A well-known result of Wigner's classification of relativistic particles is that massless particles transform with helicity $h \oplus -h$ under $ISO(2)$. Thus, such particles have two helicity states.
Wigner's classification in this form, while explaining why all massless particles have two helicity states, does not guarantee that all quantized massless fields have only two degrees of freedom (d.o.f.). The true d.o.f. are usually determined in a case-by-case basis. For instance, we achieve two on-shell degrees of freedom for a gauged vector field in 3+1 dimensions through (1) the equations of motion and (2) the choice of gauge.
Question: Does there exist a constructive proof for why all massless quantum fields have only two degrees of freedom?