### Infrared infinities and bound states

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Few months back I promised to deliver a Lamb shift calculation within the dressed particle approach. Well, this appeared to be not so trivial. It was easy to deal with ultraviolet infinities. The dressed particle approach is designed to deal with them. However, the infrared infinities are really nasty. The fundamental point is that in order to use the dressed particle approach one needs to have a well-defined S-operator. It appears that standard QED does not provide a clean S-operator in the presence of massless photons. Strictly speaking, the asymptotic time evolution of colliding electrons is not free due to the long-range Coulomb effects, and the standard scattering theory is not working. The problem is easy to understand: In QED photons are massless, so, the minimum energy to create one photon is zero. When two charges interact, they can easily create an infinite number of "soft" photons. It is impossible to consider a clean interaction of two charges. You should always take into account all those parasitic photons. An infinite number of them!

One known way to deal with this issue is to redefine scattering theory by considering the long-range Coulomb interaction as a part of the free asymptotic Hamiltonian. This has been done by Kulish and Faddeev [1]. Then, one can formally define the S-operator. However, I am still not sure if this presription is unique and if I am allowed to fit my dressed particle Hamiltonian to this redefined S-operator. I had some ideas, but everything started to look so complicated...

I decided to take a simpler route. Let us forget for the moment about those nasty massless photons. There are respectable physical theories where all particles are massive. For example, in theory of nuclear forces one has protons, neutrons, and pions. There is a number of simple QFT models designed to describe nuclear forces. Possibly, the simplest one is the scalar Yukawa model. So, I decided to first apply my approach to this model. There are no infrared problems to worry about.

Of course, I will not get such a clean experimental number as Lamb shift. But getting radiative corrections to nuclear potentials is a worthy goal by itself. Moreover, there are tons of theoretical works in this field to compare with.

That's what I am doing now, i.e., the radiative corrections to the Yukawa potential. The paper is moving ahead rather well, and I don't anticipate any surprising difficulties. I'll keep you posted.

[1] P. P. Kulish and L. D. Faddeev, "Asymptotic conditions and infrared divergences in quantum electrodynamics", Theor. Math. Phys., 4 (1970), 745.

Few months back I promised to deliver a Lamb shift calculation within the dressed particle approach. Well, this appeared to be not so trivial. It was easy to deal with ultraviolet infinities. The dressed particle approach is designed to deal with them. However, the infrared infinities are really nasty. The fundamental point is that in order to use the dressed particle approach one needs to have a well-defined S-operator. It appears that standard QED does not provide a clean S-operator in the presence of massless photons. Strictly speaking, the asymptotic time evolution of colliding electrons is not free due to the long-range Coulomb effects, and the standard scattering theory is not working. The problem is easy to understand: In QED photons are massless, so, the minimum energy to create one photon is zero. When two charges interact, they can easily create an infinite number of "soft" photons. It is impossible to consider a clean interaction of two charges. You should always take into account all those parasitic photons. An infinite number of them!

One known way to deal with this issue is to redefine scattering theory by considering the long-range Coulomb interaction as a part of the free asymptotic Hamiltonian. This has been done by Kulish and Faddeev [1]. Then, one can formally define the S-operator. However, I am still not sure if this presription is unique and if I am allowed to fit my dressed particle Hamiltonian to this redefined S-operator. I had some ideas, but everything started to look so complicated...

I decided to take a simpler route. Let us forget for the moment about those nasty massless photons. There are respectable physical theories where all particles are massive. For example, in theory of nuclear forces one has protons, neutrons, and pions. There is a number of simple QFT models designed to describe nuclear forces. Possibly, the simplest one is the scalar Yukawa model. So, I decided to first apply my approach to this model. There are no infrared problems to worry about.

Of course, I will not get such a clean experimental number as Lamb shift. But getting radiative corrections to nuclear potentials is a worthy goal by itself. Moreover, there are tons of theoretical works in this field to compare with.

That's what I am doing now, i.e., the radiative corrections to the Yukawa potential. The paper is moving ahead rather well, and I don't anticipate any surprising difficulties. I'll keep you posted.

[1] P. P. Kulish and L. D. Faddeev, "Asymptotic conditions and infrared divergences in quantum electrodynamics", Theor. Math. Phys., 4 (1970), 745.

## 17 Comments:

Hi Eugene,

In my approach - which extends a method devised by Stueckelberg in 1934 - I find that a massless photon is not possible. Given the experimental bounds on the photon mass (< 10^-14 eV, I believe) this creates a hierarchy problem that puts the others in the shade (although, I am not convinced that hierarchies are necessarily problems). However, I don't think that I am wrong in saying that a photon mass is used in more canonical developments of QFT than yours or mine to solve exactly the problem you mention.

Hi Chris,

I think most people believe that photons are truly massless, though we may never know for sure.

Introducing a fictitious photon mass is a trick similar to regularization. It is used to temporarily make integrals finite. However, in the end of calculations one should take the photon mass to zero in order to be consistent with experiment. So, non-zero photon mass is just a temporary relief. It doesn't solve the problem of infrared infinities. There are two real solutions of this problem that I know of.

The first (older) solution removes infinities from scattering cross-sections. It says that we never see pure scattering of charged particles, like

e + e -> e + e (1)

(QED scattering amplitude for this process is infrared-infinite). In experiments we really have to deal with collisions

e + e -> e + e + soft photons (2)

These soft photons normally escape observation. So, the realistic cross-section for (1) should be obtained by summing amplitudes (2) and taking square of the sum. This result is finite.

The other solution (usually associated with names of Kulish & Faddeev) is to change the definition of the S-matrix, so that instead of the free asymptotic Hamiltonian H_0 one uses the Hamiltonian

H_0 + long range Coulomb interaction

This allows one to define some kind of finite S-operator. However, this solution looks rather troubling to me. All we did is to move some part of interaction from the full Hamiltonian H_0 + V to the free Hamiltonian H_0. How can we decide which part of the interaction V can be moved like that? Why don't we move entire V to the asymptotic Hamiltonian? Then the S-operator will become trivial S=1, and all our problems will be solved? This doesn't make sense to me.

I don't want to say that there is something wrong with this approach. I am simply saying that I don't understand it yet. Maybe somebody more educated can help me to figure that out.

Hi Eugene,

Thanks for the comments. I'm still very much a novice when it comes to theoretical physics. Can you recommend a rigorous yet accessible (is that a contradiction in terms?) introductory text for special relativity and general relativity?

James

Hi James,

it is difficult for me to recommend you a book for two reasons.

First, I don't know your current knowledge level in physics and math. Special relativity is quite simple mathematically. High school algebra might be enough to get the main idea. General relativity is a different matter. It requires quite sophisticated differential geometry.

The second difficulty is that I learned these subjects a quarter-of-century ago by reading Russian books. I simply do not recall which book I liked more. One book I do remember is Landau and Livshitz vol. 2 "Field theory" (I am sure there are English translations). It has classical electromagnetic theory, special and general relativity all in one tome. As you can imagine, it presents the material in a very condensed form. So, perhaps, this is not the best pick for a beginner.

My advise would be to go to a university library and find a shelf with books on relativity. A good library would have dozens of textbooks: from popular to advanced. You can browse these books and pick the one which you like the most.

Eugene.

Hello Eugene,

I think that the true dressing happens as follows: you find a solution of electron in the free electromagnetic quantized field. Such an électron (that I call a real electron) is not pointlike but smeared due to field fluctuations. So no UV divergencies arise. In addition, when you push such an electron, you perturbe its state with respect to EM field, so it radiates in the first Born approximation. No IR divergencies either as the radiation is taken already into account. Why not to try this way?

Vladimir Kalitvianski.

Hi Vladimir,

I haven't seen these ideas before. How are they different from the usual dressing approach? Do you have any references?

As I wrote, IR divergences are ultimately related to zero photon mass. I don't see how your proposal can avoid that.

Hello Eugene,

It's me who develops these ideas and this approach.

Similar ideas were developped by Welton in 1948 year (Phys. Rev. V. 74, N. 9).

Your dressing is carried out perturbatively.

I take the EM vacuum fluctuations into account exactly.

Whatever are particular masses of particles constituting a compound system, when you give to one of them a push, the system can get excited. If you neglect the binding force, then the excitation will happen in the second (not first) Born approximation. The correction can be big to show that you should not have neglected the binding force.

Vladimir Kalitvianski.

Hi Vladimir,

Thank you very much for the reference. I'll check it out next time I visit the library. Have you published any of your own ideas?

Eugene.

Hello Eugene,

Many years ago I encountered divergences in a Sturm-Liouville problem (eigenfunctions and eigenvalues). At that time I managed to get rid of them by choosing a more appropriate initial approximation for the eigenfunctions. I understood that the correction values depended on the quality of initial approximation. The better it is, the smaller the corrections are. I published an article in the journal of Academy of Science in Russia (On perturbation theory for Sturm-Liouville problem with variable coefficients. Journal of Computational Mathematics and Mathematical Physics, N 3, 1994, pp. 491-494). I advanced some ideas how it could be related to QED divergences.

I also published an article where I showed importance of taking the bound states into account exactly rather then perturbatively (Attenuation of the Rutherford scattering and atom exciting by fast charged particles for large-angle scattering. Ukrainian Journal of Physics, V. 38, N 6, 1993, pp. 851-854). Again, I noted a close analogy with QED.

The idea of taking into account the vacuum fluctuations is not my own. Many others (Bethe, Welton, etc.) have already published such things. But I think I have found a non perturbative way to take them into account in the initial approximation in QED (real electron) that seemingly leads to finite corrections. Now I am looking for financing my future calculations since I cannot continue to work on weekends on my own. This subject is not a play. I need civilised conditions to finish my calculations.

Do you have an idea how to get some grant/contract/etc. to finalise my research?

Vladimir Kalitvianski.

Vladimir:

"Now I am looking for financing my future calculations since I cannot continue to work on weekends on my own. This subject is not a play. I need civilised conditions to finish my calculations.

Do you have an idea how to get some grant/contract/etc. to finalise my research?"

That's a very good question. I wished I knew the answer myself. Kak govoritsja, "znal by prikup, zhil by v Sochi".

Seriously, I don't know your personal circumstances, but I suspect that you already missed the opportunity to go by the "normal" route: graduate study -> PhD -> postdoc -> postdoc -> ... -> assistant prof. -> prof. This is how people survive in academia in a very tough, moneyless, and competitive environment. Even if you went on this path, you would need to wait many years until you had a chance to obtain "civilized conditions" in which you can freely work on questions that interest you personally. Even being a prof. doesn't give you much freedom. Most of your time will be eaten by teaching, coaching students, writing grant applications, political fights, ... So, you'll end up doing physics on weekends anyway. I spent a few years at US university in a field which is much less poisonous than today's theoretical high energy physics (Department of Chemistry at the University of Utah), and I think I know what I am talking about.

I decided for myself that I don't need all this aggravation. It is, actually, not so bad to work on evenings and weekends. This gives me the most important advantage - the freedom to do exactly what I like at my own pace without pressure to publish, etc.

There is a chance to get some funding for your work. Check this website http://www.fqxi.org They distribute some grant money to independent researches who are not necessarily affiliated with any institution. Good luck.

Eugene.

Hi Eugene,

Thanks for the link. It sounds promising. I will try when they announce the next request for proposals.

Best regards,

Vladimir Kalitvianski.

I too think that most people believe that photons are truly massless, though we may never know for sure. Great post! I am a college sophomore with a dual major in Physics and Mathematics @ University of California, Santa Barbara. By the way, i came across these excellent physics flashcards. Its also a great initiative by the FunnelBrain team. Amazing!!!

Dear Eugene,

Can you express yourself about the so called adiabatic hypothesis in QED when the charge-field interaction is switched off in asymptotic states (decoupled, free equations).

I am glad to report that I've found a solution for the infrared problem in dressed particle QED. This solution is even simpler than I expected. It involves cancellation of divergences in rather nasty loop integrals. Please see new chapter 15 (Radiative corrections) in the most recent version (v14) of my book http://arxiv.org/abs/physics/0504062

You can find there dressed particle calculations of major QED renormalization effects: the Uehling potential, the electron's anomalous magnetic moment, and the Lamb shift.

Eugene, as soon as you work with the same QED Hamiltonian, you are bound to get the same result for IR cancellation and other effects in the same way.

If you could avoid IR divergences, that would be a progress.

Vladimir,

In my approach infrared divergences are real physical effects (unlike ultraviolet divergences). The physics is that in the 4th perturbation order the direct interaction potential between two charges has infrared-divergent contribution (15.26). So, the force acting between the electron and the proton is, actually, infinite. However, as explained in section 15.3, there is also an opposite divergent force, which comes from bremsstrahlung radiation of (soft) photons. These two infinite forces almost cancel each other. The resulting force is only slightly different from the usual Coulomb force, and this difference is responsible for the Lamb shift seen in experiments.

In bound states there may not be IR divergences, that is why they cancel. It does not mean that there is no physics behind it. The physics is simple - averaging over the quantized field oscillator variables coupled to the electron coordinate makes an additional smearing effect.

In scattering they may not cancel since the radiation exists. IR divergence is just a big amplitude of a weak oscillator, if one speaks classically. http://fishers-in-the-snow.blogspot.com/2011/08/objasnenie-perenormirovok.html

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