in general this is complicated but we only care about it for � μ G μ a 0 then

In general this is complicated but we only care about

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in general, this is complicated — but we only care about it for η μ G μ a = 0 — then the first term on the RHS vanishes and the change in η μ G μ a is independent of G — just a constant — infinite but uninteresting — NO GHOSTS this is called “temporal gauge” if η μ η μ > 0 , it is called “axial gauge” if η μ η μ < 0 , and it is called “light-cone gauge” if η μ η μ = 0 — so you see that there are many gauges in which gauge fixing works very simply — you just impose a condition on G μ to break the symmetry and calculate — for gauge invariant quantities, the results are independent of which gauge you chose 21
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Δ( G ) = Π x,a fl fl fl fl [ dη μ G μ Ω a ] [ d Ω] fl fl fl fl η μ G μ Ω a =0 because we are evaluating at η μ G μ a = 0 we only need infinitesimal transformations Ω = 1 + idω a T a G μ = T a G μ a → Ω G μ Ω - 1 - i Ω ∂ μ Ω - 1 → dG μ a = i [ dω a T a , T b G μ b ] - T a ∂ μ dω a = - T a ∂ μ dω a - i [ T b G μ b , dω a T a ] = - D μ ( T a dω a ) d ( η μ G μ a ) = η μ dG μ a = i [ dω a T a , T b η μ G μ b ] - T a η μ ∂ μ dω a in general, this is complicated — but we only care about it for η μ G μ a = 0 — then the first term on the RHS vanishes and the change in η μ G μ a is independent of G — just a constant — infinite but uninteresting — NO GHOSTS this is called “temporal gauge” if η μ η μ > 0 , it is called “axial gauge” if η μ η μ < 0 , and it is called “light-cone gauge” if η μ η μ = 0 — so you see that there are many gauges in which gauge fixing works very simply — you just impose a condition on G μ to break the symmetry and calculate — for gauge invariant quantities, the results are independent of which gauge you chose 22
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funny propagator — look at one G μ a and drop the a — assume η μ η μ 6 = 0 L = - 1 4 ‡ ∂ μ G ν - ∂ ν G μ ·‡ ∂ μ G ν - ∂ ν G μ · + Kη μ G μ + s μ G μ + · · · K equation of motion η μ G μ = 0 G ν equation of motion ∂ μ ∂ L ∂ μ G ν = ∂ L ∂G ν - ∂ μ ‡ ∂ μ G ν - ∂ ν G μ · = Kη ν + s ν 0 = ∂ ν ‡ Kη ν + s ν · ⇒ K = - ( ∂s ) ( η∂ ) ( η∂ )( ∂G ) = - η ν ∂ μ ‡ ∂ μ G ν - ∂ ν G μ · = η ν ‡ Kη ν + s ν · = - ( ηη ) ( ∂s ) ( η∂ ) + ( ηs ) ( ∂G ) = ( η∂ )( ηs ) - ( ηη )( ∂s ) ( η∂ ) 2 - ( ∂∂ ) G ν = s ν - η ν ( ∂s ) ( η∂ ) - ∂ ν ( η∂ )( ηs ) - ( ηη )( ∂s ) ( η∂ ) 2 = - - g νμ + ∂ ν η μ + η ν ∂ μ ( η∂ ) - ( ηη ) ∂ ν ∂ μ ( η∂ ) 2 s μ 23
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another example of gauge fixing functions — f ( G, φ ) = Π x,a δ ‡ ∂ μ G μ a ( x ) · I call this “Landau gauge” (also called “Lorenz gauge” or “Lorentz gauge”) K a ∂ μ G μ a so that the equation of motion for K a is ∂ μ G μ a = 0 what is Z f ( G Ω , φ Ω ) [ d Ω] ? Δ( G, φ ) - 1 = Z f ( G Ω , φ Ω ) [ d Ω] = Z Π x,a δ ‡ ∂ μ G μ Ω a ( x ) · [ d Ω] = Z Π x,a δ ‡ ∂ μ G μ Ω a ( x ) · 1 `fl fl fl fl [ d∂ μ G μ Ω a ] [ d Ω] fl fl fl fl [ d∂ μ G μ Ω a ] = Π x,a ˆ 1 , fl fl fl fl [ d∂ μ G μ Ω a ] [ d Ω] fl fl fl fl ∂ μ G μ Ω a =0 !
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