Question

How to calculate the Stern-Gerlach term from the charged spin one anti-symmetric field equation?

Answer 1

How does one calculate the Stern-Gerlach term from the charged spin one anti-symmetric field equation?

I first denoted the following momentum operator Da=(iℏ∂a−qAa)Da=(iℏ∂a−qAa), and then I wrote down the anti-symmetric field

ϕa b=Daψb−Dbψaϕ ba=Daψb−Dbψa

and then computed its divergence

Daϕa b==Da(Daψb−Dbψa)DaDaψb−DaDbψaDaϕ ba=Da(Daψb−Dbψa)=DaDaψb−DaDbψa

Next I introduced a gauge term and the commutator

Daϕ ba=DaDaψb+[Db,Da]ψa−DbDaψaDaϕa b=DaDaψb+[Db,Da]ψa−DbDaψa

and then I set the gauge term to zero Daψa=0Daψa=0.

Daϕ ba=DaDaψb+[Db,Da]ψaDaϕa b=DaDaψb+[Db,Da]ψa

I assumed the commutator [Db,Da][Db,Da] was the Stern-Gerlach term for Spin One particles, however I discovered

[Db,Da]≠iℏq(∂bAa−∂aAb)=iℏqFba[Db,Da]≠iℏq(∂bAa−∂aAb)=iℏqFba

was not the Electromagnetic Field Tensor FbaFba.

Answer 2

Provided that [Aμ,Aν]=0[Aμ,Aν]=0, i.e. AμAμ is the EM vector potential and not an arbitrary gauge field, [Dμ,Dν][Dμ,Dν] has exactly the value Fμν=∂μAν−∂νAμFμν=∂μAν−∂νAμ (constants omitted). How would you get a wavefunction into [Dμ,Dν][Dμ,Dν] at all? It's an operator, not a wavefunction. There's just ∂mu∂mu and AμAμ in there, and since [∂μ,∂nu]=0[∂μ,∂nu]=0, all that remains from the commutator are the mixed ∂μAν∂μAν terms in FμνFμν.