When trying to quantize Maxwell's equations \begin{equation} \partial_{\mu}F^{\mu\nu}=0, \end{equation} one takes into account gauge invariance \begin{equation} A^{\mu}(x) \rightarrow A^{\mu}(x)+ \partial \Lambda(x). \end{equation} From this gauge, we know that mass terms are forbidden in the Lagrangian and thus photons are massless. this is clear to me.

However, Nobel laureate M. Veltman says: Due to this gauge choice, we have unphysical photons in the formalism that cannot be observed. Thus we choose to give photons a little mass in order for the Lagrangian not to be gauge invariant anymore and thus getting rid of all unphysical photons.

Further, in order to recover the 2 states of polarization for the photons, we must cancel out two of the 4 components of $A^{\mu}(x)$. One is done by the Coulomb gauge \begin{equation} \nabla \cdot \boldsymbol{A}=0, \end{equation} which is a standard procedure to do in QFT. The second state we want to remove is due to the scalar function $\Lambda(x)$, which has the property: \begin{equation} \partial^{\mu} \partial_{\mu} \Lambda(x)=0. \end{equation} Now, all redundancies are removed and we recover 2 states of polarization. Again, M.Veltman uses a different method to get rid of the latter redundant. He takes the limit of the mass to zero and recovers two states of polarization. I think both ways have same physical meaning, but I don't see the connection

  • 2
    $\begingroup$ What is your question? $\endgroup$
    – Andrew
    Jun 23, 2021 at 12:44
  • 2
    $\begingroup$ Link to Veltman? Which page? $\endgroup$
    – Qmechanic
    Jun 23, 2021 at 12:45
  • $\begingroup$ isn't it forbidden for photons to have a mass due to gauge invariance of the Lagrangian? $\endgroup$
    – M91
    Jun 23, 2021 at 12:53
  • $\begingroup$ My reference is: Diagrammatica: The Path to Feynman diagrams. page 11 $\endgroup$
    – M91
    Jun 23, 2021 at 12:54
  • 3
    $\begingroup$ I think he's just regulating an infrared divergence. $\endgroup$ Jun 23, 2021 at 13:06


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