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1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f.
2) For gauge theories with massless gauge bosons like QED and QCD it is well known that the 4-vector Aμ carries two unphysical d.o.f. which can be eliminated by gauge fixing (a la Dirac, Gupta-Bleuler, BRST, ...). An obvious way to see this is to
i) use the temporal gauge A° = 0 to eliminate one unphysical d.o.f. A° (∏° = 0 b/c there's no ∂°A° in the Lagrangian ~ F²)
ii) keep the corresponding Euler-Lagrange equation (Gauss law G) as constraint to define the physical Hilbert space as its kernel G|phys> = 0 which fixes the residual gauge symmetry of time-indep. gauge transformations ∂°θ = 0
b/c we have 4 components in Aμ and 2 gauge fixing conditions A° = 0 and G ~ 0 we arrive at 4-2 = 2 d.o.f.
The method 2) gives us exactly the two helicity states described in 1) But 1) is using Poincare invariance whereas 2) is using gauge invariance w/o ever looking at Poincare invariance. So it seems that it's sheer coincidence that 1) and 2) arrive at the same results.
Where's the relation?
2) For gauge theories with massless gauge bosons like QED and QCD it is well known that the 4-vector Aμ carries two unphysical d.o.f. which can be eliminated by gauge fixing (a la Dirac, Gupta-Bleuler, BRST, ...). An obvious way to see this is to
i) use the temporal gauge A° = 0 to eliminate one unphysical d.o.f. A° (∏° = 0 b/c there's no ∂°A° in the Lagrangian ~ F²)
ii) keep the corresponding Euler-Lagrange equation (Gauss law G) as constraint to define the physical Hilbert space as its kernel G|phys> = 0 which fixes the residual gauge symmetry of time-indep. gauge transformations ∂°θ = 0
b/c we have 4 components in Aμ and 2 gauge fixing conditions A° = 0 and G ~ 0 we arrive at 4-2 = 2 d.o.f.
The method 2) gives us exactly the two helicity states described in 1) But 1) is using Poincare invariance whereas 2) is using gauge invariance w/o ever looking at Poincare invariance. So it seems that it's sheer coincidence that 1) and 2) arrive at the same results.
Where's the relation?