# DoF of a gauge boson

1. ### ismaili

161
As we know, the number of physical degrees of freedom(DoF) for a photon is 2.

I can understand this by gauging away redundant DoF's by gauge fixing.

For example, in QED, by fixing the Lorentz gauge $$\partial_\mu A^\mu = 0$$,

we could get rid of one DoF, moreover, the residual gauge symmetry, which is

$$A^\mu \rightarrow A^\mu + \partial^\mu f(x)$$

with $$\partial^2 f = 0$$ could allow us to remove another DoF.

This means the physical DoF of a photon is 2.

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However, on the other hand, we know that the virtual photon

which appearing in the internal legs of Feynman diagrams could have some longitudinal component.

And this longitudinal DoF could interact with other particles in a Feynman diagram.

However, this means the above symmetry argument in the first part of my post could NOT apply

to virtual photons. I don't know why. We could always gauge away two DoF's,

however, consideration of Feynman diagrams says that we could only gauge away 1 DoF of

virtual particles, why is that?

Thanks!

2. ### tiny-tim

26,043
hi ismaili!

virtual photons aren't real

they're mathematical inventions with an extra degree of freedom

3. ### tom.stoer

5,489
In QED you can fix the gauge in such a way that only the two physical polarizations do survive. I like the A°=0 and div A = 0 gauge better b/c A° is unphysical (its conjugate momentum is zero and it therefore acts as a Lagrange multiplier generating the Gauss law). Once you chose this gauge A° has been eliminated by constrcution and you can solve the Gauss law such that unphysical (longitudinal) polarizations are restricted to an unphysical Hilbert space which is (and remains under time evolution) orthogonal to the physical Hilbert space.

Constructing the physical, gauge-fixed Hamiltonian you only see two polarizations.

I know that in standard QFT textbooks this gauge is rarely discussed as Lorentz covariance is no longer visible and has to be checked for explicitly in all the calulations. Nevertheless it is useful in order to study physical degrees of freedom. In QCD you can show that both longitudinal gluons and even ghosts are absent in such physical gauges.

So the reason behind different number of degrees of freedom is a gauge artefact only.