Sunday, 10 June 2018

quantum field theory - Counting massive degrees of freedom after gauge fixing


Consider the theory of scalar QED with the Lagrangian L=14FμνFμν+(Dμϕ)(Dμϕ)m2ϕϕ where ϕ is a complex scalar field with mass m. Counting the degrees of freedom, we have




  • two massless real degrees of freedom from Aμ

  • two massive real degrees of freedom from ϕ


Now, even though there's no symmetry breaking going on here we can still choose to go to unitary gauge, i.e. fixing the gauge so that ϕ is real. We now have the gauge-fixed Lagrangian L=14FμνFμν+(Dμφ)(Dμφ)12m2φ2 where φ is a canonically normalized real scalar field, and there is no gauge symmetry. Then we have



  • three real degrees of freedom from Aμ

  • one massive real degree of freedom from ϕ


where I know there are three real degrees of freedom in Aμ, because gauge fixing always removes one and we have no gauge fixing here.



What confuses me is that there must be two massive degrees of freedom, just as there were in the original theory. So that somehow means that one of the degrees of freedom in Aμ is massive while the other two aren't -- but how can that be? There's no mass term for Aμ in sight.



Answer



What you can remove with a field redefinition (of the form of a U(1) gauge transformation) is the phase of ϕ. But the total number of degrees of freedom (dof's) is not going to change since the resulting lagrangian is not longer gauge invariant. The new counting is 3+1 where the 3 dof's come from Aμ (with a covariant constraint, see below, so that really 3=41), and the 1 dof comes instead from the scalar radial mode of ϕ.


More explicitly, write ϕ(x)=eieπ(x)h(x)2 (with both π and h(x) real scalar fields) and the covariant derivative Dμϕ=(μieAμ)ϕ becomes Dμϕ=eieπ[ie(μπAμ)+12μh] so that the lagrangian reads L=14FμνFμν+e22h2(μπAμ)2+12(μh)2m22h2. Now, defining the new variable Aμ=Aμμπ (which is the gauge invariant combination), the lagrangian becomes L=14FμνFμν+e22h2A2μ+12(μh)2m22h2 in full analogy to the abelian Higgs mechanism except that h has vanishing vacuum expectation value. Now, this last lagrangian depends on the fields Aμ and h: how many dof's are there? Well, the h(x) certainly counts 1. The Aμ on the other hand counts 3 and neither 4 (even though μ=0,1,2,3) nor 2. Indeed, from the equations of motion μFμν=e2h2Aν we see the covariant constrain μ(Aμh2)=0 which sends us from 4 to 3. But notice that there is no gauge invariance for Aμ in its lagrangian above (the e2h2A2μ-term breaks it) and no longitudinal mode can be therefore removed in that way: Aμ is a different physical configuration than say AμμΩ (in particular one solves the equation of motion, the other doesn't). So the counting ends here and it matches that for massive spin-1 plus a real scalar, that is 3+1 as expected. Notice however, that the spin-1 particle is not massive and all this gymnastic is just artificial as you wanted to move around one scalar dof π inside Aμ.


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