Spin-Orbit interaction in nucleus' rest frame

[ShayanJ]

In all the places where Spin-Orbit interaction is discussed, the equations are derived by going to electron's rest frame and considering the interaction of nucleus' magnetic field with electrons spin magnetic moment. But from SR, we know that there as to be an explantion from the nucleus's rest frame too. But I have problem coming up with something!
Any ideas?
Thanks

 

[VantagePoint72]

In the nucleus' rest frame, spin-orbit interaction is the electron's spin magnetic moment interacting with the magnetic field that the electron itself is producing due to its orbital angular momentum (hence the name). Classically, you'd need to use the Liénard-Wiechert formula for retarded potentials to compute the magnetic field at the electron's location at any time. I'm not entirely sure if that can be adapted quantum mechanically.

 

[ShayanJ]

But Liénard-Wiechert potentials are singular at the source's location!

 

[VantagePoint72]

Hm, you're right...

 

[ShayanJ]

Also the explanation in the electron's rest frame, is based on the magnetic field coming from transforming nucleus' electric field. So the explanation in the nucleus' rest frame should rely on some field produced by the nucleus.
The only thing that comes to my mind is a kind of interaction between nucleus' electric field with electron's spin magnetic moment but I don't remember any such interaction in EM!

I found it man!
A moving magnetic dipole acquires an electric dipole moment. And this electric dipole moment interacts with nucleus' electric field.

 

[vanhees71]

Using the Dirac Equation, which is relativistically covariant let's you write down the Hamiltonian directly in any reference frame you like. Case closed. The interesting point of the derivations in the non-relativistic realm is that you get, even for an elementary particle as the electron, where the Dirac Equation is applicable, you get gyro factor that is wrong by a factor of 2 when using the Galilei transformation at this point of the derivation. This is known as Thomas precession and marks the discovery of the Wigner rotation in the composition of two Lorentz boosts that are not in the same boost direction, which is completely missing in the Lorentz transformation.

Of course, the true gyro factor of the electron (and the other leptons) is not exactly 2 but (g-2) is one of the most accurately measured quantities which are (for the electron) in astonishing agreement with QED (at the four-loop order of perturbation theory!), while for the muon there may be a discrepancy with the standard model since the most accurate measurement of (g-2) performed at BNL shows a deviation by somewhat more than 3 standard deviations. This is not a discovery but only an evidence for physics beyond the standard model, and for this reason the entire experiment has been moved to Fermilab now and will be driven to even higher precision very soon. The main theoretical problem is the contribution from the strong interaction to the radiative corrections, which are somewhat larger for the muon than for the electron. These corrections have thus to be measured with high accuracy too. Such experiments at very low energies (in contrast to the largest energies in the experiments at LHC) are done in Mainz, and also their efforts towards even higher precision is under way. It's a very exciting issue in elementary-particle physics!

 

[ShayanJ]

contribution from the strong interaction to the radiative corrections

Is this referring to the process that the muon spontaneously radiates a photon, that photon decays to(I guess there is a better word for it!) a quark-antiquark pair and they annihilate each other to give a photon which is again absorbed by the muon?

 

[vanhees71]

The leading-order contribution is the QED triangle diagram with the exchanged photon dressed by self-energies involving hadrons and "light-by-light" scattering diagrams. For pictures of these diagrams and a review on the status, see

http://arxiv.org/abs/1311.2198