What Are the Implications of a New Relativistic Quantum Theory?

[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.

 

[A. Neumaier]

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.

See
http://arnold-neumaier.at/ms/lightslides.pdf
http://arnold-neumaier.at/ms/optslides.pdf
for the standard quantum optics view of this matter. It is all explainable by the field picture - particles entering only in a semiclassical view.

 

[meopemuk]

See
http://arnold-neumaier.at/ms/lightslides.pdf
http://arnold-neumaier.at/ms/optslides.pdf
for the standard quantum optics view of this matter. It is all explainable by the field picture - particles entering only in a semiclassical view.

So, photons are "localized lumps of energy". Do they pass through one slit or through both slits at once in the double-slit experiment?

Eugene.

 

[A. Neumaier]

So, photons are "localized lumps of energy". Do they pass through one slit or through both slits at once in the double-slit experiment?

As any wave, localized or not, through both slits.

Only upon recording the photons, they materialize - at a single spot only.

 

[A. Neumaier]

Suppose now that we constructed a relativistic interacting theory in which the interacting Hamiltonian is *not* a product of fields. For example, it can be V= a^{\dag}a^{\dag}aa.

This still is a product of nonrelativistic local fields. I guess you mean: not (a linear combination of) products of relativistic, local fields.

Then [...] two important conditions of Haag's theorem will not be satisfied, and we will not be able to prove that the Fock space is excluded.

Yes.

As a result of this exercise we will obtain a non-trivial interacting theory in the Fock space. "Dressed particle" theories are exactly of this form. Their only problem is that interacting fields are non-covariant and non-commuting. Could you please explain why you think that this is an important problem? Is there any measurable property that proves the impossibility of non-covariant and non-commuting interacting fields?

First, you now have infinitely many interacting terms. So additional problems about well-definedness and self-adjointness arise.

Second, observed physics is Poincare invariant, to very high accuracy. Your Poincare representation is only approximate at any order.

Third, lack of covariance makes all calculations (especially those of higher order) much more messy. This is the main reason, I think, why very few people today work with noncovariant methods.

Finally, you do everything at zero temperature. However, most application of QED where the time evolution is important take place at finite temperature. There everything is different again - in place of the asymptotic particles at T=0 one now has only effective particles, which are different.

 

[meopemuk]

As any wave, localized or not, through both slits.

Only upon recording the photons, they materialize - at a single spot only.

For the double-slit experiment with visible light the distance between two slits can be macroscopic, e.g. 0.1 millimeter, or something like that. This means that the photon "lump" should be no smaller than this size. So, you are saying that the energy lump associated with a single visible-light photon can be as big as 0.1 millimeter or so? And that all this volume is filled with a time-changing electromagnetic field? Do I understand your model correcly?

Eugene.

 

[A. Neumaier]

For the double-slit experiment with visible light the distance between two slits can be macroscopic, e.g. 0.1 millimeter, or something like that. This means that the photon "lump" should be no smaller than this size. So, you are saying that the energy lump associated with a single visible-light photon can be as big as 0.1 millimeter or so? And that all this volume is filled with a time-changing electromagnetic field?

Of course. Photons can be very delocalized. This is precisely what happens in a double slit experiment. Each photon in a laser beam has the same shape as the classical field by which this beam is described.

My views about what photons are turned by almost 180 degrees after I had begun to talk to the experimentalists in Zeilinger's group (who moved to Vienna in 1999)...

 

[meopemuk]

This still is a product of nonrelativistic local fields. I guess you mean: not (a linear combination of) products of relativistic, local fields.

In our example we've considered a \Phi^4 theory. So, I am saying that V = a^{\dag} a^{\dag}aa cannot be expressed as a linear combination of integrals

\int d\mathbf{r} \Phi^n(\mathbf{r}, 0)

First, you now have infinitely many interacting terms. So additional problems about well-definedness and self-adjointness arise.

What do you mean by "many interaction terms"? My interaction has only one term V = a^{\dag} a^{\dag}aa.

Second, observed physics is Poincare invariant, to very high accuracy. Your Poincare representation is only approximate at any order.

So, you are saying that it is impossible to formulate a Poincare-invariant theory with interaction V = a^{\dag} a^{\dag}aa? Can you prove that?

You are making the claim that Haag's theorem does not leave any chance for using Fock space in relativistic quantum theories. So, it is your job to prove that interaction V = a^{\dag} a^{\dag}aa violates relativity.

Third, lack of covariance makes all calculations (especially those of higher order) much more messy. This is the main reason, I think, why very few people today work with noncovariant methods.

Messy or not messy is a matter of taste and convenience. This is not a scientific argument.

Finally, you do everything at zero temperature. However, most application of QED where the time evolution is important take place at finite temperature. There everything is different again - in place of the asymptotic particles at T=0 one now has only effective particles, which are different.

I am interested in what happens when two electrons (or other particles) move close to each other. I think it is important to understand this simplest event first. Only then we can switch to systems with many particles, non-zero temperature, etc.

Eugene.

 

[A. Neumaier]

What do you mean by "many interaction terms"? My interaction has only one term V = a^{\dag} a^{\dag}aa.

So you are talking about a harmonic oscillator?

Or do you mean V = \int dx a^*(x) a^*(x)a(x)a(x)? This is already a linear combination of field products, not only one product.

So, you are saying that it is impossible to formulate a Poincare-invariant theory with interaction V = a^{\dag} a^{\dag}aa? Can you prove that?

No. What I was saying is spelled out in post #74.

What you though I was saying is probably possible in the style of Bakamjian, at least when one allows an additional energy shift. But then you don't have cluster separability.

You are making the claim that Haag's theorem does not leave any chance for using Fock space in relativistic quantum theories. So, it is your job to prove that interaction V = a^{\dag} a^{\dag}aa violates relativity.

No. I am only claiming that Haag's theorem says what it says. This is in the literature, freeing me from any further obligation.

If you create a theory that violates the assumptions of Haag's theorem, you are not bound to its conclusions.

Messy or not messy is a matter of taste and convenience. This is not a scientific argument.

Of course it is. Kepler's theory was in his time not more accurate than Ptolemy's. But it was far less messy, and made things so much easier that it was adopted as the standard.

I am interested in what happens when two electrons (or other particles) move close to each other.

Whereas I an interested to explain standard QFT to those who want to understand it.

I think it is important to understand this simplest event first. Only then we can switch to systems with many particles, non-zero temperature, etc.

2-electron systems are useless approximations, once one has interactions terms changing particle number.

Once one wants to insist on a dynamical view (rather than the asymptotic scattering view), one needs to prepare the particles at finite time, hence they'll never be exact 2-particle states (which make sense only asymptotically). But even if they were exact for one moment, they'd lose that property the very next moment.

Thus you are chasing a chimera. I prefer to chase the insights of the community of high energy physicists, who all work in the quantum field paradigm.

 

[meopemuk]

So you are talking about a harmonic oscillator?

Or do you mean V = \int dx a^*(x) a^*(x)a(x)a(x)?

No, I meant a 2-particle interaction whose full expression involves momentum-space a/c operators

V = \int d\mathbf{p} d\mathbf{q} d\mathbf{k} V(\mathbf{p}, \mathbf{q}, \mathbf{k}) a^{\dag}(\mathbf{p-k}) a^{\dag}(\mathbf{q+k}) a(\mathbf{p}) a(\mathbf{q})

Here V(\mathbf{p}, \mathbf{q}, \mathbf{k}) is a numerical coefficient function, which can be selected in such a way that interaction is translationally and rotationally invariant. I also claim that one can find a corresponding interacting boost operator, so that entire theory becomes relativistically invariant.

My other claim is that in this theory interacting quantum fields constructed by usual formulas do *not* transform covariantly. So, the main condition of Haag's theorem is not satisfied, the theorem is not applicable, and we are permitted to work in the Fock space.
More details on this example can be found in

H. Kita, "A non-trivial example of a relativistic quantum theory of particles without divergence difficulties", Progr. Theor. Phys., 35 (1966), 934.

If you create a theory that violates the assumptions of Haag's theorem, you are not bound to its conclusions.

Great! I am glad that we agree about that.

Eugene.

 

[meopemuk]

Of course. Photons can be very delocalized. This is precisely what happens in a double slit experiment. Each photon in a laser beam has the same shape as the classical field by which this beam is described.

OK, so we have a macroscopic lump of electromagnetic energy, which falls on the double slit and interferes with itself according to Maxwell equations. Then this macroscopic lump reaches the photographic plate and suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion. The photon energy, that was previously spread up in a macroscopic lump now gets released within a group of few atoms.

How does Maxwell equation explain this collapse?

Eugene.

 

[A. Neumaier]

No, I meant
V = \int d\mathbf{p} d\mathbf{q} d\mathbf{k} V(\mathbf{p}, \mathbf{q}, \mathbf{k}) a^{\dag}(\mathbf{p-k}) a^{\dag}(\mathbf{q+k}) a(\mathbf{p}) a(\mathbf{q})

OK, not a product but an integral over a product. It is difficult to understand you if you shorten formulas to an extent that they become unrecognizable.

I also claim that one can find a corresponding interacting boost operator, so that entire theory becomes relativistically invariant.

My other claim is that in this theory interacting quantum fields constructed by usual formulas do *not* transform covariantly.

The quantum fields of interest are those constructed from
\phi(0)=\int dp (a(p) +a^*(p))
by conjugating with the representation of the Poincare group you claim exists. Since the stabilizer of zero is the Lorentz group, the translations gives a Hermitian field phi(x) which transforms covariantly. But the field is not local, and hence not of interest for particle physics. One can construct plenty of similar field with Bakamjian's construction.

So, the main condition of Haag's theorem is not satisfied, the theorem is not applicable, and we are permitted to work in the Fock space.

Yes, but you get something lacking cluster separation.

More details on this example can be found in

H. Kita, "A non-trivial example of a relativistic quantum theory of particles without divergence difficulties", Progr. Theor. Phys., 35 (1966), 934.

Probably this is the reason why this pseudo-breakthrough only generated 7 references in over 50 years.

 

[A. Neumaier]

OK, so we have a macroscopic lump of electromagnetic energy, which falls on the double slit and interferes with itself according to Maxwell equations. Then this macroscopic lump reaches the photographic plate and suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion. The photon energy, that was previously spread up in a macroscopic lump now gets released within a group of few atoms.

How does Maxwell equation explain this collapse?

The Maxwell equations are valid in vacuum and need not explain their failure in the presence of a detector.

The behavior of the detector in the presence of the incident classical electromagnetic field is fully explained by the detector's quantum structure. See Chapter 9 of the quantum optics book by Mandel & Wolf.

That the photon ''suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion'' is pure fantasy. The photon is absorbed by the detector, and doesn't survive as a localized photon.

 

[meopemuk]

The Maxwell equations are valid in vacuum and need not explain their failure in the presence of a detector.

The behavior of the detector in the presence of the incident classical electromagnetic field is fully explained by the detector's quantum structure. See Chapter 9 of the quantum optics book by Mandel & Wolf.

That the photon ''suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion'' is pure fantasy. The photon is absorbed by the detector, and doesn't survive as a localized photon.

So, is it correct to say that the macroscopically delocalized lump of the photon's EM field is absorbed by the entire photographic plate, and then all this absorbed energy gets channeled somehow to a single grain of photoemulsion?

If this model is not correct, then how should I visualize the interaction between the photon and the photographic plate? Do I need to take the quantum structure of the photographic plate into account?

Eugene.

 

[A. Neumaier]

So, is it correct to say that the macroscopically delocalized lump of the photon's EM field is absorbed by the entire photographic plate

I answered in the thread https://www.physicsforums.com/showthread.php?p=3165407 where this part of the discussion belongs.

 

[meopemuk]

OK, not a product but an integral over a product. It is difficult to understand you if you shorten formulas to an extent that they become unrecognizable.

Sorry about that.

The quantum fields of interest are those constructed from
\phi(0)=\int dp (a(p) +a^*(p))
by conjugating with the representation of the Poincare group you claim exists. Since the stabilizer of zero is the Lorentz group, the translations gives a Hermitian field phi(x) which transforms covariantly.

You've probably meant

\Phi(\mathbf{0}, 0) = \int \frac{d \mathbf{p}}{\sqrt{\omega_{\mathbf{p}}}}\left(a(\mathbf{p}) + a^{\dag}(\mathbf{p}) \right)

We went throgh this exercise in posts #135, #137, #139 of the thread
https://www.physicsforums.com/showthread.php?t=388556&page=9 and we found there that covariant transformation law and space-like commutativity can be proven if the interaction is written in the form

\int d \mathbf{r} \Phi^n(\mathbf{r}, 0)

My interaction does not have this form, so I don't see how you're going to prove the covariance and commutativity?

But the field is not local, and hence not of interest for particle physics.

I don't think there is a proof that particle physics can be explained only in terms of local quantum fields. In my opinion, there is some historically motivated prejudice, but not a proof.

Yes, but you get something lacking cluster separation.

The cluster separability of the "dressed particle" approach is proven in section 10.2. Interaction of the type presented above is cluster-separable if the coefficient function V(\mathbf{p}, \mathbf{q}, \mathbf{k}) is smooth, i.e., non-singular. This is not difficult to achieve.

Eugene.

 

[A. Neumaier]

You've probably meant
\Phi(\mathbf{0}, 0) = \int \frac{d \mathbf{p}}{\sqrt{\omega_{\mathbf{p}}}}\left(a(\mathbf{p}) + a^{\dag}(\mathbf{p}) \right)

Well, as I had said in our other discussion, I normalize the annihilator fields differently in order to get rid of the sqrt factors. (Anyway, it doesn't matter in this case, since no matter which p-dependent factor we use, we don't get a causal field.)

My interaction does not have this form, so I don't see how you're going to prove the covariance and commutativity?

This is precisely the point, with my recipe to construct a covariant field you don't get causality. With your recipe to construct a causal field you don't get covariance. But one needs both. Thus either field is useless. And the lack of citations of Kita's paper is empirical proof of that - only 7 people in over 50 years found it useful enough to merit a citation.

The cluster separability of the "dressed particle" approach is proven in section 10.2. Interaction of the type presented above is cluster-separable if the coefficient function V(\mathbf{p}, \mathbf{q}, \mathbf{k}) is smooth, i.e., non-singular. This is not difficult to achieve.

Please give the page; I didn't see it. I only found a passing reference on p.356, which referred to p.271, which establishes cluster separability only in a preferred frame of reference. You declare this to be enough on p.428 - but very few people will follow you in that. You discard the basic insight of 20th century physics. There are no experimental hints that Nature has preferred frames.

While searching for information, I came across the following unrelated remark on p. 426: ''We can record time even if we do not measure anything, even if there is no physical system to observe.'' I wonder how you record time without observing (i.e., measuring) a clock. If there is no physical system to observe, there is in particular no clock and no observer.

And on p.427; ''the formally quantum nature of clocks and rulers does not play any role in experimental physics.'' But they play a big role if the times and distances become short enough. Accurate experiments become impossible (or extremely difficult) when these are so short that the quantum nature starts playing a role.

 

[meopemuk]

And the lack of citations of Kita's paper is empirical proof of that - only 7 people in over 50 years found it useful enough to merit a citation.

If you want to make fun of Kita's paper, I can help you. One of these 7 citations is in my book, two of them are by my friend Shirokov in Dubna, and three are Kita's self-citations. So, we are not talking about mainstream here. You are right.

Eugene.

 

[meopemuk]

This is precisely the point, with my recipe to construct a covariant field you don't get causality. With your recipe to construct a causal field you don't get covariance. But one needs both. Thus either field is useless.

I agree that the interacting field built in this manner is useless. It is useless not because it is non-covariant or non-commuting, but because actual calculations of the S-matrix, bound states, time evolution, etc. do not involve this field at all. All we need for practical calculations is the Hamiltonian, and there is no need to worry about the properties of the interacting field.

Please give the page; I didn't see it. I only found a passing reference on p.356, which referred to p.271, which establishes cluster separability only in a preferred frame of reference. You declare this to be enough on p.428 - but very few people will follow you in that. You discard the basic insight of 20th century physics. There are no experimental hints that Nature has preferred frames.

I agree that the point about cluster separability is not well emphasized. I will think how to rewrite section 10.2 to make this point more clear. But the idea is very simple. Interaction is guaranteed to be cluster-separable if the coefficient function is smooth. This is Statement 7.7 on page 271. This is the same condition as in Weinberg's sections 4.3-4.4.

In building the "dressed particle" version of QED I start from the usual QED Hamiltonian, whose interaction V satisfies the above separability condition already. (Well, strictly speaking, this condition is satisfied only if we ignore the singularity associated with the zero photon mass. But I don't want to open this can of worms again here.) I obtain the dressed particle interaction by applying an unitary transformation e^{i\Phi} to the QED Hamiltonian. As I claim in Theorem 10.2, the Hermitian operator \Phi must be smooth (=separable) in order to preserve the S-matrix. So, I pay a special attention to make sure that \Phi satisfies this condition. Then, when I do dressing transformation order-by-order in section 10.2, I obtain new interaction terms in the form of multiple commutators involving operators V and \Phi. Both of these operators are smooth (=separable), so by theorem 7.11 all their commutators are smooth (=separable) too. This proves that the dressed particle Hamiltonian is separable in each perturbation order.

You are right that these calculations have been done in a single (but arbitrary) reference frame. If we want to obtain the Hamiltonian in a moving frame, we would need to apply the boost generator to it. Due to the principle of relativity, the Hamiltonian in the moving frame will have the same expansion coefficients with respect to to a/c operators in the moving frame as the coefficients of the rest-frame Hamiltonian with respect to rest-frame a/c operators. So, if the rest-frame Hamiltonian is separable, then the moving-frame Hamiltonian is separable too.

I am not sure where you found any evidence that I adhere to the preferred frame idea. I am 100% behind the principle of relativity. All inertial frames are equal. It is true that specific calculations can be done more easily in a specific frame. And that's what I do. But this should not be a problem, since there exist well-defined rules about how to translate our descriptions between different frames.

Eugene.

 

[meopemuk]

While searching for information, I came across the following unrelated remark on p. 426: ''We can record time even if we do not measure anything, even if there is no physical system to observe.'' I wonder how you record time without observing (i.e., measuring) a clock. If there is no physical system to observe, there is in particular no clock and no observer.

I've tried to explain my views on measurements (and on time measurements, in particular) on pages xxix and xxx that refer to Figure 1. In this figure there is a clear separation between the measuring device and the observed physical system. The measuring device and the clock are not parts of the physical system. They are parts of the experimental equipment, which every observer must have. The clock plays a specific role in a sense that it records something (time), which has no relevance to the observed physical system. Time is a quantity that exists by itself, without any connection to the observed system. And time can be recorded even if there is no system to observe, i.e., if the space between the preparation device and the measuring apparatus in Fig. 1 is empty. This allows me to say in subsection 11.3.4 that there cannot be a "time observable"

Of course, we may decide to treat our clock as a physical system, i.e., put it between the preparation device and the measuring apparatus in Fig. 1. Then we can find all kinds of quantum uncertainties associated with the clock. E.g., we can find that the leads cannot have certain velocities and positions simultaneously. But then, in order to keep time labels of our measurements, we would need to choose some other (reference) clock in our laboratory. The readings of this reference clock will be considered as true classical time labels.

So, any laboratory has a clock associated with it. The readings of this clock are postulated to be exact and classical time labels that are attached to all measurements that we do in the laboratory.

It might also happen that our reference clock behaves irregularly due to quantum fluctuations. This simply means that we've chosen a bad instrument to serve as a clock. We would need to replace it with some other (e.g., more massive) device whose ticks are more regular and predictable.

This is kind of philosophical point, and I am still struggling to formulate it in a coherent fashion.

And on p.427; ''the formally quantum nature of clocks and rulers does not play any role in experimental physics.'' But they play a big role if the times and distances become short enough. Accurate experiments become impossible (or extremely difficult) when these are so short that the quantum nature starts playing a role.

I agree that this is not a well-formulated sentence. I am going to replace it with the following: "So, for theoretical purposes, it is reasonable to assume the availability of ideal clocks and rulers, whose performance is not affected by quantum effects."

Eugene.

 

[A. Neumaier]

If you want to make fun of Kita's paper, I can help you. One of these 7 citations is in my book, two of them are by my friend Shirokov in Dubna, and three are Kita's self-citations. So, we are not talking about mainstream here.

I had no intention to be funny here. ''useful'' is a sociological term with a different meaning of ''mainstream''. Mainstream is the dominant view on a subject; useful is something if people start using it once its exist.

I wouldn't regard Polyzou's work on covariant few-particle systems as mainstream. But it has about 200 citations, and looking at some of them, one finds that is useful to some extent. On the other hand, something that hasn't been used in many years can hardly be called useful.

 

[A. Neumaier]

I agree that the interacting field built in this manner is useless. It is useless not because it is non-covariant or non-commuting, but because actual calculations of the S-matrix, bound states, time evolution, etc. do not involve this field at all.

As can be seen by Weinberg, who odes scattering calculations via LSZ and bound state calculations via Bethe-Salpeter, these calculations instead involve the usual covariant fields. So do time evolution calculations in the CTP framework.

All we need for practical calculations is the Hamiltonian, and there is no need to worry about the properties of the interacting field.

You calculations with explicit Hamiltonians are less practical than those employed by Weinberg and CPT. As with ''useful'', ''practical'' means that it is reasonably easy to practice. This is not the case with your complex formulas.

In building the "dressed particle" version of QED I start from the usual QED Hamiltonian, [...] obtain the dressed particle interaction by applying an unitary transformation e^{i\Phi} to the QED Hamiltonian.

This transformation is not unitary; see the remark at the bottom of p.4 of Shirokov's paper at http://lanl.arxiv.org/abs/math-ph/0703021 , that you quoted in one of your recent mails. Indeed, the paper is very nice in that it makes explicit much of what we had been talking about without introducing unnecessary details.

In the terminology of Shirokov's paper (but in my words), Haag's theorem is essentially saying that if you have a dressing transform W of the kind described in the paper then W is not unitarily implementable since the physical Hilbert space (which is the image of the dressed Fock space under the mapping W) is not a Fock space.

Note that _all_ rigorous constructions in constructive field theory that we had been talking about (though not the Wightman approach, which is considered ''axiomatic'', not ''constructive'') are precisely about the rigorous construction of a (nonunitary) dressing transformations that provides a definition of the physical Hilbert space on which the Heisenberg fields act.

The mistake in your experimental interpretation of your formalism is that you take as the physical Hilbert space the Fock space rather than its image under W. This gives all your ''physical'' operators a touch of surrealism revealed by the strange deviations from standard relativity reported e.g. in your Section 11.4. (E.g., at the bottom of p.437, you talk about a non-covariant boost transformation law - a contradiction in itself under the usual interpretation.)

As I claim in Theorem 10.2, the Hermitian operator \Phi must be smooth (=separable) in order to preserve the S-matrix.

More importantly, it must be self-adjoint, which is not the case because of Haag's theorem (in the form mentioned above).

 

[meopemuk]

As can be seen by Weinberg, who odes scattering calculations via LSZ and bound state calculations via Bethe-Salpeter, these calculations instead involve the usual covariant fields. So do time evolution calculations in the CTP framework.

As we agreed already, my approach is not a "field theory". So, I don't use fields in any of those calculations. I use the old-fashioned quantum mechanics, which is based on the Hamiltonian.

This transformation is not unitary; see the remark at the bottom of p.4 of Shirokov's paper at http://lanl.arxiv.org/abs/math-ph/0703021 , that you quoted in one of your recent mails. Indeed, the paper is very nice in that it makes explicit much of what we had been talking about without introducing unnecessary details.

First, my approach is different from the one used by Shirokov. The difference is explained in subsection 10.2.10. I apply the dressing transformation to the Hamiltonian (and other Poincare generators). Particle states are not transformed, so the Fock space structure remains the same as in the free theory. Second, if the dressing operator e^{i \Phi} were non-unitary I would get a non-Hermitian dressed Hamiltonian as a result. However, this Hamiltonian is explicitly represented as a sum of commutators of Hermitian operators, so it is bound to be Hermitian itself. See eq. (10.8). It might happen that this infinite series of Hermitian terms converges to something non-Hermitian in the infinite perturbation order limit. I am not sure about that. But I am not claiming that I have a non-perturbative approach. So, for my purposes it is safe to say that the dressed Hamiltonian is Hermitian and the dressing transformation is unitary.

In the terminology of Shirokov's paper (but in my words), Haag's theorem is essentially saying that if you have a dressing transform W of the kind described in the paper then W is not unitarily implementable since the physical Hilbert space (which is the image of the dressed Fock space under the mapping W) is not a Fock space.

I have a different understanding of what this paper says. It is best captured by the abstract itself:

It is demonstrated that the ”dressed particle” approach to relativistic local
quantum field theories does not contradict Haag’s theorem. On the contrary,
”dressing” is the way to overcome the difficulties revealed by Haag’s theorem."

The mistake in your experimental interpretation of your formalism is that you take as the physical Hilbert space the Fock space rather than its image under W. This gives all your ''physical'' operators a touch of surrealism revealed by the strange deviations from standard relativity reported e.g. in your Section 11.4. (E.g., at the bottom of p.437, you talk about a non-covariant boost transformation law - a contradiction in itself under the usual interpretation.)

As I said above, in my approach the Fock space is *not* transformed by W. You are right that the non-covariant boost transformation law is one of the most important conclusions of my work. If you found a specific mistake in the derivation of this law, I would appreciate your pointing it to me. In subsection 11.3.1 I claim that existing derivations of Lorentz transformations in special relativity are not correct, because they disregard interactions between particles. If you see a hole in my arguments, please let me know.

Eugene.

 

[A. Neumaier]

Particle states are not transformed, so the Fock space structure remains the same as in the free theory. Second, if the dressing operator e^{i \Phi} were non-unitary I would get a non-Hermitian dressed Hamiltonian as a result.

The first statement is the same as what Shirokov does. Therefore, the second statement is true in a sharpened form: By (Haag's theorem or) Shirokov's remark (p.4 bottom)

the expression exp(R(a^*a) is not an operator, as it fails to map vectors of the Hilbert space in the Fock representation of operator alpha to vectors in the same space

your e^{iPhi} is undefined (except as a formal power series). Thus your dressed Hamiltonian is also undefined (except as a formal power series), and any claims about preserving properties based on the unitarity od e^{iPhi} are vacuous.

As I said above, in my approach the Fock space is *not* transformed by W.

Precisely, and this is the reason why you work in an unphysical representation that loses both the covariant meaning of the field coordinates and all field information. It is the source of all the strange things you report in Chapter 11.

You are right that the non-covariant boost transformation law is one of the most important conclusions of my work. If you found a specific mistake in the derivation of this law, I would appreciate your pointing it to me. In subsection 11.3.1 I claim that existing derivations of Lorentz transformations in special relativity are not correct, because they disregard interactions between particles.

They don't disregard anything since they stick to the physical representation (on the image of Fock space under W) rather than insisting (without sufficient reasons) that Fock space is physical.Your most important conclusion is based on this false identification of the physics.

Note that relativity was discovered through the Maxwell equations, and until after QED was first conceived, it was (apart from the single other observed fact - the mercury anomalies) the only reason to look for a marriage between quantum mechanics and relativity. The arguments of the maxwell fields have the traditional physical interpretation essential to give relativity a meaning. And the meaning of the arguments is preserved in the standard formulation of QED, while it is mutilated in your approach.

 

[meopemuk]

your e^{iPhi} is undefined (except as a formal power series). Thus your dressed Hamiltonian is also undefined (except as a formal power series), and any claims about preserving properties based on the unitarity od e^{iPhi} are vacuous.

I have stated repeatedly that I don't claim a non-perturbative solution. Everything I do is within perturbation theory, and only few lowest orders have been explored explicitly.

Precisely, and this is the reason why you work in an unphysical representation that loses both the covariant meaning of the field coordinates and all field information. It is the source of all the strange things you report in Chapter 11.

They don't disregard anything since they stick to the physical representation (on the image of Fock space under W) rather than insisting (without sufficient reasons) that Fock space is physical.Your most important conclusion is based on this false identification of the physics.

Discussion in Section 11.2, where I derive the non-covariance of boost transformations, is completely unrelated to the Fock space or dressing transformations or other subtle matters. This discussion even doesn't require quantum mechanics, because all arguments can be repeated for classical particles moving along trajectories. The only essential thing is that there exists an interacting representation of the Poincare group (in the Hilbert space or in the phase space). In this representation the generator of boosts is interaction-dependent (see Weinberg's (3.3.20)). From this it follows immediately that boost transformations cannot have the same universal covariant form as in the non-interacting case.

Eugene.

 

[A. Neumaier]

I have stated repeatedly that I don't claim a non-perturbative solution. Everything I do is within perturbation theory, and only few lowest orders have been explored explicitly.

Then why bother with Haag's theorem at all? It says nothing in perturbation theory. Also the distinction between Fock representations and non-Fock representations does not even exist in perturbation theory. So you should stop claiming anything about that all states are representable in Fock space! They are so only in perturbation theory.

But perturbation theory is known to be very inadequate for particle physics - with exception of the processes where all external lines are massive elementary particles. QED is not among these theories, since the photon is massless. Even there, you need some nonperturbative tricks such as a resummation of propagators, in order to get the correct form factors and self-energies (which are apparently missing in your perturbative treatment).

Thus if you want to claim to have a theory of QED, you need to go beyond perturbation theory and beyond Fock space!

Discussion in Section 11.2, where I derive the non-covariance of boost transformations, is completely unrelated to the Fock space or dressing transformations or other subtle matters. This discussion even doesn't require quantum mechanics, because all arguments can be repeated for classical particles moving along trajectories.

I didn't spent enough time on the details of your theory to figure out where exactly is the mistake. (I am not interested in understanding the details of a theory whose conclusions violate so obviously the demands of relativity.)

The no-go theorem by Currie, Jordan, and Sudarshan proves that the multi-particle view for classical point particles is inconsistent with relativity. You comment this on p.418 with ''Of course, it is absurd to think that there are no interactions in nature.'' But it only proves the absence of point particles, not the absence of interactions between fields. Your conclusion on p.419, ''for us the only way out of the paradox is to admit that Lorentz transformations of special relativity are not applicable to observables of interacting particles'' may satisfy you, but it doesn't satisfy anyone who understood the meaning of covariance. A field theory has no need to make such queer assumptions.

The only essential thing is that there exists an interacting representation of the Poincare group (in the Hilbert space or in the phase space).

And your existence claim for this representation in the quantum case is solely based on the unitarity of the operator e^{iPhi}, that, according to Shirokov (or Haag) doesn't exist.

 

[A. Neumaier]

Note that relativity was discovered through the Maxwell equations, and until after QED was first conceived, it was (apart from the single other observed fact - the mercury anomalies) the only reason to look for a marriage between quantum mechanics and relativity. The arguments of the Maxwell fields have the traditional physical interpretation essential to give relativity a meaning.

See post #27 in https://www.physicsforums.com/showthread.php?t=474571

 

[meopemuk]

Thus if you want to claim to have a theory of QED, you need to go beyond perturbation theory and beyond Fock space!

First, we agreed that my approach is *not* a field theory. The entire Part II of the book has the title "The quantum theory of particles". So, it is not the same as QED. The claim is that within this particle-based formalism it is possible to reproduce the same experimental observations as in QED and even more (e.g., the time evolution). I understand that this claim sounds hollow, because I haven't presented a single loop calculation within my approach. This is related to infrared difficulties, as we discussed earlier.

Second, I am not going to back away from the Fock space, because this structure of the Hilbert space (with orthogonal n-particle sectors etc.) is postulated in my approach. I don't see any fundamental problem in representing infinite number of soft photons in the Fock space. This is not a bigger problem than placing the infinity on a real line. We've talked about that already.

(I am not interested in understanding the details of a theory whose conclusions violate so obviously the demands of relativity.)

I am pretty sure that my conclusions do not violate the "principle of relativity", i.e., the equality of all inertial frames. I am also sure that my conclusions do violate conclusions of Einstein's special relativity, such as the covariance, Minkowski space-time, etc. However, there is no logical contradiction, because covariance and Minkowski space-time were derived in special relativity by using a tacit assumption of the absence of interactions. I am working in the interacting case. I go through all these arguments in detail in Chapter 11.

And your existence claim for this representation in the quantum case is solely based on the unitarity of the operator e^{iPhi}, that, according to Shirokov (or Haag) doesn't exist.

I would like to steer our discussion away from things like the difference between Hermitian and self-adjoint operators. The existence of the unitary dressing transformation e^{iPhi} is not the only foundation of my approach. Actually, this transformation serves only as a demonstration of the connection betwen the old theory (QED) and the new approach. But this new approach can stand on its own as well. For its viability the only important thing is the existence of 10 Poincare generators with usual commutators. These commutators have been proven in Appendix O. And this proof is completely unrelated to the issue of self-adjointness of Phi.

Eugene.

 

[A. Neumaier]

First, we agreed that my approach is *not* a field theory. The entire Part II of the book has the title "The quantum theory of particles". So, it is not the same as QED.

It is even designed as a theory to _replace_ field theory,: On p.353 you write:''The position taken in this book is that the presence of infinite counterterms in the Hamiltonian of QED H^c is not acceptable and that the Tomonaga-Schwinger-Feynman renormalization program was just a first step in the process of elimination of infinities from quantum field theory. In this section we are going to propose how to make a second step in this direction: remove infinite contributions from the Hamiltonian H^c and solve the paradox of ultraviolet divergences in QED.'' This is a claim that you have something _better_ than QED, not just something _different_.

If it should not be a replacement for QED, why do you argue so heavily against QED? But you beat a straw man only:

You begin it with Section ''10.1 Troubles with renormalized QED'', although QED has none of the trouble you have to fight with. On p.349 you introduce a straw man with the words ''The traditional interpretation of the renormalization approach is that infinities in the Hamiltonian H^c (9.38) have a real physical meaning. The common view is that bare electrons and protons really have infinite masses ˜m and ˜M and infinite charges +-~e''. But this is far from the traditional interpretation, which (in the more careful treatments) declares all bare stuff to be devoid of meaning, and the infinities to be a limit that is to be taken only at the end of the renormalization calculations. Moreover, working with a cutoff Lambda and taking m,M,e finite but large gives formulas that are essentially as accurate as the infinite limits, and certainly as accurate as you low order approximations.

Another strawman is introduced on p.350: ''traditional QED predicts rather complex dynamics of the vacuum and one-particle states.'' This is not the case; the vacuum is stable under the renormalized dynamics, and the space of all 1-particle states is invariant under the dynamics in the (formal) Wightman representation. You get things wrong since you consider the dynamics of unphysical bare vacuum and 1-particle states, which no quantum field theorist considers.

Your complaints about the finite-time behavior of QED on p.352 are unfounded, as we have discussed already. Lots of applications of QED on the kinetic or hydrodynamic level are time-dependent, and consistent with experiments.

On p.383, you write: ''In spite of its dominant presence in theoretical physics, the true meaning of QFT and its mathematical foundations are poorly understood.'' This is not ttrue; except if you talk about your personal understanding (but then you should say so). The mathematics of perturbative QED is perfectly understood, in full mathematical rigor (see Steinmann's book ''Perturbative quantum electrodynamics and axiomatic field theory''). What you offer as replacement is much is less rigorous. As to the true meaning - this is a subjective term -- if you think your noncovariant conclusions are in any reasonable sense truer than the covariant conclusions of QED, you are mistaken.

On p.384, you write ''If (as usually suggested) fields are important ingredients of physical reality, then we should be able to measure them. However, the things that are measured in physical experiments are intimately related to particles and their properties, not to fields. For example, we can measure (expectation values of) positions, momenta, velocities, angular momenta, and energies of particles as functions of time (= trajectories). [...] All these measurements have a transparent and natural description in the language of particles and operators of their observables.'' But you ignore that the routine measurements of the macroscopic electromagnetic field are measurements of (expectation values of) the fields E(x) and B(x), on precisely the same par as the observables you list. None of these measurements has a transparent and natural description in the language of particles and operators of their observables.

You call the latter ''very questionable. When we say that we have “measured the electric field” at a certain point in space, we have actually placed a test charge at that point and measured the force exerted on this charge by surrounding charges.'' But when we measure the momentum of a particle in a scattering experiment, we have actually placed a wire chamber in its way, together with a magnetic field, and measured the energy deposited on the wires, from which we calculated the curvature of the track and deduced the momentum at the time the particle entered the chamber. - And you forget to question this indirect measurement. Measurement of angular momentum or the energy of bound states are also measured quite indirectly, and hence should be very questionable according to your standards!

Thus - a double standard wherever one looks. You praise your achievements, silently glossing over their weaknesses, and you magnify the problems of QED to an extent where you can't hope to get agreement from anyone who understands the matter better than you do.

The claim is that within this particle-based formalism it is possible to reproduce the same experimental observations as in QED and even more (e.g., the time evolution). I understand that this claim sounds hollow, because I haven't presented a single loop calculation within my approach. This is related to infrared difficulties, as we discussed earlier.

It is hollow, and sounds so. No computation of form factors, or of self energies, or of the lamb shift, or of the anomalous magnetic moment of the electron. What you calculated is almost disjoint of what is calculated (and compared with experiment) in the usual books.

Note that in the standard treatment (e.g., Peskin & Schroeder (6.59)), the computation of the anomalous magnetic moment to lowest nontrivial order is free of infrared problems.
If you run into infrared problems there, these are introduced by your problematic dressing transform.

Second, I am not going to back away from the Fock space, because this structure of the Hilbert space (with orthogonal n-particle sectors etc.) is postulated in my approach. I don't see any fundamental problem in representing infinite number of soft photons in the Fock space. This is not a bigger problem than placing the infinity on a real line. We've talked about that already.

Talked yes, but not substantiated.

I would like to steer our discussion away from things like the difference between Hermitian and self-adjoint operators. The existence of the unitary dressing transformation e^{iPhi} is not the only foundation of my approach. Actually, this transformation serves only as a demonstration of the connection between the old theory (QED) and the new approach.

No. It is the _basis_ of your whole approach. Without the e^{iPhi} conjugation, you don't even have a Hamiltonian to start with. You borrow the theory and the S-matrix from standard QED, and then you transform the nice, covariant results of the latter into a mutilated version that has all sorts of noncovariant features.

 

[meopemuk]

You begin it with Section ''10.1 Troubles with renormalized QED'', although QED has none of the trouble you have to fight with. On p.349 you introduce a straw man with the words ''The traditional interpretation of the renormalization approach is that infinities in the Hamiltonian H^c (9.38) have a real physical meaning. The common view is that bare electrons and protons really have infinite masses ˜m and ˜M and infinite charges +-~e''. But this is far from the traditional interpretation, which (in the more careful treatments) declares all bare stuff to be devoid of meaning, and the infinities to be a limit that is to be taken only at the end of the renormalization calculations. Moreover, working with a cutoff Lambda and taking m,M,e finite but large gives formulas that are essentially as accurate as the infinite limits, and certainly as accurate as you low order approximations.

By referring to calculations with finite cutoff Lambda and very large m,M,e you basically confirm my statement that standard approaches use the idea of infinite (or very large, which is basically the same) particle parameters. Just after this quote I go on to explain the usual idea of physical particles as linear combinations of bare particle states. Perhaps you are right that different textbooks use different philosophies on whether bare and virtual particles should be regarded as something real or imaginary. So, I should avoid words like "traditional interpretation" and "common view". I will replace them with "In some textbooks", "sometimes", etc.

Another strawman is introduced on p.350: ''traditional QED predicts rather complex dynamics of the vacuum and one-particle states.'' This is not the case; the vacuum is stable under the renormalized dynamics, and the space of all 1-particle states is invariant under the dynamics in the (formal) Wightman representation. You get things wrong since you consider the dynamics of unphysical bare vacuum and 1-particle states, which no quantum field theorist considers.

Thanks, I will replace this phrase with ''traditional QED predicts rather complex dynamics of *bare* vacuum and one-particle states.'' Yes, I agree that dynamics of bare states is useless. But, unfortunately, the Hamiltonian of QED is formulated explicitly in terms of bare particle operators only. So, it is not possible to describe the time dynamics of physical particles without special tricks. Moreover, when quantum field theorists calculate scattering amplitudes they are pretty happy to identify states created by bare creation operators as real observable particles. Perhaps, axiomatic QFT can deal with these problems nicely, but these things are not explained in textbooks, like Weinberg.

Your complaints about the finite-time behavior of QED on p.352 are unfounded, as we have discussed already. Lots of applications of QED on the kinetic or hydrodynamic level are time-dependent, and consistent with experiments.

I am not convinced about that. If dynamical solution does not exist even for the simplest 2-particle interacting system, then how one can be sure about the correctness on the more complicated kinetic or hydrodynamic level?

Ideally, I would like to see a time-dependent wave function for a system of two slowly colliding particles obtained in QED from first principles. This solution should satisfy unitarity, agree with simple QM and classical solutions, yield the same scattering probabilities as the S-matrix approach. There will be approximations, for sure, but they must be clearly justified. Then I will be convinced that QED can describe the time evolution.

On p.383, you write: ''In spite of its dominant presence in theoretical physics, the true meaning of QFT and its mathematical foundations are poorly understood.'' This is not ttrue; except if you talk about your personal understanding (but then you should say so). The mathematics of perturbative QED is perfectly understood, in full mathematical rigor (see Steinmann's book ''Perturbative quantum electrodynamics and axiomatic field theory''). What you offer as replacement is much is less rigorous. As to the true meaning - this is a subjective term -- if you think your noncovariant conclusions are in any reasonable sense truer than the covariant conclusions of QED, you are mistaken.

Thanks. I will replace "are poorly understood" with "remain controversial". The quotes from Wilczek and Wallace ephasize this controversy.

On p.384, you write ''If (as usually suggested) fields are important ingredients of physical reality, then we should be able to measure them. However, the things that are measured in physical experiments are intimately related to particles and their properties, not to fields. For example, we can measure (expectation values of) positions, momenta, velocities, angular momenta, and energies of particles as functions of time (= trajectories). [...] All these measurements have a transparent and natural description in the language of particles and operators of their observables.'' But you ignore that the routine measurements of the macroscopic electromagnetic field are measurements of (expectation values of) the fields E(x) and B(x), on precisely the same par as the observables you list. None of these measurements has a transparent and natural description in the language of particles and operators of their observables.



You call the latter ''very questionable. When we say that we have “measured the electric field” at a certain point in space, we have actually placed a test charge at that point and measured the force exerted on this charge by surrounding charges.'' But when we measure the momentum of a particle in a scattering experiment, we have actually placed a wire chamber in its way, together with a magnetic field, and measured the energy deposited on the wires, from which we calculated the curvature of the track and deduced the momentum at the time the particle entered the chamber. - And you forget to question this indirect measurement. Measurement of angular momentum or the energy of bound states are also measured quite indirectly, and hence should be very questionable according to your standards!

This is rather philosophical dispute, which we are not going to resolve easily. As you said, the war between particles and fields goes on for centuries. Of course, many measurements are done indirectly. My point is that when we happen to measure something in the most direct way, like the photon blackening a grain of photoemulsion, we always see countable indivisible particles.

Thus - a double standard wherever one looks. You praise your achievements, silently glossing over their weaknesses, and you magnify the problems of QED to an extent where you can't hope to get agreement from anyone who understands the matter better than you do.

Hopefully, with your help I'll make the presentation in the book less abrasive.

Note that in the standard treatment (e.g., Peskin & Schroeder (6.59)), the computation of the anomalous magnetic moment to lowest nontrivial order is free of infrared problems.
If you run into infrared problems there, these are introduced by your problematic dressing transform.

This is actually a good idea! Why didn't I think about it before? I can calculate this part of the charge-charge dressed potential separately since no infrared infinities should be involved.
Thank you very much.


Eugene.

 

[A. Neumaier]

Thanks, I will replace this phrase with ''traditional QED predicts rather complex dynamics of *bare* vacuum and one-particle states.''

But this is not true either. QED predicts nothing at all about bare states, since these disappear completely during renormalization. In the limit where the cutoff is removed, the bare stuff no longer exists in any meaningful sense!

The predictions of QED (or any other QFT) solely concern expectation values and time-ordered expectation values of products of the physical (=renormalized) Heisenberg fields. This (and only this) is what contains the physics, and this (and only this) can be compared with experiment!

Yes, I agree that dynamics of bare states is useless. But, unfortunately, the Hamiltonian of QED is formulated explicitly in terms of bare particle operators only.
No. This is only the starting point, not the final Hamiltonian.

Your theory begins with the same starting point and then performs nonexisting ''unitary transformations'' in order to translate the bare, ill-defined stuff into something perturbatively well-defined (ignoring infrared problems, which prove the lack of self-adjointness of your Fock space Hamiltonian). You then regard _this_ as the ''real'' theory - the other stuff was just scaffolding to be thrown away after you have the Hamiltonian.

In the same way, standard QED begins with the same starting point and then performs renormalization calculations in order to translate the bare, ill-defined stuff into something perturbatively well-defined. _This_ is then regarded as the ''real'' theory - the other stuff was just scaffolding to be thrown away after you have the Hamiltonian.

What you accept as only a pretext for your theory should also be treated by you as only a pretext for standard QED. This is how the bare stuff is viewed by the experts, and to be fair, you should view it in the same way.

So, it is not possible to describe the time dynamics of physical particles without special tricks.

The tricks are no worse than applying your nonexisting operator exp{iPhi} to get what you take as your basis.

Moreover, when quantum field theorists calculate scattering amplitudes they are pretty happy to identify states created by bare creation operators as real observable particles.

_Nobody_ is doing that.

Perhaps, axiomatic QFT can deal with these problems nicely, but these things are not explained in textbooks, like Weinberg.

How wrong you are! Maybe you haven't noticed that because of your aversion to filed theoretic methods (which you denounce as mere mathematical tricks), but it is in every textbook where the LSZ formula is derived. For example, in Weinberg, this is handled in Section 10.2.

Note that on p.430, Omega_0 is the true vacuum, not the bare one, and the A_i are the renormalized Heisenberg fields with space-time arguments. The latter act on the physical Hilbert space spanned by the states of the form A_1 ... A_n Omega_0, though Weinberg doesn't emphasize this explicitly. But one can see it from the fact that he takes matrix elements between such states (_not_ between bare states!). This is _precisely_ the recipe that I had given in my explanation of the Wightman approach to QFT. Wightman didn't take his approach from nowhere, but only isolated the minimal stuff from the usual, nonrigorous treatment that one would have to make clear mathematical sense of in order to have a rigorous, nonperturbatively defined theory.

I am not convinced about that. If dynamical solution does not exist even for the simplest 2-particle interacting system, then how one can be sure about the correctness on the more complicated kinetic or hydrodynamic level?

Dynamical solutions exist for the whole physical Hilbert space, and the field operators from which the kinetic and hydrodynamic equations are derived act on this space! Iit is just that the notion of a 2-particle system is no longer well-defined, except asymptotically.

Ideally, I would like to see a time-dependent wave function for a system of two slowly colliding particles obtained in QED from first principles.

It is you who is proposing a dynamical particle view of QED; so it is your obligation to substantiate that picture. Thus you have to study standard QED well enough that you can make a reasonable proposal for what in this standard framework the state of two slowly colliding particles should be. Then you get its evolution for free. The mainstream view is that the dynamics is a dynamics of fields, and particles exist only as asymptotic bound states. This is the version in which QED makes sense. Particles at finite times only exist in an approximate sense.

This solution should satisfy unitarity, agree with simple QM and classical solutions, yield the same scattering probabilities as the S-matrix approach. There will be approximations, for sure, but they must be clearly justified. Then I will be convinced that QED can describe the time evolution.

I gave you the construction of the Hilbert space and the spanning sets between which one computes the S-matrix elements. You can read Haag-Ruelle theory to find out how in simple cases (which apply to massive QED) the asymptotic particle states are constructed, and then guess from this the form of the approximation you need to make to get what you want.

Thanks. I will replace "are poorly understood" with "remain controversial". The quotes from Wilczek and Wallace emphasize this controversy.

This is foul play. If you criticize QED because it has no mathematically rigorous formulation so far, you must criticize your own theory for the same reason. Since you excuse your own theory from this demand, you have no moral right to call the understanding of QED controversial. The controversies are _only_ about the question how rigorous QED can be made. But _nothing_ about the experimental content of the theory is controversial!

This is rather philosophical dispute, which we are not going to resolve easily. As you said, the war between particles and fields goes on for centuries. Of course, many measurements are done indirectly.

You argue that momenta or angular momenta (where your theory happens to have observables) be truly observable, while electromagnetic fields (for which your theory has no observables) to be very questionably observable, though both require about the same degree of indirectness. What you actually write (since you hide the momentum indirection) is a very unfair and biased argument that no one will buy who has only a moderately realistic view of how actual measurement must be done.

My point is that when we happen to measure something in the most direct way, like the photon blackening a grain of photoemulsion, we always see countable indivisible particles.

So the position of photons is measurable in the most direct way, but you don't even have an observable for it in your theory! This shows that the observables in a theory and the naive intuition about measurements diverge quite radically! And as discussed in the other thread, you can never measure a photon while it is alive! This again shows the same thing!

Hopefully, with your help I'll make the presentation in the book less abrasive.

You'll not be able to hide the wolf (a faulty interpretation of QED and a faulty view of covariance) in sheep's clothing (aka less abrasive presentation). The second part of your book needs important corrections in the contents, not only in the presentation!

This is actually a good idea! Why didn't I think about it before? I can calculate this part of the charge-charge dressed potential separately since no infrared infinities should be involved.

If you like this sort of advice, I have two pieces more:

1. The photon self-energy is infrared finite to first nontrivial order; see Weinberg (11.2.16) and (11.2.22).

2. Why don't you postulate that the photon has a tiny mass? This is experimentally indistinguishable from real QED, and has a number of advantages:
-- Massive photons have a position operator, and hence a fully adequate Schroedinger picture. This would make your philosophical position much better grounded.
-- Massive photons save you from all infrared problems. Without the infrared problems, Fock space is perturbatively fully adequate, and all my criticism regarding the IR problem and wrong asymptotics is no longer applicable.
-- With massive photons, you can calculate radiative corrections to Compton scattering and get a finite result for the Lamb shift.