Exploring the Possibilities of a New Relativistic Quantum Theory

[meopemuk]

You'll discover that this sort of hair splitting causes real problems once you try to correctly handle soft photons in your approach. (I tried to find your blog from a few years ago where you had discussed your failure to handle the IR problem, but is seems to have disappeared.)

Try http://meopemuk2.blogspot.com/ This blog was inactive for a few years, but still alive. You are welcome to post your thoughts there.

Eugene.

 

[meopemuk]

Once one allows enough coherent states into the picture (which naturally arise in the field formulation but are orthogonal to all Fock states), the challenge disappears (on the level of rigor of theoretical physics - on the fully rigorous level, additional hurdles arise that are not yet fully overcome).

I am not quite sure that coherent states cannot live in the Fock space. This is sort of like saying that infinity does not belong to the real axis.

But bare particles are no particles at all. They become that only if one tries to give the bare fields a particle interpretation - which they cannot have, since all these particles would be infinitely heavy after renormalization. Nobody except popularizers of physics thinks of bare particles as particles.

I am glad that we fully agree on this point.

Eugene.

 

[A. Neumaier]

Let me be clear about it. I believe that Maxwell equations are fundamentally wrong. Yes, their solution can sometimes yield accurate results. But history of science knows many examples when incorrect theories were used successfully for centuries. I agree with your statement that Maxwell equations cannot be derived in my approach, but this doesn't worry me a bit.

Even in case they should be fundamentally wrong, any theory of electrodynamics must explain why they are empirically successful: Not only sometimes but always - except in idealistic situations involving particles that are so small that it is obvious that a classical treatment is no longer adequate. Indeed, so successful that our modern technology would be impossible without it. QED does this without any significant trouble; should yours be an alternative, it would have to do it as well.

Instead of Maxwell equations, I suggest to use the Darwin-Breit Hamiltonian derived in subsection 12.1.1

How will you convince engineers that they should use your formalism in place of what they are used to and can apply without any troubles?

If Maxwell equation cannot describe the dynamics of two isolated electrons (I don't care whether you call them point particles or not), then how you can be sure that dynamics of billions of electrons in wires is described correctly?

The former is clearly a matter of quantum physics, not of classical equations, while the latter is an extremely well established empirical fact. And I have given you references to papers that derive this stuff from standard QED.

In my approach, the anomalous magnetic moment would appear as a 4th perturbation order correction to the Darwin-Breit potential. I have confessed already that the dressed particle transformation has been performed explicitly only in the 2nd and 3rd orders.

The 4th order calculation is complicated by infrared divergences in loop integrals. For example, the relevant vertex renormalization integral in 9.2.6 contains expression $$\log(\lambda)$$, where $$\lambda \to 0$$. The resolution of this divergence is well understood in standard QED. It is related to consideration of soft photons. I hope that a similar solution can be found in the dressed particle approach as well. But I have not done that, and I am not ready to discuss it here.

You don't need to discuss it; but you'll probably get these problems in every computation involving radiative corrections - i.e., in all problems of QED that make the difference between the state of the art in 1930 and in 1948, and that account for the excellent reputation that QED enjoys. Nobody would want to exchange this for a theory where radiative corrections already constitute such a serious difficulty.

I can assure you that an understanding of the work of Faddeev and Kulish would tell you that these problems are intimately related to the fact that your electrons are not the right asymptotic objects, and that you need something more general than a Fock space to start with. Your dressing transformation enforces that your single electron states are also asymptotic states. But they lack the soft photon clouds with which the asymptotic electron states of QED are equipped.

I think, we simply used different terminologies. No disagreement on substance. I promise not to call electron and photon point particles anymore. I will call them "elementary particles".

Good. It is always better to conform to accurate terminology - this makes misunderstandings less likely.

 

[A. Neumaier]

As I suspected, these are just theoretical speculations. The symmetry breaking and instantons/solitons have not been observed directly in experiment. Is it true?

Oh yes. Nothing in the standard model beyond QED has been observed directly in experiment. It is all but theoretical speculations. It is highly deplorable that the Nobel price committee gave already a number of their prizes to this sort of speculative physics.

Instead they should give the prize to authors of serious work such as http://lanl.arxiv.org/pdf/physics/0504062 , which promises in its abstract:

''The developed theory is applied to realistic physical objects and processes including the hydrogen atom, the decay law of moving unstable particles, the dynamics of interacting charges, relativistic and quantum gravitational effects. These results force us to take a fresh look at some core issues of modern particle theories, in particular, the Minkowski space-time unification, the role of quantum fields and renormalization, and the alleged impossibility of action-at-a-distance. A new perspective on these issues is suggested. It can help to solve the biggest problem of modern theoretical physics – a consistent unification of relativity and quantum mechanics.''

(In the light of the many practical applications given there, they should excuse the minor difficulty that some of the old speculative computations of the Lamb shift of hydrogen and and the magnetic moment of the electron which agree only to 12 decimal places with experiment could not yet be reproduced.)

 

[A. Neumaier]

I think this is just a terminological issue. It depends on how we define "elementary particle". I decided to identify elementary particles with irreducible representations of the Poincare group. Since 1-photon space is not irreducible, then, according to my formal definition, photon is not elementary. Another point is that we cannot treat separately right-handed and left-handed photons, because one can always make a linear combination of them, which will defy such classification.

One can also make a linear combination of electron and positron states, or of proton and neutron states. This routinely happens in the applications.

Whereas superpositions of electron and photon states are forbidden by a so-called superselection rule.

 

[A. Neumaier]

Try http://meopemuk2.blogspot.com/ This blog was inactive for a few years, but still alive. You are welcome to post your thoughts there.

No. I'll discuss my thoughts on your work here in this thread. I just wanted to link to some of your entries where you write about the difficulties to handle the IR. But since you talk about these difficulties here, such a link is no longer needed.

 

[A. Neumaier]

I am not quite sure that coherent states cannot live in the Fock space. This is sort of like saying that infinity does not belong to the real axis.

Yes, and it doesn't. I haven't seen anyone place infinity somewhere on the real line.

The real line with infinity adjoined is a compact space, whereas the usual real line (without infinity) is noncompact. Also the algebraic properties become quite different. (Of course, if you add many infinities and infinitesimals, you can restore the field property, but then you lose instead completeness, which makes things even worse!)

In a similar way, adding more coherent states to Fock space (some of them live there already, but by far not enough to handle the IR problems!) radically change the properties of the latter.

Thus your analogy is excellent in this respect!

 

[meopemuk]

Even in case they should be fundamentally wrong, any theory of electrodynamics must explain why they are empirically successful: Not only sometimes but always - except in idealistic situations involving particles that are so small that it is obvious that a classical treatment is no longer adequate. Indeed, so successful that our modern technology would be impossible without it. QED does this without any significant trouble; should yours be an alternative, it would have to do it as well.

Let me repeat my logic once again. I am concerned about being consistent with experiment. I know that Darwin Hamiltonian explains experiments pretty well (we leave alone radiation at the moment). I've found that my approach reduces to the Darwin Hamiltonian in some approximation. So, I am happy.

Maxwell equations also reduce to the Darwin Hamiltonian in some approximation, and, therefore, also describe experiments pretty well. But this does not mean that it is necessary to prove the equivalence of my approach and Maxwell equations. I know that there are examples involving few interacting particles, where Maxwell's theory yields bad results. These examples include the infinite energy of the electromagnetic field associated with elementary charged particle, violation of the 3rd Newton's law in magnetic interactions, paradoxes of Kislev-Vaidman, Trouton-Noble, "hidden momentum", etc. So, I would prefer to stay away from Maxwell's theory.

How will you convince engineers that they should use your formalism in place of what they are used to and can apply without any troubles?

As I said earlier, in cases relevant to engineers, Maxwell equations and Darwin Hamiltonian yield the same results. So, I have no problem if engineers continue to use Maxwell equations in their calculations.

You don't need to discuss it; but you'll probably get these problems in every computation involving radiative corrections - i.e., in all problems of QED that make the difference between the state of the art in 1930 and in 1948, and that account for the excellent reputation that QED enjoys. Nobody would want to exchange this for a theory where radiative corrections already constitute such a serious difficulty.

Infrared divergences are present in almost all loop integrals of QED. Getting rid of them is not a trivial matter. Yes, you can blame me for not solving this problem within my theory. I am working on it, and believe that it can be done. I am not asking (yet) to replace QED with my approach. I am just saying that there is a non-traditional way to look at QFT. I agree with you that this way has not reached a level of fully developed theory yet. But so far you haven't convinced me that this way is wrong or inconsistent or disagrees with experiment.

I can assure you that an understanding of the work of Faddeev and Kulish would tell you that these problems are intimately related to the fact that your electrons are not the right asymptotic objects, and that you need something more general than a Fock space to start with. Your dressing transformation enforces that your single electron states are also asymptotic states. But they lack the soft photon clouds with which the asymptotic electron states of QED are equipped.

Yes, I know Faddeev-Kulish work, but I have a somewhat different understanding of it. Their point is that due to long-range Coulomb interaction, charged particles never move asymptotically. This means that it is wrong to describe their dynamics at large distances by the free Hamiltonian H_0. The assumption of the $$\exp(iH_0t)$$ asymptotic dynamics is the cornerstone of the S-matrix definition. This means that for charged particles the usual S-matrix cannot be formally defined. The S-matrix plays an important role in my dressed particle approach. I am fitting the dressed particle Hamiltonian to the known form of the S-matrix. If there is no S-matrix, there is no dressed particle Hamiltonian, and my approach does not work.

Kulish and Faddeev suggested a new definition of the S-matrix, which can be applied to charged particles. In this definition, the asymptotic dynamics is described not by H_0, but by H_0 + W, where W is a portion of the full interaction Hamiltonian V, which survives at large distances. The claim is that this new S-matrix can be properly defined. However, there are few issues, which I haven't understood yet.

1. The separation of the asymptotic portion W from the full interaction V cannot be made in a unique fashion. So, I am wondering if this freedom has any effect on the uniqueness of final results.

2. Replacing the asymptotic Hamiltonian H_0 by the new H_0 + W would change completely the perturbation theory. The rules for calculating Feynman diagrams should be different. Kulish-Faddeev didn't spend much time discussing that, and I didn't have enough diligence to repeat all calculations myself.

I read a lot of more recent papers regarding infrared divergences, but I am still not satisfied with my understanding of the issue. I would appreciate if you can recommend a book or review, where all this is explained for pedestrians such as myself.

Eugene.

 

[meopemuk]

Oh yes. Nothing in the standard model beyond QED has been observed directly in experiment. It is all but theoretical speculations. It is highly deplorable that the Nobel price committee gave already a number of their prizes to this sort of speculative physics.

I've noticed a hint of sarcasm. Or I was mistaken?

We were talking about instantons, solitons, and spontaneous symmetry breaking. I am not an expert in this stuff, but I guess it would be fair to say that these phenomena have not been observed in experiments. They are parts of some theoretical formalisms. The formalisms themselves can be rather successful in explaining/predicting experimental data, but this is not a guarantee that everything inside them have observable counterparts.

Perhaps we should return to this discussion after the Higgs boson is completely rejected by LHC experiments.

Eugene.

 

[meopemuk]

One can also make a linear combination of electron and positron states, or of proton and neutron states. This routinely happens in the applications.

Whereas superpositions of electron and photon states are forbidden by a so-called superselection rule.

I thought that superselection rules forbid forming linear combination of states with different electric charges. So, I wouldn't try a half-electron-half-positron state.

Eugene.

 

[meopemuk]

The real line with infinity adjoined is a compact space, whereas the usual real line (without infinity) is noncompact. Also the algebraic properties become quite different. (Of course, if you add many infinities and infinitesimals, you can restore the field property, but then you lose instead completeness, which makes things even worse!)

I heard that these kinds of issues are solved within non-standard analysis. I've also heard some people say that this is the only proper way to teach analysis and math in general. But I don't know much about that, so humbly shutting up.

Eugene.

 

[A. Neumaier]

I've noticed a hint of sarcasm. Or I was mistaken?

The smiley tells it all.

We were talking about instantons, solitons, and spontaneous symmetry breaking. I am not an expert in this stuff, but I guess it would be fair to say that these phenomena have not been observed in experiments. They are parts of some theoretical formalisms. The formalisms themselves can be rather successful in explaining/predicting experimental data, but this is not a guarantee that everything inside them have observable counterparts.

One could say this for the whole formalism of quantum mechanics, or even of classical mechanics.

Elementary particles are rather successful in explaining/predicting experimental data, but observed are only tracks on photographs, in bubble chambers or wire chambers, moving pointers on instruments, curves and surfaces on screens, etc..

You simply draw the boundary to what is theoretical speculation at the point where your understanding ends. But this is a very subjective and not very constructive position.

 

[meopemuk]

One could say this for the whole formalism of quantum mechanics, or even of classical mechanics.

Elementary particles are rather successful in explaining/predicting experimental data, but observed are only tracks on photographs, in bubble chambers or wire chambers, moving pointers on instruments, curves and surfaces on screens, etc..

You simply draw the boundary to what is theoretical speculation at the point where your understanding ends. But this is a very subjective and not very constructive position.

I totally agree with that. I was only surprised by your demand that my approach must reproduce instantons, solitons, symmetry breaking and other internal machinery of modern field theories. I don't discard the possibility that a different theory can be proposed, which does not have all these theoretical (unobservable) ingredients, and still describes the experimental world pretty well.

Perhaps even this new theory will be based on particles rather than fields. Knock-knock on wood.

Eugene.

 

[A. Neumaier]

Let me repeat my logic once again. I am concerned about being consistent with experiment. [...]So, I am happy.

Maybe your theory is consistent with experiment; this might satisfy you.

But for two reasons it will still be ignored by almost everyone:
The first is that it is so much more difficult to handle than the conceptual toolkit that has been used extremely successfully in the past. Already looking at the explicit QED Hamiltonian to second order in Appendix L is awesome: Everyone will be happy that the terms only comprise three full pages, though the derivation takes 18 tedious pages.
The second is that it does not allow to derive the conceptual toolkit that has been used extremely successfully in the past.

I know that there are examples involving few interacting particles, where Maxwell's theory yields bad results.

It would be much easier to discount these situations as just theoretical speculations, like symmetry breaking and instantons/solitons - they have not been observed directly in experiment.

I am not asking (yet) to replace QED with my approach

But the introductory quotes to your Chapters 9 and 12 come close to that. And your finale on p.568 calls traditional physical theories nuts: ''In this book we critically examined various assumptions and postulates of traditional physical theories (special relativity, Maxwell’s electrodynamics, general relativity, quantum field theory, etc.). We have concluded that many important statements of these theories are either not accurate or not valid''

I am just saying that there is a non-traditional way to look at QFT. I agree with you that this way has not reached a level of fully developed theory yet. But so far you haven't convinced me that this way is wrong or inconsistent or disagrees with experiment.

I am not trying to do that. I am trying to make you apply to your own work the motto you quote in Chapter 14: ''Many things are incomprehensible to us not because our comprehension is weak, but because those things are not within the frames of our comprehension.''

I am trying to teach you
- standards that you need to satisfy in order to get your work recognized, and
- facts about standard quantum field theory that hopefully tells you that the latter is much more powerful than you believe,
- insights that could correct a few deficiencies of your approach that come across as crackpottish, and that makes it impossible to recommend your book to anyone. Your work could be much more commendable if these features - which are not intrinsic to the dressing approach but only to your particular take on it - were absent.

One of your claims in the preface (p.xx) is that ''In modern QFT the problem of ultraviolet infinities is not solved''. This s not the case; the suggestion in Dirac's quote that infinitely large quantities are neglected by renormalization is simply false. What is done is no worse than
replacing the infinity in 1/(1-x) - x/(1-x) for x=1 by the benign expression 1 obtained by properly rearranging the terms before taking the limit. If you can't see that, you are simply not familiar with presentations of QFT that present things in more careful way than what you are accustomed to.

On p. xxi, you claim that ''Our goal here is to demonstrate that all known physics fits nicely
into this mathematical framework.'' From the omission of the maxwell equations, one can conclude that you consider the latter not to be part of the known physics - something almost everybody will find strange.

On p. xxii, you claim that the ''Usual Lorentz transformations of special relativity are thus approximations that neglect the presence of interactions. The Einstein-Minkowski 4-dimensional space-time is an approximate concept as well.'' This is a severe misunderstanding, and nobody will buy that (outside of a theory including gravity).

On p.345 you conclude: ''The existence of instantaneous action-at-a-distance forces implies the real possibility of sending superluminal signals. Then we find ourselves in contradiction with special relativity, where faster-than-light signaling is strictly forbidden''. Rather than have this open your eyes to some misconception in your assumptions (since standard QED does not allow such a conclusion, and you derive your theory from the standard QED interaction), you boldly claim your error to be a failure of all previous approaches - although what you derive is just theoretical speculation. And you top it on p.565 by saying: ''First is the principle of relativity. In spite of widely held beliefs, this principle implies that the concepts of Minkowski spacetime and manifest covariance are not exact and should be avoided in a rigorous theory.''
Write that into the abstract, and everybody will take you for a crackpot. (But some read the summary before the bulk of the work; so your modesty to put this statement at the end will not save you.)

On p. xxii, you claim that ''the rules connecting bare and physical particles are not well established'', although they are fully explained in every textbook on QFT.

On p.566, you claim as a first major advantage of your approach that ''It does not require effective field theory arguments, such as strings or Planck-scale space-time “granularity,” in order to explain ultraviolet divergences and renormalization.'' - This is a strawman argument, since standard QED has the same major advantage!

The second major advantage contains the unfulfilled claim that ''The time evolution, scattering, bound states, etc. can be calculated according to usual Rules of Quantum Mechanics without regularization, renormalization, and other tricks.'' But you were not able to calculate a single scattering cross section involving a loop integral.

The third major advantage is subjective since it depends on what one is prepared to call an observable quantity.

The fourth major advantage that ''The time evolution, scattering, bound states, etc. can be calculated according to usual Rules of Quantum Mechanics without regularization, renormalization, and other tricks'' is also shared by standard QED, together with the CTP formalism (that you never took seriously although i had mentioned it repeatedly). The latter even has the advantage that it is manifestly invariant, while your approach completely sacrifices Lorentz invariance in actual calculations. As a result, all your calculations are much more messy than the corresponding QED calculations.

On p.xxiii you praise your theory that ''All calculations with the RQD Hamiltonian can be done by using standard recipes of quantum mechanics without encountering embarrassing divergences'', although you haven't demonstrated a single calculation involving radiative corrections and are well aware that you run there into embarrassing divergences.

You repeatedly emphasize what you see as shortcomings of standard QED, but you generously overlook discussing the shortcomings of your approach. But every unacknowledged shortcoming discovered by a reader will ruin the reputation of your presentation.

I read a lot of more recent papers regarding infrared divergences, but I am still not satisfied with my understanding of the issue. I would appreciate if you can recommend a book or review, where all this is explained for pedestrians such as myself.

The stuff is somewhat scattered. You might try:
O. Steinmann,
Perturbative quantum electrodynamics and axiomatic field theory,
Springer, Berlin 2000.
but probably this is too mathematical for you. Kulish/Faddeev and Kibble are probably still the easiest treatments on the level of rigor that you are trying to achieve. Perhaps the papers by Lavelle (enter author:Lavelle infrared into http://scholar.google.com ) are helpful, too.

 

[A. Neumaier]

I was only surprised by your demand that my approach must reproduce instantons, solitons, symmetry breaking and other internal machinery of modern field theories.

I didn't demand that; I was just pointing out that the QFT description of particles allows for much stuff beyond Fock space, and needs it once you go beyond QED. For QED you also need to go beyond Fock space but in less conspicuous ways. Well, charged states break Lorentz symmetry...

But since your principles change so radically the accepted tenets of tradition (as you explain in your summary), they would have to accommodate also the deeper levels correctly described by the standard model. And it would be extremely surprising if this could be done without all the extra stuff that I had mentioned.

 

[A. Neumaier]

I thought that superselection rules forbid forming linear combination of states with different electric charges. So, I wouldn't try a half-electron-half-positron state.

Ah, you touched on an interesting problem, to which I have no answer.

In spite of the charge superselection rule, the Dirac equation for a single electron in an external field (for the relativistic hydrogen atom) treats superpositions of the electron and positron states; though the positron part is ultimately eliminated through a Foldy-Wouthuisen transformation.

On the other hand, nuclear models treat successfully superpositions of protons and neutrons as a nucleon, though these have different charges.

So it seems that the charge superselection rule is perhaps not that fundamental.

 

[A. Neumaier]

I heard that these kinds of issues are solved within non-standard analysis. I've also heard some people say that this is the only proper way to teach analysis and math in general. But I don't know much about that, so humbly shutting up

Nobody is teaching analysis and math in this ''proper'' way - nonstandard analysis is far too hard. And it lacks compactness, and with it one of the fundamental tools ordinary analysis has.
Although we have a strong research group in our department working on Colombeau algebras (one of the ways of doing nonstandard analysis) nobody ever has managed to apply it to a nontrival problem in quantum mechanics.

 

[meopemuk]

Maybe your theory is consistent with experiment; this might satisfy you.

Thanks, this is the most important thing for me.

But for two reasons it will still be ignored by almost everyone:
The first is that it is so much more difficult to handle than the conceptual toolkit that has been used extremely successfully in the past. Already looking at the explicit QED Hamiltonian to second order in Appendix L is awesome: Everyone will be happy that the terms only comprise three full pages, though the derivation takes 18 tedious pages.

This is just the normal QED Hamiltonian written explicitly in terms of particle a/c operators. Nothing fancy. I don't see any problem with these long formulas. If correct physics requires long expressions, so be it! I agree that it is much easier to work with fields and propagators than in the a/c operator representation. Dressed particle theory is forced to work in this messy representation, because important definitions of phys, unphys, and renorm operators cannot be given in terms of fields.

It would be much easier to discount these situations as just theoretical speculations, like symmetry breaking and instantons/solitons - they have not been observed directly in experiment.

The difference is that one can, in principle, collide two electrons and monitor their trajectories and measure accelerations at each time point and determine the magnitudes and directions of forces. This is difficult to do, but possible. So, a good theory *must* have a consistent description of such events. On the other hand, no instantons have been prepared in a laboratory. If a theory doesn't know what is instanton - not a big deal.

But the introductory quotes to your Chapters 9 and 12 come close to that. And your finale on p.568 calls traditional physical theories nuts: ''In this book we critically examined various assumptions and postulates of traditional physical theories (special relativity, Maxwell’s electrodynamics, general relativity, quantum field theory, etc.). We have concluded that many important statements of these theories are either not accurate or not valid''

I agree that this is a bit arrogant. Thanks for drawing my attention. Let me change the last sentence to "We have concluded that many important statements of these theories are not valid in our approach'' Then I go on to list 6 of those questionable "important statements". I would be very much obliged if you can address these specific claims.

I am trying to teach you
- standards that you need to satisfy in order to get your work recognized, and
- facts about standard quantum field theory that hopefully tells you that the latter is much more powerful than you believe,
- insights that could correct a few deficiencies of your approach that come across as crackpottish, and that makes it impossible to recommend your book to anyone. Your work could be much more commendable if these features - which are not intrinsic to the dressing approach but only to your particular take on it - were absent.

Many thanks, indeed!

One of your claims in the preface (p.xx) is that ''In modern QFT the problem of ultraviolet infinities is not solved''. This s not the case; the suggestion in Dirac's quote that infinitely large quantities are neglected by renormalization is simply false. What is done is no worse than
replacing the infinity in 1/(1-x) - x/(1-x) for x=1 by the benign expression 1 obtained by properly rearranging the terms before taking the limit. If you can't see that, you are simply not familiar with presentations of QFT that present things in more careful way than what you are accustomed to.

I didn't want to put too much details in the preface, but since you've mentioned it, I would like to add there one more sentence: ''In modern QFT the problem of ultraviolet infinities is not solved, it is "swept under the rug". The problem is that even if the infinities are "renormalized", one ends up with an ill-defined Hamiltonian, which is not suitable for describing the time evolution of states'' And I stand by this statement. The usual renormalization is not simply a proper definition of limit. Before the limit is taken, one needs to add counterterms to the Hamiltonian, and these counterterms become infinite in the limit. So, we have eliminated divergences in scattering amplitudes at the expense of adding divergences to the Hamiltonian and thus destroying any chances of proper treatment of the time evolution.

On p. xxi, you claim that ''Our goal here is to demonstrate that all known physics fits nicely
into this mathematical framework.'' From the omission of the maxwell equations, one can conclude that you consider the latter not to be part of the known physics - something almost everybody will find strange.

Let me change it to a more precise statement ''Our goal here is to demonstrate that observable physics fits nicely into this mathematical framework.'' I take it as a postulate that any exact theory must be (1) quantum and (2) relativistic. This immediately implies that any such theory must be formulated in terms of a unitary representation of the Poincare group in some Hilbert space. Maxwell theory doesn't have this form. Therefore, Maxwell theory is not exact and must be replaced by a more rigorous approach.

On p. xxii, you claim that the ''Usual Lorentz transformations of special relativity are thus approximations that neglect the presence of interactions. The Einstein-Minkowski 4-dimensional space-time is an approximate concept as well.'' This is a severe misunderstanding, and nobody will buy that (outside of a theory including gravity).

This was just a trailer. The full ad can be found in section 11.3. Read it and then see if you want to buy the product.

On p.345 you conclude: ''The existence of instantaneous action-at-a-distance forces implies the real possibility of sending superluminal signals. Then we find ourselves in contradiction with special relativity, where faster-than-light signaling is strictly forbidden''. Rather than have this open your eyes to some misconception in your assumptions (since standard QED does not allow such a conclusion, and you derive your theory from the standard QED interaction), you boldly claim your error to be a failure of all previous approaches - although what you derive is just theoretical speculation.

These introductory statements are supposed to be just appetizers. I suggest you to read the full explanation of these statements in section 11.4 and then form your judgement.

And you top it on p.565 by saying: ''First is the principle of relativity. In spite of widely held beliefs, this principle implies that the concepts of Minkowski spacetime and manifest covariance are not exact and should be avoided in a rigorous theory.''
Write that into the abstract, and everybody will take you for a crackpot. (But some read the summary before the bulk of the work; so your modesty to put this statement at the end will not save you.)

I will wait until you find the time to read the bulk of my arguments. I understand pretty well that this conclusion sounds shocking. I've put it at the end of the book intentionally, so that the reader comes prepared after reading the bulk of the book. You've spoiled my plot by skipping required reading.

On p. xxii, you claim that ''the rules connecting bare and physical particles are not well established'', although they are fully explained in every textbook on QFT.

I know only one QFT textbook, which devotes significant attention to this issue. This is S. Schweber's book. Still, only model examples are considered there.

On p.566, you claim as a first major advantage of your approach that ''It does not require effective field theory arguments, such as strings or Planck-scale space-time “granularity,” in order to explain ultraviolet divergences and renormalization.'' - This is a strawman argument, since standard QED has the same major advantage!

Why is it that now 60 years after Tomonaga-Schwinger-Feynman people are still discussing ultraviolet divergences and "effective fields"? I guess many of them are not satisfied with the renormalization solution. Perhaps, because of the ill-defined Hamiltonian, as I've mentioned earlier. The dressed particle approach allows one to have finite scattering amplitudes and a well-defined Hamiltonian at the same time. This has not been done in standard QED.

The second major advantage contains the unfulfilled claim that ''The time evolution, scattering, bound states, etc. can be calculated according to usual Rules of Quantum Mechanics without regularization, renormalization, and other tricks.'' But you were not able to calculate a single scattering cross section involving a loop integral.

This is a fair point. I will add there a sentence: "However, the full realization of this program requires solution for the problem of infrared infinities, which remains a challenging mathematical task."

The fourth major advantage that ''The time evolution, scattering, bound states, etc. can be calculated according to usual Rules of Quantum Mechanics without regularization, renormalization, and other tricks'' is also shared by standard QED, together with the CTP formalism (that you never took seriously although i had mentioned it repeatedly). The latter even has the advantage that it is manifestly invariant, while your approach completely sacrifices Lorentz invariance in actual calculations. As a result, all your calculations are much more messy than the corresponding QED calculations.

I stand by my claim that unitary time evolution in quantum mechanics requires a well-defined Hermitian Hamiltonian. I don't understand how CTP can do time evolution without a Hamiltonian. I would appreciate very much if you can explain that to me in simple terms.

On p.xxiii you praise your theory that ''All calculations with the RQD Hamiltonian can be done by using standard recipes of quantum mechanics without encountering embarrassing divergences'', although you haven't demonstrated a single calculation involving radiative corrections and are well aware that you run there into embarrassing divergences.

Thanks. I will change it to ...embarrassing *ultraviolet* divergences...

The stuff is somewhat scattered. You might try:
O. Steinmann,
Perturbative quantum electrodynamics and axiomatic field theory,
Springer, Berlin 2000.
but probably this is too mathematical for you. Kulish/Faddeev and Kibble are probably still the easiest treatments on the level of rigor that you are trying to achieve. Perhaps the papers by Lavelle (enter author:Lavelle infrared into http://scholar.google.com ) are helpful, too.

Thanks. I've tried Kulish/Faddeev and Kibble previously. Maybe I'll be more lucky with Lavelle.

I really appreciate your taking time to read through my book draft and raising your concerns. I see that you've focused on Introduction and Summary so far. Hopefully, we will have even more substantive discussions when you reach the heart of my arguments.

Thanks.
Eugene.

 

[meopemuk]

I didn't demand that; I was just pointing out that the QFT description of particles allows for much stuff beyond Fock space, and needs it once you go beyond QED. For QED you also need to go beyond Fock space but in less conspicuous ways. Well, charged states break Lorentz symmetry...

But since your principles change so radically the accepted tenets of tradition (as you explain in your summary), they would have to accommodate also the deeper levels correctly described by the standard model. And it would be extremely surprising if this could be done without all the extra stuff that I had mentioned.

I haven't ventured into the fields of weak and strong nuclear interactions (though I've made one humble attempt in http://arxiv.org/abs/1010.0458 [/URL]). It may well happen that I will find my method not applicable in these cases. Then I will be forced to change my ways. But something tells me not to worry.

Eugene.

 

[meopemuk]

In spite of the charge superselection rule, the Dirac equation for a single electron in an external field (for the relativistic hydrogen atom) treats superpositions of the electron and positron states; though the positron part is ultimately eliminated through a Foldy-Wouthuisen transformation.

I thought that Dirac sea, zitterbewegung, Foldy-Wouthuisen and all that stuff is not taken seriously anymore. Or they are still alive?

On the other hand, nuclear models treat successfully superpositions of protons and neutrons as a nucleon, though these have different charges.

Proton charge does not contribute much to nuclear forces. So, it is often ignored in approximate nuclear models. But I think that rigorously we are not allowed to take superposition of a proton and a neutron.

Eugene.

 

[meopemuk]

Nobody is teaching analysis and math in this ''proper'' way - nonstandard analysis is far too hard. And it lacks compactness, and with it one of the fundamental tools ordinary analysis has.
Although we have a strong research group in our department working on Colombeau algebras (one of the ways of doing nonstandard analysis) nobody ever has managed to apply it to a nontrival problem in quantum mechanics.

I've read only popular accounts of nonstandard analysis. And I thought to myself that this is exactly what is needed to cure many problems in quantum mechanics and QFT. All these problems with unbounded operators, closed subspaces, domains, convergence, etc, which I hate so much. Unfortunately, my math background is insufficient for that.

Eugene.

 

[A. Neumaier]

I've read only popular accounts of nonstandard analysis. And I thought to myself that this is exactly what is needed to cure many problems in quantum mechanics and QFT. All these problems with unbounded operators, closed subspaces, domains, convergence, etc, which I hate so much. Unfortunately, my math background is insufficient for that.

It is less difficult than creating an alternative foundation for QED...

You can make a start understanding things by studying the paper mentioned in my post #32.

 

[A. Neumaier]

I thought that [...] Foldy-Wouthuisen and all that stuff is not taken seriously anymore. Or they are still alive?

Only the Dirac sea is obsolete. F/W is still needed to understand in quantitative detail bound states of a particle in an external field. Also, more refined versions of the stuff are used in quantum chemistry of heavy atoms, where a relativistic treatment is essential.

Proton charge does not contribute much to nuclear forces. So, it is often ignored in approximate nuclear models. But I think that rigorously we are not allowed to take superposition of a proton and a neutron.

Well, it is this sort of things that require one to have an eye at the more general context while developing a special theory. One cannot undo the superpositions of a proton and a neutron by simply adding electromagnetic interactions. But one can renounce the superselection rule.

Thus the likely solution is that Charge is a superselection sector in QED but no longer in the standard model.

 

[A. Neumaier]

Thanks, this is the most important thing for me.

No reason to thank - I only said ''perhaps'', in order to make the argument. I believe the basic dressing approach is sound, but the details are not since you misunderstand the nature of a single electron.

This is the reason why you get IR divergences in any loop calculation, while the standard QED approach gets them only in calculations involving external photon lines. Choosing the wrong description of the physical particles makes things worse in your dressing rather than better.

Moreover, your interpretation of relativity is faulty, since you overrate the relevance of the Newton-Wigner representation. it forces the framework into a nonrelativistic mold, and is the source of all the strange effects that you take for real but I had labelled as crackpottery.

They do not occur in Weinberg's setting for QED (on which the CTP formalism is based), from which you start. Since they occur in your version, it can only be because somewhere along the way you made a mistake. The mistake iseems to be either in the wrong features of your electrons, or in your interpretation of Newton-Wigner.

TIf correct physics requires long expressions, so be it!

People will compare two formalisms for the same physics, and choose the one that leads to the shortest calculations. The power of a superior mathematical representation has always shaped what was regarded as more fundamental.

Then I go on to list 6 of those questionable "important statements". I would be very much obliged if you can address these specific claims.

I addressed everything in my post. I chose to quote your summarizing statements since that was much easier. But I don't think there is any substance at all in where you deviate from the standard interpretation of relativity.

I didn't want to put too much details in the preface, but since you've mentioned it, I would like to add there one more sentence: ''In modern QFT the problem of ultraviolet infinities is not solved, it is "swept under the rug". The problem is that even if the infinities are "renormalized", one ends up with an ill-defined Hamiltonian, which is not suitable for describing the time evolution of states'' And I stand by this statement. The usual renormalization is not simply a proper definition of limit. Before the limit is taken, one needs to add counterterms to the Hamiltonian, and these counterterms become infinite in the limit. So, we have eliminated divergences in scattering amplitudes at the expense of adding divergences to the Hamiltonian and thus destroying any chances of proper treatment of the time evolution.

Good renormalization is nothing but a careful limiting procedure. The precise nature of the limit
is spelled out informally for quantum field theories in Chapter B5 ''Renormalization'' of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#B5 . As a complement, it is spelled out in complete detail for simple quantum mechanical toy models in my paper ''Renormalization without infinities - a tutorial'' http://arnold-neumaier.at/papers/physpapers.html#ren , where I took great pains to avoid even the slightest taint of hand-waving. Reading the latter, you should be able to see why there is nothing at all wrong with proper renormalization, and then you'd be able to see from Chapter B5 in my FAQ that in QFT nothing fundamentally different happens.

any such theory must be formulated in terms of a unitary representation of the Poincare group in some Hilbert space. Maxwell theory doesn't have this form. Therefore, Maxwell theory is not exact and must be replaced by a more rigorous approach.

You are completely mistaken. Maxwell's theory in vacuum is the theory of the irreducible massless spin 1 representation of the full Poincare group (discrete symmetries included). Thus it is fully identical with the theory of a single photon.

The full ad can be found in section 11.3. Read it and then see if you want to buy the product.

What you write in the preface and the summary is an accurate reflection of what your book contains. I read it a number of years ago. In the mean time, the product hasn't changed, apart from a few cosmetic operations in the advertising. I have every reason not to buy a book that turns serious mistakes into a call for a change of the foundations.

I suggest you to read the full explanation of these statements in section 11.4 and then form your judgement.

I know, and I formed my judgment accordingly.

I will wait until you find the time to read the bulk of my arguments.

I won't read your faulty derivations a second time.

Why is it that now 60 years after Tomonaga-Schwinger-Feynman people are still discussing ultraviolet divergences and "effective fields"?

The divergences serve as an introduction to the otherwise incomprehensible need for renormalization. Effective fields have nothing to do with QED, but can be used to show why any low energy theory of the universe must be approximately renormalizable in the classical sense. This is justification enough to discuss these things.

 

[meopemuk]

This is the reason why you get IR divergences in any loop calculation, while the standard QED approach gets them only in calculations involving external photon lines. Choosing the wrong description of the physical particles makes things worse in your dressing rather than better.

The loop calculations in Chapter 9 are not different from Weinberg or any other QED textbook. Both vertex loop and electron self-energy loop are infrared divergent. See eqs. (11.3.11) and (11.4.14) in Weinberg.

You are completely mistaken. Maxwell's theory in vacuum is the theory of the irreducible massless spin 1 representation of the full Poincare group (discrete symmetries included). Thus it is fully identical with the theory of a single photon.

First, Maxwell's theory claims to be something more general than the theory of a single photon. Second, I still insist that *any* relativistic theory must possesses 10 generators of the Poincare group with corresponding commutation relations. Maxwell's theory is not formulated in such a way. So, it violates the most basic requirements of quantum mechanics and relativity.

I won't read your faulty derivations a second time.

Thanks for your comments anyway.

Eugene.

 

[dextercioby]

First, Maxwell's theory claims to be something more general than the theory of a single photon.

Maxwell fields are classical objects, photons are concepts from a quantum theory. Maxwell theory couldn't claim to be <more general> than a theory of one photon, because there's no such thing as a theory of one photon. There's a theory of photons, or of photon quantum states.
As per my understading, Maxwell's theory leads through quantization to a theory of massless, free, relativistic spin 1 quantum fields. The Fock space of this theory comprises as many multi-photon states as you want.

Second, I still insist that *any* relativistic theory must possesses 10 generators of the Poincare group with corresponding commutation relations.

OK.

Maxwell's theory is not formulated in such a way.

Of course not, because it's purely classical.

So, it violates the most basic requirements of quantum mechanics and relativity.

Of course it violates the requirement of QM, because it's a classical field theory. Relativity violated? Of course not, since relativity itself was built to explain Maxwell's theory...

EDIT: Apparently your use of <Maxwell theory> is different from mine and gives rise to confusions. For me Maxwell's name cannot be associated to a quantum theory.

 

[A. Neumaier]

First, Maxwell's theory claims to be something more general than the theory of a single photon.

In terms of quantum mechanics, Maxwell's theory is what you get if you restrict the Photon Fock space (or rather Kibble's far bigger space) to the manifold defined by all coherent states. The coherent states are in 1-1 correspondence with the solutions of the maxwell equations, and behave essentially classically, as you can glance from any quantum optics book. Mandel&Wolf show that this very well accounts for classical optics.

Second, I still insist that *any* relativistic theory must possesses 10 generators of the Poincare group with corresponding commutation relations. Maxwell's theory is not formulated in such a way. So, it violates the most basic requirements of quantum mechanics and relativity.

Maxwell's theory possesses 10 such generators, since it is manifestly covariant, and constitutes an irreducible representation of the full Poincare group. Perhaps you can't see this in your favorite QFT book by Weinberg, but he derives it in one of his any spin papers: Phys.Rev. 134 (1964), B882-B896

In contrast, you theory possesses only approximate such generators, to the order that you do your expansion.

 

[meopemuk]

Maxwell's theory is not formulated in such a way.

Of course not, because it's purely classical.

So, it violates the most basic requirements of quantum mechanics and relativity.

Of course it violates the requirement of QM, because it's a classical field theory. Relativity violated? Of course not, since relativity itself was built to explain Maxwell's theory...

A proper relativistic classical theory must be derived as a limit of a relativistic quantum theory. So, the classical theory must keep the most important features of its quantum counterpart. Namely, it must possesses 10 generators of the Poincare group. The quantum commutator should be represented by its classical analog, such as the Poisson bracket. And the 10 classical generators must satisfy the well known Poincare bracket relations. I don't see these features in Maxwell's theory, so I have a reason to doubt that this theory is a classical limit of a more general quantum approach.

Eugene.

 

[dextercioby]

A proper relativistic classical theory must be derived as a limit of a relativistic quantum theory. So, the classical theory must keep the most important features of its quantum counterpart. Namely, it must possesses 10 generators of the Poincare group. The quantum commutator should be represented by its classical analog, such as the Poisson bracket.

Agree so far.

And the 10 classical generators must satisfy the well known Poincare bracket relations. I don't see these features in Maxwell's theory, so I have a reason to doubt that this theory is a classical limit of a more general quantum approach.

Eugene.

Well, in a classical field theory in the Lagrangian formulation (which has the advantage of being manifestly covariant), we have the famous Noether's theorem which gives us the 10 generators of the Poincare Lie algebra in terms of the classical fields, the 1-forms $A_{\mu} (x)$. You must know that...

 

[meopemuk]

Maxwell's theory possesses 10 such generators, since it is manifestly covariant, and constitutes an irreducible representation of the full Poincare group. Perhaps you can't see this in your favorite QFT book by Weinberg, but he derives it in one of his any spin papers: Phys.Rev. 134 (1964), B882-B896

You've possible mistaken this paper for another Weinberg's work

Weinberg, S. Photons and gravitons in perturbation theory: derivation of
Maxwell's and Einstein's equations. Phys. Rev. 138 (1965) B988-B1002

where he discusses Maxwell equations in section VI. This work is the basis of my subsection 8.1.2 and Appendix N. It is true that Weinberg has 10 quantum generators of the Poincare group. It is also true that his fields $$A^{\mu}$$ and $$J^{\mu}$$ satisfy relationships that formally resemble the set of Maxwell equations. However, this is only a superficial resemblance.

First, Weinberg stresses many times that his goal is to calculate the S-matrix. So, he is not interested in interacting time dynamics of charges and electromagnetic fields (which is the subject of Maxwell's theory). He is right not to go there, because the field Hamiltonian (eqs. (8.10) - (8.14) in my book) is incapable of describing the time evolution even for simplest one-charge states. I've explained that failure in section 10.1 of my book. So, Weinberg's fields are just formal mathematical objects that are not related to anything observed in experiments.

In contrast, you theory possesses only approximate such generators, to the order that you do your expansion.

This is true. Higher perturbation orders require more work.

Eugene.

 

[meopemuk]

Well, in a classical field theory in the Lagrangian formulation (which has the advantage of being manifestly covariant), we have the famous Noether's theorem which gives us the 10 generators of the Poincare Lie algebra in terms of the classical fields, the 1-forms $A_{\mu} (x)$. You must know that...

See my response to Arnold.

Eugene.

 

[dextercioby]

See my response to Arnold.

Eugene.

My post was about the classical theory by itself and not thinking of it as a limit of a quantum theory.

Your statement <A proper relativistic classical theory must be derived as a limit of a relativistic quantum theory> sets conditions on the quantum theory rather than on the classical counterpart. So it's ok.

Further you claim that <the classical theory must keep the most important features of its quantum counterpart>. I think it's again a matter of perhaps wrong wording: the quantum theory must generalize the classical one and could(and should have novel features compared to it. So it's likely that some (if not all) of the <the most important features> of the quantum theory couldn't have any classical analogue. Just think of the H-atom. Also your statement suggests the opposite of the first one: it looks like it sets conditions on the classical theory based on knowledge of the quantum one. But that's wrong, the quantum theory is the issue, the classical one is to be postulated as it's simply a particular case of the specially relativistic dynamics which we already know to be correct.

Futher <Namely, it must possesses 10 generators of the Poincare group>. The classical theory does that, of course.

Next: <The quantum commutator should be represented by its classical analog, such as the Poisson bracket>. It does (actually this is a classical gauge theory, so there's some ambiguity at the classical level in the definition of fields which forces us to use a different simplectic structure on the classical state space than the normal PB). But again, you reverse the logics: to the classical Poisson bracket one must find a proper quantum commutator and not viceversa.

Next: <I don't see these features in Maxwell's theory>. Can you justify your statement ?

 

[meopemuk]

Arnold, dextercioby,

Yes, you've convinced me that Maxwell's theory has a set of 10 generators, e.g., those defined as classical analogs of QED functions $$H, \mathbf{K}, \mathbf{P}, \mathbf{J}$$ of operator fields $$A^{\mu}, J^{\mu}$$ in subsection 8.1.2. So, formally, it is OK as a relativistic theory.

However, I think, it is important to note that this alleged quantum-classical correspondence is only formal. The point is that quantum QED generators in subsection 8.1.2 do not form a viable physical theory. So, its classical limit cannot be viable too. The bad features of QED in 8.1.2 are seen from the fact that the S-matrix computed with the Hamiltonian (8.10) is divergent. Of course, in QED this problem is fixed by adding counterterms, which effectively result in infinite masses and charges. If we were to take a classical limit of QED with counterterms, we wouldn't get the familiar Maxwell's theory.

However, even QED with counterterms is not satisfactory as a quantum theory of interacting charges and photons. As I discuss in section 10.1, the time evolution of states is not acceptable in this approach. The next step should be taken, which is a unitary transformation of the QED Hamiltonian with counterterms to the dressed particle form. Then, finally, we obtain an acceptable quantum theory with a finite Hamiltonian, realistic time evolution of particle states, and experimentally confirmed S-matrix. But the classical limit of this theory is not going to look as Maxwell's theory at all. It looks more like the Darwin-Breit Hamiltonian.

[...] the quantum theory must generalize the classical one and could(and should have novel features compared to it. [...] the classical one is to be postulated [...] to the classical Poisson bracket one must find a proper quantum commutator and not viceversa.

I strongly disagree with the idea that first we must postulate a classical theory and then "quantize" it in order to get a quantum-mechanical counterpart. The most general and exact theory of nature must be both quantum and relativistic. So, if we don't want to make mistakes, we must first postulate a self-consistent fully quantum approach with Hilbert space, commutators, and all that. Then, the classical analog should be obtained in the limit $$\hbar \to 0$$. In this limit we may lose some fine quantum features, but, at least, we can be confident that our classical approach has a solid foundation in quantum postulates. If for some reason we find that the classical limit doesn't work or disagrees with experiment, then we should modify our quantum theory and try again.

The idea of "quantization" can possibly work as a heuristic tool for guessing the form of yet unknown quantum theory in the absence of other theoretical options. But I wouldn't consider "quantization" as a rigorous theoretical mechanism.

Eugene.

 

[A. Neumaier]

you've convinced me that Maxwell's theory has a set of 10 generators, e.g., those defined as classical analogs of QED functions $$H, \mathbf{K}, \mathbf{P}, \mathbf{J}$$ of operator fields $$A^{\mu}, J^{\mu}$$ in subsection 8.1.2. So, formally, it is OK as a relativistic theory.

With time, we'll convince you that, formally, all the things you didn't like about field theory and that you claimed before to be actually false or unproved, are OK. But though all of that stuff is empirically validated, you'll always say that these are only theoretical speculations that have no weight compared with those unvalidated practical speculations that figure in your book...

However, I think, it is important to note that this alleged quantum-classical correspondence is only formal.

Your judgment reveals a serious lack of knowledge of the state of the art. This quantum-classical correspondence is extremely well established and backed up by lots of experimental evidence.

For example, the quantum optics book by Mandel and Wolf discusses in Chapters 5-9 classical Maxwell theory and its optical consequences. Chapter 10-20 do the same for the quantum version, starting with standard QED. They find that the quantum theory of coherent fields is essentially identical with that of classical fields, except for corrections of the order of hbar (that give highly interesting nonclassical effects).

 

[meopemuk]

It is possible to make a source of light, which produces single photons one-by-one on demand. If I shine this light on a double-slit, I get a famous picture on the screen, where each photon makes a separate tiny spot, and only after long exposure the interference pattern emerges. I don't see other way to explain this behavior but the quantum mechanical picture in which light particles are described by a probabilistic wave function. The Maxwellian field representation of light is incapable of describing this experiment at all. So, I conclude that electromagnetic field is just a crude approximation, which works only for high-intensity light, where large numbers of photons are present at once and measuring devices are not sensitive enough to distinguish each individual photon.

Eugene.