Adiabatic hypothesis in QFT

[Bob_for_short]

Can you give me references, if any, to the justification of adiabatic hypothesis in QFT? Why should we switch off the interaction in asymptotic states? Because we cannot solve coupled equations?

[ajb]

Philosophically, one would expect it to be almost true for interactions that get weaker as the separation increases.

 

In reality I think it largely comes down to the use of the S-matrix, the interaction picture and perturbation theory in QFT. Like you said, it comes down to the computational difficulties. We know how to work with plane waves.

 

Any QFT book will say something. You may also be interested in Haag theorem, which muddies the waters even more.

[Severian]

It is reasonably believable in normal quantum mechanics if you have a finite range force. In QFT is is more problematic since even your vacuum state changes when an interaction is present, so the effects can never be isolated.

 

I would view it on a more pragmatic level. We make predictions with QFT with this sort of difficulty in mind, and are surprised that our QFTs do so incredibly well at predicting the outcomes of our experiments.

[Bob_for_short]

It is reasonably believable in normal quantum mechanics if you have a finite range force. In QFT it is more problematic since even your vacuum state changes when an interaction is present, so the effects can never be isolated.

Thus it is clearly a "difficulty" of the present theory.

I would view it on a more pragmatic level. We make predictions with QFT with this sort of difficulty in mind, and are surprised that our QFTs do so incredibly well at predicting the outcomes of our experiments.

It is not the only difficulty, unfortunately.

 

My second question is whether it is desirable to have a theory without any conceptual and mathematical difficulties, just in the spirit of QM?

[Severian]

My second question is whether it is desirable to have a theory without any conceptual and mathematical difficulties, just in the spirit of QM?

 

Of course it would be. But I would not be willing to throw away the predictive power of our Standard Model. Predictivity is more desirable than conceptual and mathematical consistency, in my opinion.

[ajb]

Of course it would be. But I would not be willing to throw away the predictive power of our Standard Model. Predictivity is more desirable than conceptual and mathematical consistency, in my opinion.

 

I agree with this. Having an elegant mathematical description in which calculations are very clear is desirable, but if we were to loose all contact with the real word then it is not clear if we still doing "physics".

 

This is not to be confused with mathematical physics in which one is interested in the mathematical structures and their generalisations that are found in theoretical physics. This is closer to mathematics than physics. Also, there is using physical ideas in mathematics, but that is another story.

[Bob_for_short]

I agree with this. Having an elegant mathematical description in which calculations are very clear is desirable, but if we were to loose all contact with the real word then it is not clear if we still doing "physics".

Yes, I also speak of a physical theory with physical predictions. Currently, however, the theory operates with "bare" stuff, "counterterms", etc., and the final results are not really clear to most of researchers.

For example, the size of a real (or dressed) electron, what is it according to QED? I mean theoretical estimation in terms of e, m, h-bar (not an experimental one).

[ajb]

Renormalisation is an issue with many QFTs.

 

QED has had great success in predicting the anomalous magnetic moment of the electron and the Lamb shift for example. This means that at some level we are doing "physics" even if there are still some sticky issues.

 

As for the size of an electron, it depends what you mean. If I define it in terms of the total cross section of Thomson scattering we get

 

[math]\sigma_{T} = \frac{8 \pi}{3} \left( \frac{\alpha \hbar}{mc} \right)^{2}[/math],

 

for small energies for the incoming photon.

[Bob_for_short]

As for the size of an electron, it depends what you mean.

I mean the same thing as an atomic size - the electron elastic form-factor determining the charge distribution in space for ealstic scattering. When we write 1/r, we mean a point-like electron with Coulomb law. In QED the charge is in permanent interaction with the quantized electromagnetic field that smears the charge over space. I wonder - what size of charge smearing does QED predict for a real electron? In order to calculate the corresponding cross section (expressed via elastic form-factor), one has to add the quantized EMF in the electron equation.

Edited May 20, 2010 by Bob_for_short