Why Does a Photon Not Have a Spin Component Sz=0?

[ndung200790]

Please teach me this problem:
Photon has spin=1,so there are three values of Sz=-1,0,1.But I know that a real photon has only left and right handed polarizations.There is not longitudinal polarization.Because of that there not exist states with Sz=0.I don't understand this problem.
Please give me a favor to explain this for me.
Thank you for advanced.

 

[tom.stoer]

There are two ways to discuss this problem.

1) For massless particles one should replace spin by helicity; the symmetry is then an E2 rotation in the spatial (transversal) 2-plane which results in two helizities +1 and -1 (this is what you call usually spin +1 and spin -1).
For a good reference see Ryder's book on quantum field theory

2) For massless photons (quantum) electrodynamics has an U(1) gauge symmetry. Usually one eliminates one A-component via gauge fixing but essentially one can show that one eliminates two! This is best understood in the A°=0 gauge. This gauge is nice b/c a) A° is an unphysical field b/c there is no time derivative and therefore no canonical conjugate momentum (the diagonal terms in the F-tensor are zero). So let's set A°=0. A° acts as a lagrangian mutiplier; the equation of cosntraint is the Gauss law which has to be solved for. This is equivalent to eliminate the longitudinal photon. The Gauss law reflects the fact that after gauge fixing A°=0 there is still a gauge symmetry left, namely the gauge symmetry with time-independent gauge functions which do not spoil A° = A'° = 0. So after A°=0 and solving the Gauss law two polarizations have been eliminated.