Undergrad Are there asymptotic QFT/QED states in a constant magnetic field?

[gentzen]

It is "easy" to produce experimental setups that could and should for all practical purposes be described as having a constant background magnetic field everywhere, especially in the "asymptotic region" where the detectors are located.

You can do this both in vacuum, and inside a solid sample. In a solid sample, the effect of the constant magnetic field is "basically clear" and "well studied": you get a circular motion, and if the radii of that motion are bigger than the sample, then you get "scattering effects" at the boundary of the sample. In vacuum, the meaning of such asymptotic states seems less clear to me than in a solid sample, because there is no sample size and no boundaries, for "comparison".

On the ofther hand, I guess that QFT/QED has no problems in principle to describe such "easy to produce" experimental situations. But how is this actually done?

 

[vanhees71]

You just have a "semiclassical model", i.e., you introduce the classical em. field as an classical "c-number background field". Then you can do perturbation theory for particles (in QED usually electrons and positrons) and the em. field (photons) as usual. You only have to solve for the modes (energy-eigenmodes) including the classical background field.

 

[gentzen]

You just have a "semiclassical model", i.e., you introduce the classical em. field as an classical "c-number background field". Then you can do perturbation theory for particles (in QED usually electrons and positrons) and the em. field (photons) as usual. You only have to solve for the modes (energy-eigenmodes) including the classical background field.

Good to hear that this "can be done in principle", like I guessed.

But how do you deal with the circular (or spiral) motion of the asymptotic states caused by the magnetic field? Do you just take the resulting quantization into account, like for the quantum Hall effect in a "sufficiently big" solid sample? Or do you "cut open" the circles at the half-space defined by the (flat) detector surface? (Or is it enough to just focus on spiral motion sufficiently different from circular motion to not cause relevant quantizations? That would somehow feel "wrong" to me.) Or ... ?

 

[vanhees71]

In a magnetic field the energy eigenmodes represent particles that in the classical analogue move on spirals. Asymptotic free states represent particles interacting only with the classical background-magnetic field but are not subject to collisions.

 

[gentzen]

In a magnetic field the energy eigenmodes represent particles that in the classical analogue move on spirals. Asymptotic free states represent particles interacting only with the classical background-magnetic field but are not subject to collisions.

Ok, so "you just take the resulting quantization into account". I guess it makes sense, because the typical "experimental scattering in vacuum setup" will probably not depend on the boundaries in a systematical way. And the "spurious" quantization(s) will have nearly no impact in the results, because they are more a property of some simple canonical basis, and less relevant for the actual physical situation. (I remember "related" discussions about fast electrons in solids, and why one doesn't need to worry the ambiguity of the wavevector for Bloch waves, because it doesn't make a difference for the actual physical situation, which is represented by appropriate superpositions, instead of the basis functions themselves.)

One reason why I "guessed" that one would prefer states that don't need to satisfy global constraints was that the typical situation for a classical background is gravity, and there the background is typically defined only locally, at least the relevant background. And the situation with the constant magnetic field is somewhat similar, because again only the locally constant magnetic field is relevant. But since it is easy to continue it globally, and it probably preserves more symmetries, it will probably make the math a bit simpler to work with global states.

 

[gentzen]

I got two more replies (because I asked 3 more people, in the hope to get an answer that would help me to clear up my confusion of how asymptotic states can fit together with "only local background matters"):

In local quantum physics (=rel. QFT), everything matters only in a neighborhood of the spacetime region of interest. Asymptotic background fields are accounted for by a Bogoliubov transform that shifts the vector potential, thereby changing the superselection sector in which the theory is used.

The static magnetic field only changes the electronic base states, as van Hees also suggested in his reply.

Then you can do perturbation theory for particles (in QED usually electrons and positrons) and the em. field (photons) as usual

The combination of that last answer and the quoted part of vanhees71's initial answer finally did the trick for me. My mental mistake can be seen in the following quotes:

constant background magnetic field everywhere, especially in the "asymptotic region" where the detectors are located.

Or do you "cut open" the circles at the half-space defined by the (flat) detector surface?

The "asymptotic states" have simply nothing to do with an "asymptotic region". This sort of far away detector geometry is only caused by the fact that it allows to measure "direction", at least in the absence of a constant magnetic field. The important part of "asymptotic states" is instead that the interaction region is sufficiently far away in space or time.

There is no need to detect those asympotic states with a huge detector with a (flat) detector surface. Wilson cloud chambers or similar modern weak measurement setups are totally appropriate, and don't noticably disturb the spiral trajectories or the constant magnetic field.