How do renormalization and dressed particles eliminate self-energy?

[RedX]

What is meant by self-energy? If you have an electron and it splits into an electron and a photon, and then recombines, this is called the self-energy, but how is this related to energy?

This self-energy is infinite, but not after renormalization.

However, this http://hitoshi.berkeley.edu/public_html/susy/susy.html" claims that the self-energy is finite because an electron not only interacts with itself, but also the vacuum creates an electron/positron pair and a photon, so the original electron gets annihilated with the positron (along with the photon), leaving the vacuum electron which then becomes a real particle. These two processes cancel, so there is no self-energy.

But how does this interpretation come out of quantum field theory? To me, diagrams b) and c) of the link are exactly the same, so not only can they not cancel, you can only take one of them.

The only thing that I can think of when talking about canceling, are counter-terms. But this implies they're saying that renormalization has to do with creation of particles in vacuum, something I'm uncomfortable with - I think that's making up more than the equations actually say.

Another thing that's problematic is that you aren't really calculating scattering wahen dealing with one initial particle and one final particle. You are calculating unscattering and I don't think the QFT scattering formulas apply. Isn't the probability just 1?

 

[meopemuk]

Another thing that's problematic is that you aren't really calculating scattering wahen dealing with one initial particle and one final particle. You are calculating unscattering and I don't think the QFT scattering formulas apply. Isn't the probability just 1?

Yes, this is exactly the main ultraviolet divergence problem in QFT (before application of the renormalization prescription). The probability must be just 1, but it turns out to be infinite in all perturbation orders. The problem is that QFT interaction has a non-trivial action on 1-particle states. (This is unheard of in regular classical or quantum mechanics, where there should be at least 2 particles to interact). The energy corresponding to this self-interaction is called "self-energy". In order to compensate this infinite self-interaction contribution one adds (infinite) counterterms to the Hamiltonian, thus restoring the correct value (=1) of the particle-particle scattering probability and making sure that amplitudes for other scattering events are finite and accurate too. The infinite counterterms make the Hamiltonian ugly and ill-defined, but nobody cares about the Hamiltonian, because time-dependent processes are not easily observable anyway.

 

[Bob_for_short]

What is meant by self-energy?

It is a perturbative correction (addendum) to the particle mass. Photon mass (=0) has perturbative addenda and the electron mass too. In the early QED and QFT these addenda were just discarded. Then physicists invented the notions of bare masses and counter-terms. It results in the same thing however it is done.

The reason of appearing self-energy terms is the self-interaction term in the initial Hamiltonian. Renormalizations remove the effect of self-action. I explained it on a very simple example in "Reformulation instead of Renormalizations" available in arXiv. Thus, one can formulate the theory without self-action and without renormalizations (a short-cut to finite perturbative series).

The link statement is wrong. It is an attempt of popular explanation without renormalizations. It does not work without renormalizations.
Supersymmetry exists since long ago. It was "promising" direction at that time but it failed. There is no physical need in supersymmetry. It is a mathematical play without physical results.

Bob_for_short.

 

[jostpuur]

another question about "what is the self-energy"

I thought that I had understood what self-energy is, when I learned about it in classical context. The infinite energy in an electric field created by a point charge. I was left confused when a completely different looking thing in QFT was called with the same name. Are these self-energies somehow related?

 

[meopemuk]

I thought that I had understood what self-energy is, when I learned about it in classical context. The infinite energy in an electric field created by a point charge. I was left confused when a completely different looking thing in QFT was called with the same name. Are these self-energies somehow related?

In my personal (minority) opinion, both these self-energies are signs of badly formulated theories. Both theories (QFT and classical electrodynamics) must be modified so that self-energies do not appear there. This is indeed possible. In the dressed particle approach, the QFT interaction can be rewritten so that there is no self-interaction of particles, while all predictions of the standard renormalized theory regarding scattering amplitudes remain valid. Classical electrodynamics can be also reformulated, so that fields (electric and magnetic) do not play any role there, and charged particles directly interact with each other via the Darwin-Breit potential. In this formulation, there is no (self-) energy associated with fields, and many paradoxes of Maxwell's theory are easily resolved.