### 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.