# longitudinal polarization of the electromagnetic field

I'm just trying to quantize the electromagnetic field. It has four independent components, where two of them are physical and belong to the (transverse) polarization. But why does it not have a longitudinal component. Is it because photons are massless?

Yes, because photons are massless. A 4-vector describing a photon gauge field has initially 4 degrees of freedom. Temporal degree of freedom is fictitious, because it (edit: its time derivative) does not appear in the kinetic term of the Lagrangian (thus it does not propagate). Then, you can fix gauge symmetry (say, $$\partial_\mu A^\mu=0$$) to eliminate one other d.o.f. The two that are left are the transverse ones. You can check this by using the Lorentz gauge $$\partial_\mu A^\mu=0$$ in momentum space: $$k_\mu A^\mu=0$$. Since there is no rest frame, you can only choose your frame so that the photon moves, say along the $$x$$ ($$\mu=1$$) axis. That means $$k_\mu=(-E,E,0,0)$$ and from the Lorentz condition we have $$-EA^0+EA^1=0$$. Thus, longitudinal mode is fictitious too.