DoF of a Gauge Boson - Why Only 1 for Virtual Photons?

In summary, the physical degrees of freedom for a photon is 2, and this can be understood by using gauge fixing in QED to eliminate redundant DoF's. However, virtual photons in Feynman diagrams can have a longitudinal component and interact with other particles, meaning that the symmetry argument used for real photons does not apply. This is due to the choice of gauge, which can restrict unphysical polarizations to an unphysical Hilbert space. Therefore, the different number of degrees of freedom is a result of gauge artefacts.
  • #1
ismaili
160
0
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 [tex] \partial_\mu A^\mu = 0 [/tex],

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

[tex] A^\mu \rightarrow A^\mu + \partial^\mu f(x) [/tex]

with [tex] \partial^2 f = 0 [/tex] could allow us to remove another DoF.

This means the physical DoF of a photon is 2.

------

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!
 
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  • #2
hi ismaili! :smile:

virtual photons aren't real

they're mathematical inventions with an extra degree of freedom :wink:
 
  • #3
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.

http://adsabs.harvard.edu/abs/1994AnPhy.233...17L
 

1. What is the DoF (Degree of Freedom) of a Gauge Boson?

The DoF of a Gauge Boson refers to the number of independent physical properties or modes of motion that the particle can possess. In the Standard Model of particle physics, gauge bosons are considered to have a DoF of 2, which means they can have two distinct states or polarization modes.

2. Why is there only 1 DoF for Virtual Photons?

This is due to the nature of virtual photons, which are not observed as free particles but rather as intermediate particles in interactions between other particles. Virtual photons are not subject to the same physical laws as real photons and therefore their DoF is not constrained by the same rules.

3. How does the DoF of a Gauge Boson affect its interactions with other particles?

The DoF of a Gauge Boson can affect its interactions with other particles in a number of ways. For example, the DoF can determine the strength of the interaction, the type of interaction, and the range of the interaction. The DoF also plays a role in determining the scattering cross section of the particle.

4. Why is it important to understand the DoF of a Gauge Boson?

Understanding the DoF of a Gauge Boson is crucial in the study of particle physics as it allows us to accurately predict and explain the behavior of particles in different interactions. It also helps in the development of new theories and models that can better explain the fundamental forces of nature.

5. Are there any exceptions to the DoF of a Gauge Boson?

While the DoF of a Gauge Boson is generally accepted to be 2 in the Standard Model of particle physics, there are some exceptions to this rule. For example, in some theories beyond the Standard Model, gauge bosons may have a DoF of 3 or even 4. However, these theories are still being researched and are not yet widely accepted.

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