How does the dressed electron look like in QED?

[Bob_for_short]

Is there any solution ψ to understand how the real electron looks like after renormalization (dressing)?

 

[meopemuk]

In the "dressed particle" approach to QED, the electron is simply a point (structureless) particle with usually measured mass and charge.

O. W. Greenberg and S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim., 8 (1958), 378.

E. V. Stefanovich, "Quantum field theory without infinities", Ann. Phys. (NY) 292, (2001), 139.

E. V. Stefanovich, "Relativistic quantum dynamics",
http://www.arxiv.org/abs/physics/0504062

 

[Bob_for_short]

In the "dressed particle" approach to QED, the electron is simply a point (structureless) particle with usually measured mass and charge.

And according to QED, the external lines have no corrections after all. So the effect of renormalization (dressing) boils down to zero, doesn't it?

 

[RedX]

And according to QED, the external lines have no corrections after all. So the effect of renormalization (dressing) boils down to zero, doesn't it?

Well there's also the field strength renormalization. So I think the external lines are corrected by Z^(-1/2).

 

[meopemuk]

And according to QED, the external lines have no corrections after all. So the effect of renormalization (dressing) boils down to zero, doesn't it?

Yes, the electron self-energy (or electron mass renormalization) counterterm cancels exactly the contribution coming from electron-photon loops attached to external electron lines. So, in renormalized QED, there is no need to draw loops in external electron lines.

However, the electron self-energy counterterm does not cancel loops in internal electron lines, because these lines are "not on the mass shell". The residual terms are responsible for small radiative corrections.

Though, I am not sure how this question is related to your previous one (about the dressed electron).

 

[Bob_for_short]

Yes, the electron self-energy (or electron mass renormalization) counterterm cancels exactly the contribution coming from electron-photon loops attached to external electron lines. So, in renormalized QED, there is no need to draw loops in external electron lines.

However, the electron self-energy counterterm does not cancel loops in internal electron lines, because these lines are "not on the mass shell". The residual terms are responsible for small radiative corrections.

Though, I am not sure how this question is related to your previous one (about the dressed electron).

If the free (incident and scattered) particles are the same after renormalizations, that means only interaction Hamiltonian modification (removing self-interaction) in course of perturbative renormalizations. What sense to speak of dressed or renormalized particles if they are the same?

 

[meopemuk]

If the free (incident and scattered) particles are the same after renormalizations, that means only interaction Hamiltonian modification (removing self-interaction) in course of perturbative renormalizations. What sense to speak of dressed or renormalized particles if they are the same?

In standard renormalized QED there is a distinction between "bare" and "dressed" particles. "Bare" particles are those whose creation/annihilation operators are used to formulate the theory, i.e., to write the Hamiltonian. The lines in Feynman diagrams correspond to the "bare" particles. However, the major inconsistency in QED is that "bare" particle states cannot correspond to real observable states seen in nature. One-bare-particle states are not eigenstates of the full interacting Hamiltonian. So, they don't have well-defined energies. Even worse, the "bare" vacuum state in QED is not an eigenstate of the full Hamiltonian too.

You are right that this unfortunate situation results from the presence of self-interaction (e.g., tri-linear) terms in the Hamiltonian.

All these problems can be fixed in the "dressed particle" approach, where the Hamiltonian of QED (with renormalization counterterms) is modified (via an unitary transformation that preserves the scattering matrix) so that self-interaction terms get removed. Then the difference between "bare" and "dressed" particles disappers. The theory is formulated in terms of real observable particle states. Another good thing is that it is guaranteed (by construction) that the scattering matrix in the "dressed" approach is exactly the same as in the renormalized QED, i.e., agrees with experiment very well.

 

[Bob_for_short]

All these problems can be fixed in the "dressed particle" approach, where the Hamiltonian of QED (with renormalization counter-terms) is modified (via an unitary transformation that preserves the scattering matrix) so that self-interaction terms get removed. Then the difference between "bare" and "dressed" particles disappears. The theory is formulated in terms of real observable particle states. Another good thing is that it is guaranteed (by construction) that the scattering matrix in the "dressed" approach is exactly the same as in the renormalized QED, i.e., agrees with experiment very well.

Thank you, Eugene, for your exhaustive explanation. I wonder if on can start from the modified Hamiltonian (without self-action term) from the very beginning rather than modify it perturbatively?

 

[meopemuk]

Thank you, Eugene, for your exhaustive explanation. I wonder if on can start from the modified Hamiltonian (without self-action term) from the very beginning rather than modify it perturbatively?

Yes, it should be possible, in principle. However, there are two major difficulties. First, how are you going to derive this modified Hamiltonian? From which principles? Second, you need to demonstrate that this Hamiltonian describes correctly already known scattering processes and spectra (e.g., the Lamb shift).

I couldn't find answers to these questions in your works.

On the other hand, in the usual "dressed particle" approach these questions have satisfactory answers. The "dressed particle" Hamiltonian is derived by applying an unitary transformation to the Hamiltonian of the standard renormalized QED. The unitarity of this transformation guarantees that the scattering matrix of the "dressed" approach is exactly the same as the scattering matrix of the orthodox renormalized QED. I.e., it agrees with experiment very well.

Eugene.

 

[Bob_for_short]

Yes, it should be possible, in principle. However, there are two major difficulties. First, how are you going to derive this modified Hamiltonian? From which principles? Second, you need to demonstrate that this Hamiltonian describes correctly already known scattering processes and spectra (e.g., the Lamb shift). I couldn't find answers to these questions in your works.

Too bad, they are there with clear physical and mathematical motivations.

On the other hand, in the usual "dressed particle" approach these questions have satisfactory answers. The "dressed particle" Hamiltonian is derived by applying an unitary transformation to the Hamiltonian of the standard renormalized QED. The unitarity of this transformation guarantees that the scattering matrix of the "dressed" approach is exactly the same as the scattering matrix of the orthodox renormalized QED. I.e., it agrees with experiment very well. Eugene.

So where are your results on scattering and the Lamb shift?

 

[meopemuk]

So where are your results on scattering and the Lamb shift?

In the references in an earlier post there is a proof that the unitary dressing transformation does not change the S-matrix. Therefore, all scattering results obtained in standard QED must remain valid. The same should be true for the Lamb shift, because energies of bound states are recorded in the S-matrix (as positions of poles).

I agree that it would be nice to perform directly high-order calculations of scattering amplitudes and Lamb shifts in the "dressed particle" approach, rather than rely on a formal theorem. I continue working on that. The main stumbling block is the treatment of infrared divergences (the emission of soft photons, etc.). I haven't found a convincing way to do that yet.