Proving the Spin Half Nature of Dirac Quanta

[quantumfireball]

but I am confused
how do you proof that the dirac field describes spin half quanta when quantized?
please refer me to a link on the net where this derivation is shown if possible
i can't find it in any of the books on quantized field theory

 

[blechman]

One way you can do it is to take the nonrelativistic limit of the Dirac equation and recover the spin-term in the hamiltonian. then you'll see explicitly that it's s=1/2. see any text on relativistic QM.

In QFT, you usually go the other way around: assume that it's a fermion and you get the Dirac equation. Fermions have to have half-integer spin thanks to the famous "Spin-Statistics Theorem". I believe this was proved first by Pauli, but I'm not sure. I'm sure Weinberg does it in his QFT tome.

 

[quantumfireball]

well what about spin a1 fields,spin 3-2 fields,spin 2 fields
how does one go about showing that they describe quanta of that much amount of intrinsic angular momenta

 

[blechman]

count the number of polarization states (this follows from the field equations); that tells you how many components the field has. You know that a field with N components must transform under Lorentz transformations the same way as an object with spin (N-1)/2 - this is just ordinary quantum mechanics.

 

[samalkhaiat]

You know that a field with N components must transform under Lorentz transformations the same way as an object with spin (N-1)/2

Not correct! spinor-vector has six components, it transforms as spin-3/2 object not 5/2.

regards

sam

 

[quantumfireball]

but how to show it using quantum field theory?
that is in terms of an operator which acts on a fock space having eigenvalue=sqrt(s(s+1))
times the number of quanta in the fock space
im totally confused

 

[blechman]

Not correct! spinor-vector has six components, it transforms as spin-3/2 object not 5/2.

regards

sam

you're right. i was too sloppy with this comment. i retract it.

QFB: you can compute the angular momentum by studying the action of the spin operator on the field. Weinberg Vol 1 talks about this.

 

[quantumfireball]

thanks Dr Belchman