### What is "dressed particle" approach to QFT?

I promised to tell about it several times. Now is the time.

Recall our discussion of the QED Hamiltonian where we found that the presence of terms like a*c*a makes this Hamiltonian useless for studying the time evolution. The problem was that these terms act non-trivially on 1-particle states. The main idea of the "dressed particle" approach is that there can be no interaction in the vacuum state and in 1-particle states. These states should evolve in time as if there were no interaction at all. Mathematically this requirement means that all terms in the interaction Hamiltonian should have at least two annihilation operators. In order to maintain the Hermiticity, there should be at least two creation operators in each interaction term. So, in a theory of interacting electrons and photons, the simplest normally ordered terms allowed in the interaction Hamiltonian are

a*a*aa

a*c*ac

a*a*c*aa

a*a*a*aaa

...

The first term here describes a direct electron-electron interaction; the second term is an electron-photon potential responsible for the Compton scattering; the third term describes bremsstrahlung, i.e., emission of photons in electron-electron collisions; the fourth term is a three-body electron-electron-electron potential,...

It is important that all these terms yield zero when acting on the vacuum and 1-particle states. This means, for example, that the electron mass is not affected by interaction, so there is no need for renormalization.

As I mentioned here , a number of relativistically invariant and cluster separable models of this sort were constructed by H. Kita. There is also a "dressing approach" by Greenberg and Schweber which allows one to make a "dressed particle" theory out of virtually any usual QFT. What is really exciting is that this approach works and one can construct a "dressed" version of QED and other popular theories. This provides a completely new perspective on foundations of relativistic quantum physics.

Recall our discussion of the QED Hamiltonian where we found that the presence of terms like a*c*a makes this Hamiltonian useless for studying the time evolution. The problem was that these terms act non-trivially on 1-particle states. The main idea of the "dressed particle" approach is that there can be no interaction in the vacuum state and in 1-particle states. These states should evolve in time as if there were no interaction at all. Mathematically this requirement means that all terms in the interaction Hamiltonian should have at least two annihilation operators. In order to maintain the Hermiticity, there should be at least two creation operators in each interaction term. So, in a theory of interacting electrons and photons, the simplest normally ordered terms allowed in the interaction Hamiltonian are

a*a*aa

a*c*ac

a*a*c*aa

a*a*a*aaa

...

The first term here describes a direct electron-electron interaction; the second term is an electron-photon potential responsible for the Compton scattering; the third term describes bremsstrahlung, i.e., emission of photons in electron-electron collisions; the fourth term is a three-body electron-electron-electron potential,...

It is important that all these terms yield zero when acting on the vacuum and 1-particle states. This means, for example, that the electron mass is not affected by interaction, so there is no need for renormalization.

As I mentioned here , a number of relativistically invariant and cluster separable models of this sort were constructed by H. Kita. There is also a "dressing approach" by Greenberg and Schweber which allows one to make a "dressed particle" theory out of virtually any usual QFT. What is really exciting is that this approach works and one can construct a "dressed" version of QED and other popular theories. This provides a completely new perspective on foundations of relativistic quantum physics.

## 2 Comments:

Dear Eugene,

How does a dressed electron look like after full dressing? What does it resemble in your opinion?

Thanks,

Vladimir.

Hi Vladimir,

I think that the terminology "dressed electron" is very unfortunate. It appeared for historical reasons, because originally QFT was formulated in terms of "bare particles". In the "bare particle" representation physical electrons look like a complicated mess of virtual photons, electron-positron pairs etc. The "dressed particle" approach cuts all this nonsense, but sadly the old terminology remains.

Ideally, we should stop using terms like "bare electron" or "dressed electron". It is better to talk about "physical electron" or simply "electron". I imagine it as a point particle characterized only by its (measurable) mass, spin and charge. There is nothing more to talk about: no virtual clouds, no mass/charge renormalization. So, the notion of the "physical electron" is pretty much the same as the seemingly naive picture, which we first learned in the beginners course on non-relativistic quantum mechanics.

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