, e.g., Halliday "Fundamentals of Physics", or any other introductory physics textbook which cover modern physics).

ix) Quark model (3 quarks) explain only 2% of the mass of a proton (or neutron). See works by R.Ent, T. Ullrich and R. Venugopalan

x) Recent observation of universe expansion yields a very precise energy density for the vacuum: a value  which is 10^118 times less than the theoretical prediction found by S. Weinberg using Einstein's General Relativity [Re. Mod. Phys. 61, p.1 (1989)]. A huge disagreement between experiment and theory, isn't it?

Prof. Fabio M. S. Lima

Well, I assume that these are questions for the community, not just for me.We all are busy doing something currently..Hopefully, there will be such that will find some time to look into your itemized list.Remember in 1900 David Hilbert spell out his famous problems in mathematics. Let us hope, that yours are of  commensurate quality

Dear Arkady and Fabio,

Arkady, I am looking for those references that you have provided. I hope they contain information, as you said, that provide answers to my question.

Fabio, your list is impressive. But if you look at my work you will see that the electron does not have any kinetic energy associated with it when it moves around the nucleus of a hydrogen-like atom. Therefore, it can be a wave or it can be floating along a wave in stable orbits. And I think its state should be a wave-like state. The differences in the predictions are obvious because the current formulation of quantum physics is not complete and coherent. As also shown in my work, dark matter may be associated with a new dynamics, which is the temporal dynamics. Just like Newtonian dynamics, we need some kind of matter to be associated with a dynamics. I think it is too early to say that quantum mechanics and general relativity are incompatible and therefore at least one of them must be wrong. To my opinion, we need to uncover what is a quantum of energy first, otherwise there would be no progress in physics, regardless of how advancedly you represent it mathematically. How would you know that a photon does not have an internal structure? General relativity is not exclusive to cosmology. The world of a quantum of energy could be a general relativistic one. And I can show you that.

Regards,

Vu.

There aren't 100 fundamental particles-but, even if there were, the number itself, indeed, isn't relevant, but whether there are known relations between the particles. Indeed there are: Just three families of quarks and three of leptons, with very well-defined properties, along with the Higgs scalar and the gauge bosons suffice to explain all known properties of known matter.  And the relations of their charges is understood. That of their masses is not-but it is understood *why* this is the case, for the moment.

The Standard Model doesn't impose that the neutrinos' mass is zero; it's compatible with it being non-zero. Depending on further properties of neutrinos, the discovery that they're massive provides hints of, hitherto, unknown particles, whose effects can only be indirectly probed.

Neutrinos, as massive particles, travel at less than the speed of light; though, being very light, it's not at all trivial to measure this speed. However the measurements, whatever popular accounts might have claimed, don't imply that it is greater than the speed of light, to discovery precision; indeed the initial claim, even neglecting the material defect, was wrong, since the sampling rate of the detector, 20 MHz, corresponds to 50 ns, which means that the advance claimed, 60 ns, had an uncertainty of 50 ns, so the original measurement was 60 ± 50 ns, which isn't greater than 0 to discovery precision. All experimental measurements have statistical and systematic errors that must be under control, in order for any measurement to make sense at all. No one doubts that special relativity is an approximation-just that how it breaks down is the issue. The known properties of matter imply that, if special relativity did appear to break down when measuring the speed of neutrinos, the experiment must have had issues, because similar break downs ought to have been observed in other measurements-and they hadn't been. 

Entanglement is a property of non-relativistic quantum mechanics, that's well understood. Nothing spooky about it, it's a standard part of courses on the subject. The non-trivial part was how to probe it experimentally, but this known for more than 34 years now. And since non-relativistic quantum mechanics isn't Lorentz invariant, the fact that it has properties that aren't compatible with Lorentz invariance isn't a surprise-it's expected. 

The Standard Model provides a quantitative description of all properties of known matter, in particular electrons and photons, as point particles. It is the subject of courses in physics all over the world for decades and there are many on-line course offerings, e.g. http://ocw.mit.edu/courses/physics/8-325-relativistic-quantum-field-theory-iii-spring-2003/

More particularly, atom trap experiments can measure properties of atoms and photons to extraordinary precision and agreement with theoretical calculations is quite non-trivial, but it is understood, cf. http://www.nobelprize.org/nobel_prizes/physics/laureates/2012/

How quarks and their interactions provide the mass of the proton (and the neutron) is understood-cf. for instance, http://arxiv.org/abs/1406.4088 

Dark energy can be quantitatively described by the cosmological constant in general relativity. While this constant, also, describes vacuum energy, it is known that it is wrong to identify it with the vacuum energy of known matter-first of all, because the backreaction of spacetime to matter isn't known (i.e. the quantum description of gravity isn't known), so claiming that this is the useful comparison to make is incorrect  and, in addition, because all possible forms of matter aren't known quantitatively. How to probe dark matter isn't known, beyond its large scale gravitational effects. This doesn't mean there's some obstacle of principle, it's just a very hard problem and one must learn, what are the useful questions to ask. If dark matter just has gravitational interactions, it will be very difficult to discover its particle content, so, for the moment, the effort is on trying to check, whether it has, direct,  weak interactions  (it's known that it doesn't have direct electromagnetic interactions); it may have new kinds of interactions, which, hopefully, can be discovered in future experiments. 

In particular, regarding the cosmological constant, until it is possible to calculate transition probabilities using spacetime configurations, that are not solutions to Einstein's equations, it's known that it makes no sense to bemoan the fact that any solution of the equations of classical gravity can only be sensitive to  the ratio of the (appropriately normalized) Newton's constant to the cosmological constant and not determine either of them-a well known fact of classical equations of motion in general.  

However, instead of incoherent cries, it would be more useful to study what is known, assuming one is interested in contributing to advancing human knowledge. Science isn't a spectator sport: one either contributes, or one doesn't. Nobody owes answering questions, especially not if they're incoherent. But, at least, one should be aware of what is known, before making statements that are *known* to be wrong. And, in any event, what's important to keep in mind is that it is understood that the description of natural phenomena is approximate, that this approximation has limitations and the description that takes over from a previous one must be always worked out. However what has been already learned imposes constraints on what might be thought to have been discovered-which many people tend to forget. 

And all these issues, of course, don't have absolutely anything to do with the hydrogen atom, whose properties are known to extraordinary precision, experimentally and theoretically. 

The list is impressive-it illustrates that knowing the terms is a far cry from knowing what they mean. Some study might be more appropriate, since the claims illustrate personal, rather than general, ignorance-one shouldn't assume that if (s)he doesn't know something, nothing is known about the subject. Knowledge is impersonal-that's something many tend to forget. It's nonsense to complain that Einstein, or Planck or whoever, were wrong-that's not interesting; what's interesting is what's the correct answer, and how to be sure it is. That's the difference between science and debating. 

So complaining that to use the Coulomb potential is somehow incorrect and classical electrodynamics would be better is known to be a meaningless statement-especially since it's known how to generalize Fock's calculation  since the 1940s, to take into account the fluctuations of the electromagnetic field;  the experiments were, first,  done by Lamb and Retherford and the calculations by Feynman, Schwinger and Tomonaga; and in atom traps in many different ways since by many others in many different ways. The names are part of history; it's the work, though, that matters. And any textbook on quantum field theory presents the calculations. 

Dear Nicolis,

I'd like to comment on each paragraph of your above text.

a) I know well all terms cited in the list, otherwise I wouldn't talk about them. On feeling that some readers could not understand one or other point, I cited some references. However, more complete, additional references could be indicated.  For instance, for the (well-known persistent!) problem of the proton mass (not difference between its mass and that of neutron), read:

http://www.scientificamerican.com/article/the-mysteries-of-the-world-s-tiniest-bits-of-matter/

P.S.: In order to avoid very long texts, the complete answer will be divided in some parts.

b) With respect to the large number of distinct fundamental particles, among other additional problems I've not listed in my original comment, you should read:

https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model

c) With respect to neutrino mass in the Standard Model, at Section "Phenomena not explained" of the link given in my previous message one reads:  "Neutrino masses. According to the standard model, neutrinos are massless particles."

d) With respect to neutrinos speed, which is very hard to be determined experimentally, it is clear that both luminal and superluminal speeds were already attained, but publishing such relevant results in high impact scientific journals is another point. Most editorial boards are defensors of Einstein's Relativitty!  For instance, the paper

http://journals.aps.org/prd/abstract/10.1103/PhysRevD.92.052005

has to be modified in many parts in order to 'accomodate' the presentation to the desires of the Physical Review editorial board.

Please, Nicolis, wait a few to write, until I conclude my answers. This will put order in the discussion.

e) With respect to the very recent experiment showing instantaneous action at a distance, read:

http://www.nature.com/news/quantum-spookiness-passes-toughest-test-yet-1.18255

and the more complete paper (also published in Nature) "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres", by Hensen group:

http://www.nature.com/nature/journal/v526/n7575/full/nature15759.html

f) You said "point particles"? That is the point, i.e. sizeless particles with finite physical quantities (mass, charge, spin, etc.) do not make sense!!!

g) You said: "How quarks and their interactions provide the mass of the proton (and the neutron) is understood". Not true, this is just the topic studied by the group mentioned in  the link in my answer "a)"

h) Your comments on the dark energy and dark matter reveal a very great confidence in General Relativity. Maybe you have not noted that it is much more simple to admit that Einstein's theory is incomplete (i.e., wrong) and experiment is ok (there is no dark matter, nor dark energy).

Now, here in Brazil, is about 23:00, so I have to sleep. I'll continue the comments tomorrow.

It's not useful to try to learn technical subjects from non-technical sources. Publications that are for the general public, inevitably, omit the technical points that are of substance and, thereby, mislead. They, also,tend to make sensational claims, that appeal to feelings and are, thus, either irrelevant, or wrong. The same holds for encyclopedias. However simply providing links to technical articles is pointless, if their content hasn't been understood: one example is the statement about the article on Bell's inequalities-that implies, in fact,  the opposite of what's being claimed it does. There's no problem, with the results it presents, since, once more, non-relativistic quantum mechanics is a non-local theory-first of all, because it's non-relativistic and, second, because of the specific properties it has.  What the experiments show are these non-local effects, indeed-that are quite challenging to isolate, that's why it's interesting to understand how this can be achieved, beyond theoretical models, that are textbook material. That's all. 

General relativity may have been invented by Einstein-but it's not his property, its content is what matters. Therefore it's not useful constantly bringing his name up, in a technical discussion, since it's not relevant. So that ``dark energy'' can be identified with the cosmological constant term is an impersonal statement, whose validity can be checked by anyone, who has studied the subject. Dark matter doesn't have anything to do with general relativity, as such, since  it is a contribution to the matter content, whose effects are consistent with what can be calculated using general relativity, i.e. only the metric and the cosmological constant,  to describe the interaction of matter and geometry. What's not known is what kind of particles make it-they aren't known particles, quarks, leptons, or the Higgs. However it is known that these particles are not the only particles that can exist-they're just the particles that have been discovered to date. Any further particles will be described by a framework, that's consistent with the Standard Model,  at the energies, where it's relevant, however-that's, also, known.

So the only way to learn a technical subject is by studying it-and these days there are many real courses available, as has been linked. Making claims based on misunderstood concepts leads to either errors or nonsense-all the points made in the previous  messages show this quite clearly. So, instead of making all these complaints, why not do the work that would resolve them? Science isn't a spectator sport, so it's no use complaining that someone *else* has or hasn't done something one is interested in. It's possible to learn enough to be able to participate. Calculations are checked by the content of *other* calculations-not by the choice of words used to communicate them, but by the meaning of the words. So that general relativity is irrelevant for understanding the properties of the hydrogen atom is the result of a calculation: it suffices to compute the general relativistic corrections to its spectrum to find they're negligible. And it suffices to notice that the energy-momentum tensor of a hydrogen atom makes a negligible contribution to spacetime curvature. In fact realizing that Newton's constant has the value it does is sufficient to conclude that the corrections are irrelevant, at the energy scale of the hydrogen atom, since the corrections are proportional to powers of (E/E_Planck)~10^(-27); E~13.6 eV and E_Planck~10^(28) eV. That's all that's needed.

One should distinguish sociology from content: where a paper is published is sociology-what it says, is content. The latter is of relevance, the former much less so. And the content of the work is distinct from the ``fame'' one might acquire one way or another. The two are only loosely correlated. 

If omit some topological issues, then it can be shown, that vector potential can be split into 3 parts: scalar, longitudinal part of 3-vector and transverse part of 3-vector. First two gives longitudinal electric field, giving action on a distance. Third part gives retarded action. Choosing Coulomb potential is not reasonable step, giving infinities in solutions. From Feynman rules we can decide, that it must be some effective potential, resembling Coulomb only on some range of "r" distance.

Please consult any textbook on quantum mechanics and electromagnetism. This is the material of undergraduate physics courses, so it's astonishing that people, assuming they do have degrees in physics, can write this nonsense. If they haven't studied the subject, however, they should learn it-technical subjects aren't up to metaphysical debates, their content matters.

Dear Stam,

I have just read through your comments. I must admit that your understanding of physics is deep and your knowledge is vast. But your comments seem to be very incoherent. Sometimes it seems as though you just embrace what have been done, theoretically and experimentally. Some other times it seems as though you just reject them all. You yourselves do mention Coulomb's name so many times, so what's wrong if we want to mention Einstein's name to show respect. General relativity is Einstein's theory and it does matter when we say that because it is possible to formulate a different general relativity that is different from Einstein's theory. There are in fact many such versions in the literature. We are here not to just socialise, but to discuss ideas that are expressed in terms of mathematics, which is a language that we all use to write physical laws. Even Einstein himself once said, the introduction of the cosmological constant was the worst blunder of his. So what do you think we should do with it now? Another thing that I think you should remember, physicists have been trying to unify general relativity with quantum physics, so there must be something that is relevant to do so. The energy-momentum tensor of a hydrogen atom is used to describe the spacetime curvature of a hydrogen atom, not that of the universe. You said only the content of general relativity is what matters, and this is exactly what I have done here, unless the "content" that you want to talk about is the whole universe. I once asked a mathematician the question: Have you ever questioned yourselves why mathematics exists? His answer was quite different from mine. What is yours? At first it seems this question is too philosophical, but it is not, because it is a language that scientists use every day. 

Once more: the variations of spacetime beyond Minkowski, produced by the energy-momentum tensor of the hydrogen atom, are negligible, suppressed by powers of E/E_Planck, that's completely negligible for objects of mass smaller than that of planets; if one solves the Einstein equations with E/E_Planck on the RHS instead of zero, one can compute the variation of the spacetime metric and find it's negligible. The same holds if one tries to solve the problem of a charged particle, in a Coulomb potential, on a curved spacetime, that deviates from Minkowski by that amount. One finds results that are consistent and that are consistent with those obtained by working in flat spacetime. One, also, finds that this deviation from flat spacetime isn't amplified, which does confirm that the treatment is consistent.  That's why (usual) gravitational effects are so hard to detect and why they're negligible at collider energies-and even more so in atomic physics.

Atoms and molecules probe spacetime-they don't affect it, they're affected by it. 

So much to the question asked in this thread. Regarding the rest: one should distinguish history of physics from physics. While it is appropriate, for historical  reasons,  to call the electrostatic potential the Coulomb potential, it makes no sense to make any statement about whether Coulomb was right or wrong about something, if one isn't interested in history. Similarly, while it's appropriate to call the equations of motion of general relativity the Einstein equations, it doesn't make sense to make any statement about whether Einstein was right or wrong about something, if one isn't interested in history. And in technical discussions the history is irrelevant. What's relevant is the content of the right question and the right answer, not *who* was right or wrong. Personal issues aren't relevant. Similarly, of course, while giving names to equations is useful shorthand, it doesn't imply anything for their content. The latter matters.

While Einstein himself may have introduced the cosmological constant without understanding what its meaning was, as it turned out and then wrote about blunders, this is part of history, not of physics. It wouldn't have mattered, had he said it wasn't a blunder, either. That's just words-the technical aspects matter. What is relevant is that, since that time, technical understanding has improved. The technical statement that can be proved, in a mathematical sense of the word, is that the cosmological constant is a contribution to both the equations as to the (Einstein-Hilbert) action, that is consistent with all symmetries of the theory, therefore it's not at the discretion of anyone to include or exclude by whim. This is, now, a homework exercise in courses on general relativity. What is, also, a consequence of the mathematical treatment, that's taught in all courses in mechanics, is that the solutions of the Einstein equations don't determine the numerical value of either Newton's constant or of the cosmological constant-solutions exist for any values and no classical effects can change this. Only quantum gravitational effects, that it isn't, for the moment, known, how to describe, can select the value of the cosmological constant. Therefore any discussion that doesn't deal with them, in a controlled way, is pointless. In particular, it is known that any attempts to identify the cosmological constant with the contribution to the vacuum energy of the fields of the Standard Model only serves to show that these fields are not the whole story, since their contribution, extrapolated to the Planck scale, in order to make it relevant at all, is not under control and the backreaction of spacetime can't be computed  in a controlled way and, whatever it is, their contribution, in any event, doesn't exclude an ``intrinsic'' contribution, whose value can only be fixed by consistently treating the backreaction. Therefore even had the result of the vacuum energy contribution been comparable to the measured value of the cosmological constant, this wouldn't have implied that there wouldn't have been a problem-only its wording would have been different: what mechanism would have suppressed the ``intrinsic'' contribution; its substance would have been the same.  

The solution to any problem comes after having placed it in the appropriate context, by identifying ``simpler'' , but ``similar'', problems and solving them.  That's what the gauge/gravity correspondence has achieved, for example,  for many, though not all,  aspects pertaining to the relation between quantum effects and gravitational effects and trying to extend it goes, indeed, in the direction of obtaining a consistent description of quantum gravitational effects, though there are many issues, yet, to resolve and, undoubtedly, there are many surprising insights to discover.

Hi All,

I have developed my works and have re-written them to form a more complete and coherent theory. It has been posted on RG entitled: A TEMPORAL DYNAMICS: A GENERALISED NEWTONIAN AND WAVE MECHANICS. In this work it is shown that a temporal dynamics can be derived directly from the theory of special relativity using the general principle of relativity. You will also find that the spin of a particle in quantum mechanics can also be expressed naturally in a 3-dimensional space which is associated with a 3-dimensional temporal manifold.  We all have different views on most of formulations of physics, therefore to have them discussed thouroughly, even though to make our views more different, is always helpful. Thank you for your participation.

Kind regards,

Vu.

Dear Vu B Ho,

 There is a simple way to remove the difficulty you mention: “the electron does not radiate when it orbits the nucleus in stationary orbits”. I suppose that the interaction is the result of an exchange of mass in the form of very small grains between the proton and the electron. I take this hypothesis to satisfy the Einstein approach in its special relativity: “The law of physics must be independent from the place of observation”. See “Quantum state and periodicity” on my ResearchGate site.

``Views'' are irrelevant-especially if the technical content is meaningless. There's no meaning that can be given to the ``temporal dynamics'' claimed-and how to deduce the technical fact that the electron does not radiate, when in a bound state of the Coulomb potential, when quantum effects are taken into account, doesn't have anything to do with any exchange of mass-it's in all the textbooks and taught in all courses for decades and, as stressed, the explanation was discovered by Pauli and Fock,  While the classical system is, also, integrable, the quantum system has an additional property that renders the spherically symmetric state stable, despite the fact that, apparently, there isn't any centrifugal barrier in that case. Quantum integrability isn't enough, however. The reason is that it can't explain the spherical symmetry of the ground state. It plays a role, when relativistic effects become relevant, in ensuring that pair production doesn't occur. Relativistic effects can't stabilize a state by themselves. (It's useful to recall, to dispel some confusion, that the problem of the hydrogen atom atom can be exactly reduced, in the non-relativistic approximation-and, also, when relativistic effects become relevant, to that of a single particle, of mass equal to the reduced mass, in an effective potential. The center of mass motion completely factors out. That's where integrability is used. This breaks down, when the electromagnetic field itself can fluctuate, where the ``backreaction'' of the charged particle on the field can't be neglected and the field can no longer be treated as external.)

The existence of the spherically symmetric ground state, for this quantum system, can be deduced  in many different, though equivalent ways, of course. The new ingredient, introduced by quantum effects, is the uncertainty principle: the kinetic term tends to spread the wavefunction, in position space, since the wavefunction, in position space, isn't an eigenfunction of the momentum operator; the potential term tends to concentrate it, since it is an eigenfunction of the potential energy; and the existence of a ground state means that the two effects can be shown to cancel-this is what happens here. In an energy eigenstate, in addition, the particle doesn't have either a well defined position, or a well defined momentum. This is a standard exercise in quantum mechanics. In classical mechanics this doesn't happen, because momentum and position can have both delta-function distributions in classical mechanics, not, though, in quantum mechanics. 

There do exist potentials, that, classically, do have a ground state, that's rendered unstable by quantum fluctuations, of course, this way and these are standard homework exercises. 

Dear coleagues,

As you have seen in the above comments by Stam Nicolis, physics for him is an ended science!  All we have to do is to understand some standard textbook exercises (which most of us have teached to our students by years). He do believe that modern physics has no problems at all, is a perfect building!

My approach to physics is completely different!  I believe that theoretical physics still present some (open) problems, which are impossible to be solved with standard model (Einstein's relativity + quantum mechanics) simply because these theories are incorrect, i.e. their postulates are deficient. This is why we actually cannot solve the ten (among others) paradoxes I've listed above.

It seems that Stam is so strongly tied to the currently accepted paradigms that he cannot even imagine that the things could be done in another way. In the words of my young students, he is immersed in the matrix (as in the famous movie).

Note, on doing physics in the manner proposed by Stam, clearly Einstein (certainly his great idol) never would have proposed a radical theory such as Relativity.  Einstein's relativity theory (I fully agree to V. B. Ho, we must cite the name of the author of a theory for many reasons) was entirely OUT of the physics mainstream in that time.

So, Stam's approach to physics is good for those scientists who do not want to develop any great, revolutionary work, just traditional works (those which do not put in disguease the current paradigms).

Well, each person can do his own choice: traditional physics (more of the same) or revolutionary, new physics.

Prof. Fabio M. S. Lima

Dear Xavier,

I have read your article. I agree with you that the interaction is the result of an exchange of mass in the form of very small grains between the proton and the electron as you said. Actually I worked it out but then I removed it from my article. My reasons and calculations are as follows. According to my model, the interaction between the electron and the proton of a hydrogen-like atom is repulsive and the Yukawa term of the potential is dominant for the distance less than Rn. Therefore in this case it is possible to use Yukawa potential with a spectrum of massive photons that act as force carriers. Using the formula for Rn=h/(2*pi*mpc) we can work out mp=2*pi*mkq2/(chn2). Then using the uncertainty relation, the lifetime of these massive photons is found as 1.77×10-19n2 s.

Kind regards,

Vu.

Dear Stam and Fabio,

Stam, it is good to keep hold of our beliefs, but I think scientific beliefs need to be more open and flexible than religious beliefs. I agree with Fabio that your belief in the mainstream physics is unchangeable that may lead to an undesirable effect, such as, nothing else has a meaning but yours. Your description of the whole thing is great, but the most important question of all is still missing: Is quantum mechanics comprehensible in its current formulation and interpretation? Even Einstein could not make any sense out of it.

Kind regards,

Vu.

Beliefs are irrelevant-and there's no such notion as ``mainstream physics''-there's just physics. Physics isn't a matter of opinion, but of calculation. So the statement that quantum mechanics,  in its present formulations (there are many, but they're equivalent, so which one is used is a matter of personal taste),  is comprehensible is, of course a true statement-and has been known to be true for decades. There's no such notion as an ``interpretation'' of quantum mechanics-or of any physical theory. That's just words-that may have been used, but they're empty of content, since no calculation can give them any meaning. The content is in the calculation. Whatever formulation is used, all the calculations give the same answer, and that's what matters. That's all there's to it. 

So the statement that the Standard Model makes incorrect predictions can be tested-and found to be false. Similarly for general relativity. It is a correct description of known gravitational effects. While physics isn't an ``ended science'', the topics discussed on this thread are: they're homework exercises, that must be done-but their answer is known. If they're not understood, of course, then any hope of developing the tools for describing quantum gravitational effects is futile. 

Dear Stam,

Even something that has been known to be true for thousand years can still be proved to be wrong, let alone decades. I agree with you that whether quantum mechanics is comprehensible or not depends very much on personal viewpoint of the subject. According to you, in order for a word to have any meaning there must be some kind of calculations to support it? So next time when you put down a comment, please do not use words, but write them down by calculations then.

Kind regards,

Vu.

Dear Fabio and Vu;

Many thanks for your comments.

The introduction of the exchange of mass between the nucleuses and the electrons of the atoms allows to understand the periodic table. Indeed whatever the interactions with the neighbors’ atoms the trajectories of the electrons will be easily modified in such a way to keep the action of their quantum state. As a result the different electrons of an atoms can keep their quantum states as established with the calculation of the magnetic moments of several compounds. See: “Total angular momentum and atomic magnetic moments” J. Magn. Magn. Mater., with Lochack on my ResearchGate site.

Kind Regards

Xavier

Well the calculations that are relevant for the hydrogen atom, particularly in relation to Fock's work,  are reviewed here: http://math.umn.edu/~karl0163/docs/fock.pdf to quote just one example. This is the basis of understanding the solution of the non-relativistic particle in a Coulomb potential. It's  a misuse of such fora to attempt to use them to solve homework problems, that are set in all the textbooks on the subject. And how to extend the formulation to take into account relativistic corrections for the particle and, further, to take into account fluctuations of the electromagnetic field is reviewed in, for instance, in the introductory chapter to S. Weinberg's textbook ``Quantum Theory of Fields'' vol 1. 

Whether quantum mechanics is ``comprehensible'' depends on the level of learning-someone that has studied the equivalent material of what's taught in undergraduate physics courses  is expected to understand it, someone that hasn't can be expected not to; but its technical results are unambiguous. In particular, it's no longer considered newsworthy to compute the energy levels of the hydrogen atom; or the structure of the periodic table, that was worked out by H. G. Moseley in 1913, as reviewed here: http://www.chem.ox.ac.uk/news/H-Moseley-by-Russ-Egdell.pdf

and, of course, further insights were gained by the work of Pauli. That's why there's no point in rehashing such results, unless the new presentation, while equivalent with the old one, can offer any new insights in another way. No such new insights are offered, however. Special relativistic effects are known and general relativistic effects are known to be negligible-so much that they are used as calibration for the search of *new* effects: http://www.npl.washington.edu/eotwash/sites/www.npl.washington.edu.eotwash/files/webfiles/publications/pdfs/prl86-1418.pdf for instance. 

Good Job, Stam ! For a change this reference is good. However, the guy is saying this :  

"Fock calculation explains SO(4) symmetry.........Why this should be the case, remains mysterious...."

Well, not for me ....This is what I am thinking about publishing soon... But, anyway, given some time, you had struck exactly the right nerve of this problem :-) . The amount of details needed for giving the precise answer is overwhelmingly huge  though. Be ware of this...

Well the ``mystery'' is elucidated in Bander-Itzykson, among other places, surely, namely the role played by the conformal symmetry of the 1/r potential; in addition to the fact that the free equation is the Laplace equation on S^3, but in momentum space. Bander and Itzykson, indeed, investigate the case in arbitrary dimensions-and, also, the scattering states, when the atom is ionized.  So there's a lot that's known. 

However the property that makes the whole construction relevant is describd by Planck's constant-that the equation one started with, is the Schrödinger equation, not an equation for classical quantities. Indeed, if one does try to compute the canonical partition function of the classical particle in a Coulomb potential, one will find, as ought to be expected,  a divergent answer, consistent with the fact that the classical potential is unbounded from below and that classical, thermal, fluctuations don't affect this property. One needs a new kind of fluctuations, quantum fluctuations. 

Stam, Bander and Itzykson paper is 1000 times more complicated than solution which Fock had obtained originally. Fock paper does contain a 4 dimensional Laplacian in it. However, and this is a puzzle for the guy who delivered a power point presentation , Fock is not explaining clearly why there is a need to go to 4 dimensions. Bander and Itzykson also do not explain the real rationale. Even such comprehensive book as that by Bruno Cordani(which is 1000 times better that Bander and Itzykson) is wague on this subject  http://www.amazon.com/Kepler-Problem-Regularization-Quantization-Perturbations/dp/303489421X/ref=sr_1_1?ie=UTF8&qid=1459978871&sr=8-1&keywords=cordani+bruno   It is  helpful nevertheless

Dear Stam,

Stam, what level of learning is required to comprehend Quantum Mechanics? As far as I know Einstein could not understand it, Feynman said no one could understand it and Schrodinger himself said he would give up his own wave theory if people keep talking about quantum jump. Probably, Fock is the only one who could.

Kind regards,

Vu.

Vu, remember, Einstein used to say: "It is essential to make things as simple as possible, but not too simple". Hydrogen atom is just one of those things. It is like a DNA molecule for both quantum mechanics and relativity.  It captures both of these disciplines contrary to common belief than qm and relativity are not compatible.They are compatible . They cannot exist without each other! In both  we observe point-like particles  moving along geodesics....From here comes an observation that electrons DO NOT radiate! This is not what you will read in standard textbooks .However, just read papers by Jurgen Moser which I mentioned already. Unfortunately the guy is dead. Clearest mind of out times.....

Quantum mechanics is taught, typically, at third year of an undergraduate study cycle of physics and more advanced concepts are discussed, typically, the two years that follow, so in graduate school. But the solution of the Schrödinger equation for potentials such as the Coulomb potential and the harmonic oscillator is taught in third year. Typical prerequisites are classical mechanics and classical electrodynamics and the mathematical techniques that go with them-though the path integral formulation relies, in fact, quite heavily, on a mastery of thermodynamics and statistical mechanics.

Einstein invented much of it, as did Schrödinger, of course,  and Feynman invented yet another way. While it may be fascinating to read the evolution of their insights, they, of course, lived at a particular time. Understanding such concepts does take time and how to teach them, also. While vol III of the Feynman lectures is wonderful, it's definitely much more useful if one is studying it in parallel with  a textbook with exercises, like Landau and Lifshitz, or Messiah, for instance. One shouldn't take seriously what scientists write beyond their technical papers. What Feynman wanted to stress is how far from ordinary experience quantum effects are-so, in that sense, indeed, they're beyond comprehension in terms of ordinary experience. But they are amenable to a consistent mathematical description and a well-defined experimental protocol. A major point is that while Einstein, Schrödinger, Feynman and all the others  may have had their problems with the material, since they had to find the solution for themselves, these problems have been since resolved-regarding quantum mechanics. And how they have been resolved is known-among other places from vol III of Feynman's lectures. 

As Feynman stresses in this discussion, https://www.youtube.com/watch?v=3D2RaDVkylY when one asks ``why?'' one must define carefully what's assumed known and true and what's not. While the SO(4) symmetry does imply that the 3-sphere is of relevance, a non-trivial issue is what stops this sphere from having an arbitrarily large radius, but keeps it finite. The answer to that question is that the stereographic projection of the unit 3-sphere is sufficient, if one wants to describe bound states.

So the explanation is in the other direction: From the property that the Schrödinger equation for the Coulomb potential is the Laplace equation on the unit 3-sphere-in momentum space- it follows that the ground state exists.  If it were the Laplace equation on a non-compact manifold, then this wouldn't be the case-and that's relevant for scattering problems.

Wonderful clip of Feynman !!!! Thank you for this link!

Dear Arkady and Stam,

I think it is a never-ending story when we talk about quantum mechanics and relativity. To my opinion, quantisation is not a true and proper method to be used if we want to unify quantum mechanics with relativity. The foundations of quantum mechanics are mathematical foundations, not physical foundations. People have been trying to fit physics into mathematics, and nowadays we have more mathematics in physics than physics itself. Mathematics is a Nature's language, and like all other languages, not all combinations of the alphabet will make sensical words. On the other hand, like Newtonian physics, relativity theory uses mathematics as a tool to describe the physical world. And it uses only words that make sense. 

Kind regards,

Vu.

Vu, all this is nice, however, neither physics nor mathematics can help if the problem is not formulated. What is the problem? To connect QM with general relativity?! Or what else?. Schrodinger himself and, subsequently, Dirac and many-many others tried to do just this.However, without some experimental evidence nothing much can be achieved.Thus, please, describe to all of us at least one experiment which requires new theoretical explanation.Remember though that Bohr's complementarity principle is always correct.What is known already is not all together wrong and what is new must merge with what is working already well. Alternatively, one should propose something else which is going to produce exactly the same results in the same domain of validity and  allows extension impossible if other formalism is used. History of science teaches us that such principle was always used . Unlike mathematics, in physics there are sometimes many ways to get from A to B. In mathematics too, by the way... For consumer of physics results, all this is irrelevant. They just want to have something which they know is working, e.g. to use the internet or iPhone one should not learn about computer science or physics. The biggest unexplored domain of physics  is biology. And, I am afraid, it will stay this way for a long-long time. Schrodinger wrote a book "What is life" and Feynman was doing some biological experiments being influenced by Watson with whom he was a close friend while working at CALTECH. Neither Schrodinger's nor Feynman's biology related results nobody knows nowadays, expect science historians...All living creatures are quantum computers. Try to make use out of this fact and to build your own quantum computer based either on current understanding of QM or on that which you would like to design. 

If the problem is how to describe consistently quantum and gravitational effects, then the hydrogen atom  isn't relevant. Additionally, it is known that classical effects are limiting cases of quantum effects, not the other way around. The relevant systems are black holes and the pre-inflationary phase of the Universe. For the former, the problem is that it is not possible to experimentally detect Hawking radiation (hence the activity in ``analog'' systems that attempt to capture some properties, but can, detect, by experiment, the avatar of Hawking radiation, though there are significant subtleties); on the other hand, there is some hope for experimentally measuring the parameters that determine the inflaton potential. Work is ongoing on both issues, with the first more a matter of theoretical consistency.  However neither has anything to do with the consistent description of the non-relatvistic Coulomb potential. Similarly, while work on quantum computing is relevant for describing the dynamics of black hole microstates, for actually building the ion traps and controlling the quibit operations gravitational effects are irrelevant-spacetime isn't affected by the ion traps; as mentioned, they are rather used to probe for departures from Newtonian gravity, that might be relevant at those scales.

Stam,this reference is for you   http://arxiv.org/pdf/1206.3166.pdf in connection with your statement 

.. while work on quantum computing is relevant for describing the dynamics of black hole microstates,....

The reference I am providing is demonstrating just the opposite: black hole description is useful for development of quantum computing

It's known what qubits are, in terms of atomic and molecular degrees of freedom-it's not known what the black hole microstates are-just that their dynamics can, under certain circumstances-for extremal black holes-be described by unitary evolution operators, that are, also, relevant for describing the dynamics of real qubits. The black hole description is deduced, it's not known. One reason is that it's not known what are the microscopic degrees of freedom, whose classical limit is described by a black hole spacetime, but in certain cases, for extremal black holes, with ``large'' charges-and, even then, the description is not, yet, complete. One has easier access to ion traps than to black holes, after all.

Dear Stam, for me the nicest thing of this paper is the fact that these guys found uses for Fano projective plane. https://en.wikipedia.org/wiki/Fano_plane...This is very neat...

Something reminds me about the Gell Mann Octet...

Dear Arkady and Stam,

Physics is about exploring Nature, mathematically, to find out what the unknowns are, not about accepting what have been assumed, in terms of the so-called principles. If we just accept what we have learnt then we will not be able to realise that there are many problems in physics that cannot be explained within the present formulation of quantum physics. Arkady, can you clearly explain to me what a quantum of energy is? You also mentioned about simple but not too simple! Basically, quantum mechanics is a wave version of Newtonian physics. Isn't that simple? If you read my work then you will see that I have generalised both of them and derived few equations that I could not solve because they involve Fractional Laplacians. Is that simpler? Furthermore, one of these equations will tell you what a quantum of energy is.  I have tried to seek help but so far no results. If you can help me solve these equations then I will give you a personal reward (you buy anything you want to the maximum of AUS $1,000 and I will pay for it).

Dear Stefano,

Did you mean a mixed potential?

KInd regards,

Vu.

Dear Vu,please take a look at my paper http://adsabs.harvard.edu/abs/2006hep.th....8117K

in it it is clearly stated  what the quantum of energy is and why Heisenberg's version of QM is superior to that of Schrodinger. I've noticed that you tend to stay away from mathematics.Nevertheless, give yourself a trouble to read and to understand fully Heisenberg;s original paper (whose ideas he had stolen fro Kramers, incidentally) done in 1925. In his lectures on quantum mechanics Dirac emphasized the superiority of Heisenberg way of making quantum mechanics over that by Schrodinger. Also, in the book by Hawking (I believe, on the shoulders of giants) there is full set of lecture notes by Max Born on QM done in 1926. These notes were reproduced and published by MIT press. Please, take a look at these notes BEFORE making further complaints... :-)

Heisenberg's formulation of quantum mechanics  is mathematically equivalent to Schrödinger's and both are mathematically equivalent to Dirac's; and they're all equivalent to Feynman's, too, so which one is chosen is a matter of personal taste. They all give the same results and there's no way to distinguish one from the others, in an impersonal way. 

Quantum mechanics isn't a wave version of Newtonian physics-that's wrong. Planck's constant is something new and can't be deduced from classical effects; it's the other way around.  While fractional differential operators can describe certain kinds of fluctuations, it isn't at all obvious that quantum fluctuations can be consistently described by fractional differential operators; this must be proved separately. There was quite some activity, many years ago,  on fractional differential equations for describing classical disordered systems, that, apparently, didn't prove as promising as initially hoped. However, they don't seem to provide any useful tools for describing quantum gravitational effects, or for stabilizing scales (the so-called hierarchy problem).  One should be quite specific what the problems are, that can't be resolved with what's currently known in quantum physics. 

Stan, all these well known facts do not help though. Just look how Planck's constant enters QM for the first time by reading appendices to my Honeycomb paper. This is exactly the place where Heisenberg scooped poor Kramers. Introduction of h in QM is related to derivation of optical sum rules(Tomas-Raiche-Kuhn sum rules)  Bohr assigned  to young Heisenberg more mature Kramers with whom Heisenberg made 2 papers involving QM BEFORE  new QM was officially proclaimed, at the end of 1926-1st half of 1926. They explained these rules quantum mechanically . https://en.wikipedia.org/wiki/Kramers%E2%80%93Heisenberg_formula

Then in his inagural (helgoland) paper Heisenberg used these results  to insert h into his version of QM. Read Dirac's lectures on QM about equivalence of Schrodinger's and Heisenberg's pictures of QM. And, guess what? Heisenberg sent to Cambridge, to young Dirac,  his 1925 manuscript so that Dirac would also have a piece of pie. For this gesture Dirac for the rest of his life was thankful to Heisenberg...He owes his Nobel Prize to him too...

Dear Arkady,

Thank you very much for the link to your paper. I like much this subject, the beginning of Quantum Mechanics, as well as the work of Kramers (he was not a Nobel win because, unfortunately, he died one year before).

I cannot agree to Stam when he says that "both [Heisenberg and Scd] are mathematically equivalent to Dirac's; and they're all equivalent to Feynman's".  The former (Heis and Scd) are non-relativistic, whereas the latter are relativistic. Therefore they are not equivalent, they produce distinct results, even for the eigen-energies of the H atom!  And here we return to the main point of all this discussion, i.e. the question posed by V. B. Ho on the use of the Coulomb potential in the study of the H atom.

Also, when Stam says that "Planck's constant is something new and can't be deduced from classical effects" he, in fact, shows us (again) he completely agree to Modern Physics texbooks, but note that the fact that none has found a manner to derive h constant from classical physics does not mean that this is impossible. In fact, I am working in this problem (though my group is using an electrodynamics distinct from the Maxwell-Lorentz one, namely Weber electrodynamics).

Fabio M. S. Lima

Fabio, 1st when it comes to Kramers, he had a nervous breakdown after he was scooped. He was put into mental institution for a while. After that, to repair situation Bohr made him full professor at Leiden U. where Kramers remained till the rest of his life.

2nd to my knowledge, Kramers never was even nominated for the Nobel Prize. 

3.The emergence of Coulombic potential in hydrogen atom is completely natural . For this it is sufficient  to read Guillemin and Sternberg book http://www.amazon.com/Variations-Colloquium-Publications-American-Mathematical/dp/082184184X

which, incidentally, I've mentioned already several times in these discussions

4.Weber electrodynamics   https://en.wikipedia.org/wiki/Weber_electrodynamics is not in vogue nowadays. However, I strongly recommend the book by Deriglazov  http://www.amazon.com/Classical-Mechanics-Hamiltonian-Lagrangian-Formalism-ebook/dp/B00F76F91C/ref=sr_1_1?s=digital-text&ie=UTF8&qid=1460755254&sr=1-1&keywords=deriglazov    It is full of relevant ideas and also, I strongly recommend to download some f his relevant (to this book) papers. DO NOT DISREGARD MY ADVISE!

Dear Arkady, Stam and Fabio,

Arkady and Stam, please don't rule out the important role that Fractional Laplacians may play in future development of quantum physics. You should remember that Heisenberg did not know anything about matrix representation of quantum mechanics when he wrote his paper. 

Fabio, it's good to hear that you are working on the Planck's constant. Myself, I think that Planck's constant is a composite constant in the sense that it is a product of other constants at quantum level. For example, in the inverse square law we can write V=k/r wiht k=Gq1q2 and Planck's constant may be similar to k. 

Kind regards,

Vu.

Vu, again, 1st thing is is to find a good new problem and only 2nd is to find which of big guns, like fractional Laplacians, etc. to choose. Applications of big guns to things which already work, that is produing reliable numbers which practitioners can use, is not the best of way to proceed to say the least. Thus, please, spell out the new problem which you are willing to attack with fractional Laplacians.

Dear Arkady,

It seems you have not even looked at my work, let alone read it. It's all in my work entitled A TEMPORAL DYNAMICS: A GENERALISED NEWTONIAN AND WAVE MECHANICS posted on RG. I have consulted most of articles and books that I know on Fractional Laplacians but none of them could help. I even tried to contact those experts on the subject. What I would like to obtain are solutions similar to those from Schrodinger's wave equations. By the way, I have read your work. It is a very nice paper, but as I once told you I don't particularly embrace a new mathematical method that can be used to beautifully represent an old physical idea.

Kind regards,

Vu. 

Dear Vu, I've looked at your work.You are unhappy with the fact that electron emits radiation.But you also have not looked at the references i gave you alsready. According to work by Jurgen Moser  (already mentioned) electron is actually moving on geodesics on 3-sphere. Here again the reference http://www.amazon.com/Notes-Dynamical-Systems-Courant-Lecture/dp/0821835777/ref=sr_1_1?ie=UTF8&qid=1460772363&sr=8-1&keywords=moser+notes+on+dynamical+systems I believe Jurgen Moser because i can understand what he was doing and I do not see any faults. I suggest you also to penetrate yourself with these beautiful results before you jump into fractional Laplacians. Jurgen Moser is one of the most talented people, if not the most talented, who was working in mechanics  https://en.wikipedia.org/wiki/J%C3%BCrgen_Moser

Dear Arkady,

I've just had a quick look at Moser's work and I agree with you that he is one of the best. However, to be honest, although I admire mathematicians, I would not hire any of them to do a physics job. I can arrange the English alphabet into a spectacular pattern that follows some particular rules, but none of them would make any sense. Arkady, what do you think of the electron of a hydrogen-like atom, is it a particle, or a wave, or a particle floating along a wave? According to wave mechanics, it must be related to a wave, otherwise it would not be called a wave mechanics! And they all have been confirmed by experiments, as you said. I myself think that when the electron is in a bound state it behaves like a wave, but when it is in a free state it behaves like a particle. How it changes its states is a mysterious problem that I am trying to work it out. It must be related to its environment. You should also remember that Schrodinger's wave mechanics is actually about the dynamics of the phase of a wave, but not the dynamics of a particular physical wave, like that of electromagnetic wave. From these analyses, I can say that the description of the electron as a solid particle moving on geodesics on 3-sphere can not be a correct physical description, even though it is perfectly mathematical. But as I said many times, we all have our own beliefs, and the only good thing about it is we can still talk peacefully about them. 

Kind regards,

Vu.

The statement that the electron, when bound by a Coulomb potential, moves on the geodesics of the unit 3-sphere, is incorrect. What Fock found was that the momentum space  wavefunction of such an electron satisfies the Laplace equation on the 3-sphere, of finite radius, in momentum space. From the wavefunctions one then reconstructs the corresponding probability density for momentum or position, by their moments, since one can show that these do satisfy the appropriate conditions. In particular that the domain of the momentum space wavefunction is a compact manifold implies that the ground state exists. And this is a property of the quantum system, since the relevant quantities are the wavefunctions, that aren't classical quantities.

Stam, I recommend you to learn something about twistors and then to think about what kind of result Fock had obtained and then (only then) claim that electron moving on 3 sphere statement is incorrect. 

Dear Vu, I am still suggesting you to read Moser. It never hurts to know more.You always can make it less  again. I understand that both you and Stam are staying away from mathematics. Fine with me. However, make sure that you are doing your work not just for yourself but for other people and these  people may not share your beliefs. 

It's simply wrong  to claim that a quantum particle, in a bound state of the  1/r potential in three space dimensions moves on the 3-sphere and no amount of obfuscating technical terms can make that statement mean anything correct. The correct mathematical statement is that the Fourier transform of the wavefunction is the solution of the Laplace equation on the 3-sphere. That's all the mathematics that's needed to compute any physical quantity. Mathematical tools that do not lead to equivalent results are irrelevant; and those that lead to the same results are equivalent and it's a matter of taste and nothing more, if equivalent mathematical formulations are used. One doesn't get bonus points depending on the method used to solve a problem.

Stam, recall that you were also criticizing my uses of contact geometry.Well, as I've informed you already, you are having plenty of chances now to criticize my Annals of Physics paper (available online already) openly. You will soon have your chances to criticize my new paper on Kepler problem (in fact series of papers) if and when it will published in Annals too. For now, I believe, each of us know enough about other so that there is no need to continue this exchange process.

Best of luck to you