What is the origin of QFT "difficulties": "physical" or "mathematical"?

Question

Indeed, there is a 'currently" dominating "opinion" that relativistic QFT is "dictated" by this and that, so it is unavoidable that we encounter UV and IR difficulties. This and that are considered as ultimately established things amongst which there are Poincaré invariance, locality, etc., etc.

And in my opinion, there is still a room of improving the QFT initial states and interactions so that no UV and IR difficulties may appear. I propose you to express your opinion whether the current QFT is (are) satisfactory. Especially it concerns "gauge" interactions.

My question is motivated with mathematical approaches to QFTs like algebraic, axiomatic, etc. approaches. As to me, I think we have already sufficient experience in obtaining the final QFT results that may help us construct a theory without "difficulties".

Comments

Poincare invariance is what defines relativistic physics in general, whether classical or quantum. It is not just an opinion but a demarcation criterion.

---

@ArnoldNeumaier: I have nothing against Poincare invariance, not at all. I think it is just insufficient for uniqueness of a QFT.

---

Poincare invariance (i.e., ''relativistic'') and locality (joined preparability of fields at spacelike distances) are basic requirements for physical reasons.

Together they already rules out (by Haag's theorem) interactive theories on a Fock space, and hence a rigorous dynamic particle picture. The latter must therefore always be a coarse-grained approximate view.

They therefore also imply that some sort of renormalization is needed for all such theories that start with a Fock space structure. 

---

@ArnoldNeumaier: Good! Now tell me to what the final finite results belong. Do we keep the Fock language (interpretation) for final results?

---

If the final results are partition functions they refer to expectations of macroscopic variables, not to particles. The results are time-independent, hence no dynamical information is available. 

If the final results are correlation functions (as needed for relativistic kinetic theory), they refer to fields at finite times. This provides dynamical information.

If the final results are nonequilibrium expectation values (as needed for relativistic hydrodynamics), they also refer to fields at finite times This also provides dynamical information.

If the final results are S-matrix elements, they refer to the asymptotic Fock spaces at times $\pm\infty$, not to finite time behavior, hence not to dynamical information. On this level, a particle description is possible, but not a dynamical description.

---

@ArnoldNeumaier: Thank you for your response. So we cannot get rid of particles, it is our language. Currently we stick to something (wrong interaction) that does not allow us to describe the amplitude evolutions for particles at finite times. Maybe the situation is not so hopeless, however.

---

There is nothing wrong with current QFT. The particle picture is reserved exclusively for interpreting collision experiments, but dynamically, only the fields figure. This has to be so for purely mathematical reasons.

Renormalization is not something wrong; it is the technique for getting the correct physical limits in the presence of distribution-valued operators. This is very well (and rigorously) understood in 2 spacetime dimensions, where many Poincare-invariant QFTs are rigorously known. In 4 space-time dimensions, their rigorous construction is (in a special case) one of the Clay millennium problems, a problem believed to be difficult but tractable.

---

@ArnoldNeumaier: Here you imply a certain way of interaction in which the fields are multiplied; thus "difficulties". And I insist that this implicit thing must be made explicit in order to not fool ourselves. Then, maybe, we could write another "interaction" term whose contributions are just right from the very beginning and time evolution will be defined at any time, not only asymptotically.

---

In 2D it is very explicit and nobody is fooling anyone since it is fully rigorous. In 4D things are technically more complex but it is very unlikely that there is a return to a particle picture that is already absent in the well understood 2D case. 

---

I do not understand why you use 2D cases, they are not representative, to tell the truth rigorously. We are interested in our (3+1) case with particles, as we observe them. The way we write the interaction is wrong and it is seen easily without any scattering. Just take a free electron and couple it to the EMF field. First of all, they are always coupled in nature and they must be such in our theories. Instead, we treat "our interaction" perturbatively whereas this coupling is strong and cannot be neglected, especially for soft modes. Thus I see a way of improving our understanding of how to proceed: at least the soft modes must be taken into account (coupled) exactly already in the initial and final states. You cannot exclude that this new initial/final (coupled) states might be better choice for improving the theory.

---

I use 2D cases to point out analogies. If something is the case already in 2D, a situation much better understood, there is no reason to expect that the situation in 4D should be different, or at least no reason that it should be less complicated.

If one finds something that works correctly (i.e., respecting Poincare invariance) in 4D then it is almost certain that something analogous works in all lower dimensions. Hence it must reproduce the known theory in 2D. Thus one can learn from the known 2D stuff about necessary side conditions for the 4D case. This is the way much of the theoretical progress was obtained that was made in elementary particle physics.

This is quite analogous to the situation that one studies new techniques for N-particle systems by first looking at what happens to 2-particle and 3-particle systems, for which the technical difficulties are less severe than for the general case. 

---

@ArnoldNeumaier: I disagree. Let us suppose that we have something (a QFT) working in (3+1) dimensions. In order to obtain a (2+1) dimension theory from it, we must suppose that everything is uniform in our 3+1 QFT along one of space directions. This supposition changes completely the (3+1) physics we had before. Points become lines, lines become planes, lines become volumes, etc.

---

No. If something works in 4D = 1+3 D we can do something completely analogous in 2D = 1+1 D, which respects all the mathematics although the physics is different (namely simplified). The result must be something that conforms to what is already known in 2D. 

---

There is no "completely analogous" since there are many ways of writing a 2D from 4D case. You may respect something inessential and miss something essential. More particularly, you may still consider strong (decisive) interaction "perturbatively", as if it were a small addendum to something physical. Thus you may miss the essential physics of how coupling must be done in order to have a normal theory in 4D.

---

Oh yes, one may miss something because in lower dimensions the physics is simpler. But whatever one retains must be conform to what is known in 2D, and hence to a field picture rather than a particle picture.

---

Physics is not simpler, but different qualitatively and quantitatively. That is why it is not representative. I am speaking of an essential interaction and you are deaf here since you are speaking of fields as if there were no problem of coupling.

---

But I speak the consensus of physicists while you just voice your personal opinion. 

---

I prove my conviction ("opinion") in my papers. It is something. And your "consensus" is not a proof, Arnold.

---

You prove nothing at all in the relativistic context since your toy theories don't respect Lorentz invariance. Whereas the consensus of the physicists (embodied in the standard model) is corroborated by a huge number of experiments. 

---

Quantization of relativistic QED is done via quantization of harmonic oscillators, but nobody blames QED for the harmonic oscillators being formally non relativistic. I proceed in the same way. And a consensus about SM is not the same thing as a consensus about 2D QFT.

---

Many of the insights used in the standard model (instantons, theta angles, etc.) came from preliminary investigations in 2D QFT.

A single harmonic oscillator is not relativistic but an infinite family parameterized by a dispersion relation of the form $p^2=m^2$, as used in QFT, is. 

---

I would like to attract your attention to the main question of my post, namely to methodological (or physical) errors admitted while constructing theories (equations). Strongly coupled in nature things we consider as decoupled or coupled differently, thus we encounter conceptual and mathematical difficulties.

---

Your main methodological error is that you dismiss the fact that the (Lorentz invariant) standard model passes almost all experimental tests with flying colors while your proposed alternative is nothing but a sketch of a hope, and doesn't accommodate Lorentz invariance, the most important requirement for a relativistic theory.

You also mistake the (free QFT) scaffolding for the (renormalized) final theory. It is like mistaking the power functions for the exponential function, since the latter can be defined as a power series.

The standard model is strongly coupled, due to renormalization and the summing of contributions from infinitely many Feynman diagrams, resulting, e.g., in poles in S-matrix elements indicating bound states. The latter are proof that strong coupling exists and is accounted for,.

---

I dismiss and mistake nothing, do not lye about me.

---

I don't lie. You asked about opinions, not about facts. Fact is that the standard model explains almost everything there is to be explained, in spite of your opinion that it contains methodological errors.

The standard model needs better mathematical foundations, but these must be based on basic physical principles such as Lorentz invariance, locality, and gauge invariance, and not on non-invariant ad hoc Hamiltonians (suggested by nonrelativistic toy models without taking into account the wealth of experimental data, including particle creation and crossing symmetry). 

---

Yes, you lye about me. I clearly point out errors. And I never proposed something as a final QFT, but outlined a better construction, in my opinion. My toy models should make you think critically about methodological approach practiced in QFT because they repeat those errors explicitly. They are not nothing, but something valuable.

---

You lie about yourself. Where is there any error in causal perturbation theory? The latter never encounters any infinity and produces all results of renormalized perturbative QFT. 

---

Your causal perturbation theory repeats the errors of the regular QFT since it starts from the same initial approximation and has the same interaction term. "Distribution splitting", i.e., making zero from infinity, is equivalent to subtraction (or renormalization) in the regular QFT.

---

You didn't read the material thorougly enough.

Causal perturbation theory (not my theory!) has neither mathematical nor physical errors - it is mathematically rigorous and in agreement with the experimental results. 

Distributions are the correct mathematical way to deal with singularities whose sloppy treatment by physicists would result in infinities. Distribution splitting is a rigorous mathematical technique from the functional analysis of partial differential equations. Nowhere any infinity is encountered, since only mathematically permitted operations are performed, with finite results.

Causal perturbation theory does not start with a free theory but with axioms for the S-matrix on the asymptotic Hilbert space, which is well-known to be a Fock space, even for an interacting theory. The meaning of the interaction term is conceptually very different from that in the Lagrangian approach since the coupling constant is the physical electron charge and not a bare, ill-defined number as in the Lagrangian approach. 

Thus nothing of your critique applies. 

---

You yourself recognize that there is the same IR problem in the causal perturbation theory. The physical events whose probability is equal to unity are absent in the first approximation, say, in Bhabha scattering. The essential coupling to the soft modes is not done, and this is one of my severe blames advanced against such an erroneous approach.

Any QFT with a cutoff is finite, but it gives undesirable corrections to the known constants due to wrong self-action in the total interaction, thus subtractions are in order.

Any normal physical theory is (and should be) causal by its definition, so no "microlocal analysis" is necessary to obtain physically reasonable corrections to the physically reasonable initial approximations.

---

In theories without massless bare particles there is no infrared problem, and causal perturbation theory resolves everything on the perturbative level. In particular, your critique of the distribution splitting is unfounded. Note that your friend Dirac objected only against the infinities in the UV renormalization, which are handled by causal perturbation theory in a way Dirac would most likely approve of. 

Infrared problems are present in QED, so that photons and electrons are infraparticles, which need to be treated with coherent states in place of plane waves as asymptotic states. But this has been treated long ago by Kulish and Faddeev in a covariant way, so there is no need to resort to your noncovariant methods. 

It is very unlikely that your approach will capture the (experimentally verified) Kulish-Faddeev approach since your particles are ordinary particles rather than infraparticles and lead to poles rather than the needed branch cuts.

Although I haven't seen a causal perturbation theory treatment of the Kulish-Faddeev approach (which is not completely rigorous) I don't expect this to require extensive changes; it just needs to be done by someone who has sufficient interest in the infrared properties and simultaneously knows enough about the microlocal properties of distributions to handle the mathematical technicalities. 

---

In QFT without massless particles the coupling must be done differently too. My electronuim is a compound particle with soft modes taken into account exactly, so the calculations reproduce zero for elastic scattering and quasi-coherent states (coherent states limited above due to finite energy available) like in Bloch-Nordsiek calculations.

Any Hamiltonian approach looks noncovariant, but the final results are covariant. It is not what makes a theory invalid, but the theory predictions.

---

Bloch-Nordsieck is relevant only in the presence of massless objects (e.g., electromagnetic fields). 

Your final results are not covariant - and your claims about a not yet and perhaps never existing extension of your toys are pure speculation. 

---

My "claims" are my proposals, not pure speculations. If we by construction exclude unnecessary self-action, there will be no unnecessary self-action contributions. If we by construction take into account coupling to the soft modes, there will always be soft radiation in scattering processes. You cannot prove that this way of proceeding is wrong, so do not call it a "pure speculation".

---

A proposal is just speculation if it doesn't address why it has a good chance to succeed, i.e., to account for relativity, which is the main problem in getting things right.

---

I never was against relativity explicitly or implicitly. It is a next step, which has not been developed yet.

---

Relativity is the only relevant step, as it is the obstacle that makes working with distributions and the associated complicated limits necessary. Thus your proposal misses precisely the part that would give it credibility. 

---

I disagree. There are many relativistic theories, which are good for nothing. The main thing is to catch correctly the physics of what we describe (and we describe interactions of infraparticles, if you like it better). There are many toy renormalization models, which are non relativistic, but considered useful, so it is not the only relevant feature.

But I agree, now I (or we) must try to construct a relativistic theory of electroniums. For that I need funds. Currently I am jobless (since April 2016) and I encounter all kinds of difficulties for an old person like me and for my own research subject. This subject (reformulation) is not a piece of cake by itself, but I do not find understanding amongst funding organizations. A very discouraging situation.

---

That renormalizatrion is also used in the nonrelativistic case is irrelevant for relativity theory. What counts is that Lorentz covariance forces singular 2-point functions (a rigorous result) and hence requires to correctly work with distributions, including the splitting procedures that are the mathematical way of doing this singular renormalization correctly.

To get funding, you need to convince the funding agencies that your idea is more than pure speculation. Without showing how to account for Lorentz invariance, the chances for funding are essentially zero. There are too many dead ends to fund an idea that looks like failing on this account. And your proposal seems to be failing there.

---

You write funny things. How can I convince somebody without having done this work? It will always look as a project (a "pure speculation") leaving a room for doubts. You yourself have told me many times that first I have to finish my theory and only after that people might listen to me.

---

I fully agree to your last comment, except that it is not funny but the reality.

You cannot convince anyone before you have done the most basic part of the work, namely showing how to reconcile your idea with relativity. You know this both from your own experience and from repeated feedback from others, including me.

You are like someone who proposes to build a high-rise building in a swamp (relativistic theory) by giving a toy blueprint erected on rock (nonrelativistic theory), together with the statement that you first need funding to find out how to solve the difficult problem of making the foundations stable under the swampy conditions. Only a fool would give you the funds.

You don't need to finish your theory but you need to provide the foundations that makes it plausible that you will able to finish it. The complete theory (for whose construction you probably would get funding) would fill a book in which it is shown that the foundations from which you start will reproduce everything from QED that is known to conform with experiment (which is a lot). But if the foundations are ill-conceived, nobody wants to invest into the proposed building. 

---

The foundations are ill-conceived even in the relativistic CED and QED. My non relativistic toy model is not that different from its relativistic extension, I hope, as a rock and a swamp. But people are already happy without my project. Nobody wants to take responsibility or bother themselves with supporting my project.

---

Relativistic QED is an elaborate building that stands since 1948, for almost 70 years, and produced among others the most accurate results that any physical theory ever produced. So people are convinced that whatever amendments the foundations may need to make them rigorous, they will be small, worthy of the effort, and preserve essentially everything of the superstructure erected on top of it, just made more rigorous. This is indeed what happened with the UV renormalization through causal perturbation theory, and there is every reason to expect that the same will happen for the infrared treatment.

On the other hand, your proposal is to throw away the whole building completely and replace it by something to be built based on a nonrelativistic toy model only, for which you cannot even point to arguments how it should be reconciled with Lorentz invariance. Saying ''I need funding to figure that out'' means that the problem is completely unsolved, and perhaps unsolvable!

This means that your proposal does not satisfy the most basic constraints that must be placed on a sound proposal. This is the reason why nobody wants to take responsibility or bother themselves with supporting your project.

---

Dear Arnold, my proposal is not "to throw away the whole building completely and replace it by something to be built based on a non relativistic toy model only ...". Why do you stick to this erroneous blame? By the way, many things were developed and studied under non relativistic approximation. In particular, W. Pauli and Fierz, T. Welton, etc., etc., worked in the non relativistic approximation.

Yes, saying: "I need funds to work that out" is the only and the right way of doing researches. All contracts foresee funding and nobody works without salary.

---

Your proposal is a multiparticle theory in which neither fields nor Lorentz invariance appears.Thus you throw away field theory and with it QED. You don't even reference (or learn from) any of the people who did past work on relativistic multiparticle theories because it does not fit your preconceived ideas. 

Nonrelativistic approximations are only an approximation, not the real, Lorentz invariant thing. People study them even today, but not as a substitute for real QED, but to better understand its implications.

---

Radiated field is present in my construction as a set of oscillators. I just exclude an unnecessary self-action term (self-induction effect, see a quote from Feynman below) and include coupling to soft field modes exactly. In CED and QED excluding (subtraction) and including (summation of soft modes contributions) is also done (otherwise the solutions are non physical), but it is done on the level of solutions rather than on the level of the original equations. I propose different equations, but with preserving (respecting) the known physics.

---

known physics includes Lorentz invariance. This is not included in your setting.

---

It is not yet included completely, to be exact. There is still a work to do, a hard work, not a piece of cake, but you are wrong with your being completely negative about my project. Note, small velocities and energies belong to physics too and the Lorentz transformations become the Galilelian ones.

---

Yes, and at small velocities and energies you have nonrelativistic quantum field theory, and no problems with quantum fields. People are doing already rigorous IR calculations in nonrelativistic field theories approximating QED, see, e.g., Dybalski's article in arXiv:1706.09057. Thus what you did is at best a variant of what exists already.

But the challenge is to get things correct in the presence of Lorentz invariance, and there you have no proposal at all, but only a wish to be funded to find something that might work. 

---

Arnold, constructing a relativistic theory is my explicit goal. In the end of my old paper I proposed a relativistic Hamiltonian (23), but I have not done any calculations. It is similar to the regular relativistic in all respects Hamiltonian in the canonic quantization in the Coulomb gauge, but contains some modifications. I tried to explicitly exclude self-action terms and take into account coupling to the soft modes with some (most probably wrong) corrections to get outside of dipole approximation. This Hamiltonian is most likely wrong, but it show my goal. You, on the other hand, insist that I stick to non relativistic and non Lorentz invariant theory which is not true. And asking for fund is a normal practice, stop blaming me for that!

---

Your (23) is not Lorentz invariant since the time generator H plays a distinguished role. Calling it relativistic doesn't change this severe defect of your proposal. You need to propose a representation of the Lorentz group, not just of the time translation subgroup!

---

My (23) is based on the regular Hamiltonian in the Coulomb gauge (formula (A25) in QED_Equations_by_Polubarinov). It is the oldest approach to building QED which is still applicable (see S. Weinberg, for example).

---

But unless you also give generators for space translations and the Lorentz transforms and show that they satisfy the commutation rules for the Lorentz group you have only a pseudo-relativistic setting. Simply making the dispersion relation relativistic is not enough.

Weinberg produces all generators, and finds that to get them one needs locality and all the field stuff that leads to renormalization in order to be well-defined and covariant. As you want to avoid the latter you are not allowed to refer to Weinberg as a justification of your approach.

You seem to lack all care for entering true relativistic calculations, which is the very essence of the problem of making QED better founded.

---

I do not lack of care. I have been too busy with carefully doing researches for others to make my living, so I have been lacking funds and freedom to work out everything I wanted in QED.

---

The missing care consists of talking about big projects and proposals without first making sure that there is a basis for it. It is known for a long time that once one tries to get the Lorentz generators behave correctly, all problems show up that you are currently avoiding to tackle. They are not solvable by simply accounting for photon clouds but only by taking into account all what is already known, including the renormalization stuff.

---

In other words, you insist on uniqueness of QED (and generally QFT) constructions despite evident errors pointed out by me and by other researchers. What is the use of all your "axioms" or requirements if en the end you get physically wrong results? This way does not work, that is why I advanced a physical basis for a better construction. This is exactly what my friend P. Dirac had been insisting on until his death. He had been insisting on searching better Hamiltonians and on avoiding an "axiomatic" approach. And my toy models should convince you that we may make errors in our constructions, so the question is whether we made such errors in our "realistic" cases or not.

---

The only error made in the naive discussion is to assume that the asymptotic states are given by plane waves, which is incompatible with massless fields. You simply take the naive version for the one physicists use when making real predictions, and hence you find ''errors'' that you (unlike Kulish and Faddev) don't cure but only play with using your toys. 

---

Wrong! I do not blame physicists for plane waves, but for bad coupling in the initial solution. You completely mislead our readers, Arnold.

---

<p>I don't care what you blame. The analysis of Kulish and Faddeev (and lots of other work) shows that the initial and final states must be coherent states, not plane waves. Once this is accounted for, the S-matrix elements exist, and what you consider a grave error disappears.&nbsp;</p>

---

No, you don't care, that's right. And my independent analysis and original construction ("electronium") show too that the finals states are quasi-coherent ones for soft photons. As well, as the "electronium" is a soft compound system, it is unlikely to prepare it in its "ground" state, i.e., again quasi-coherent states in the initial state. You completely missed this result in my approach. You spread lies about my approach and about me personally.