# Quantum mechanics and Minimal coupling of Dirac field

1. Aug 7, 2011

### mtak0114

Hi

I have a simple question:

We know from non-relativistic quantum mechanics that the spin of an electron couples only to the magnetic field, i.e. it processes around the magnetic field. How is this resolved in the relativistic context where it would seem that the spin should couple to both electric and magnetic fields? In particular how is this implied through minimal coupling which seems to be relativistically covariant where as the magnetic field isnt?

2. Aug 7, 2011

### Bill_K

mtak, You can either start by giving your particle a minimal coupling or else put in an anomalous moment by hand. Either way you wind up considering an interaction term of the form σμνFμν. As you point out, this is relativistically invariant and reduces to S·B in the particle's rest frame. Of course you're also right that this is not S·B in a frame in which the particle is moving. In fact in a moving frame there will be an additional interaction with the E field that looks like S·(v x E). Well you can't quantize the spin along two different directions, so the obvious thing to do is combine these two terms and write them together as S·Beff where Beff is the necessary linear combination of B and v x E. But Beff is also just the B field back in the particle's rest frame, so we just write it that way!

3. Aug 7, 2011

### mtak0114

Thanks Bill

that makes sense, but is that assuming that $F^{\mu\nu}$ is just the magnetic field ${\bf B}$ in the rest frame otherwise I can't see how $\sigma_{\mu\nu}F^{\mu\nu}$ reduces to such a term in the rest frame. Is it possible to see the converse, that QFT implies that spin does not couple in the rest frame to the electric field?

4. Aug 8, 2011

### Bill_K

S·E would mean that the particle had an electric dipole moment, in contrast with a magnetic dipole moment. This violates parity conservation. Very small electric dipole moments are predicted by the Standard Model, and larger ones by other theories, but none has ever been observed.

5. Aug 8, 2011

### mtak0114

So would it be correct to say that the Dirac equation includes effects due to both electric and magnetic dipole moments of the electron? but given that the former are not observed they are suppressed?

When one goes to QED however is such a suppression necessary or does the theory predict that the electric dipole moment is small?

do you have any good references which discuss this issue?
thanks again

Mark

6. Aug 9, 2011

### Bill_K

mtak, Minimal coupling implies a magnetic dipole moment but no electric dipole moment. If electric dipole moments exist, they don't arise from the Dirac equation, rather from internal loops of parity-violating particles.

In the Standard Model the moments predicted in this way are far smaller than can be detected. Other theories like supersymmetry predict moments near the present detection threshold. For a reference, look for "Neutron electric dipole moment" and "Electron electric dipole moment" on Wikipedia. Almost all of the information you'll find will be experiment-oriented.