# Massive photons and gauge invariance

When trying to quantize Maxwell's equations $$$$\partial_{\mu}F^{\mu\nu}=0,$$$$ one takes into account gauge invariance $$$$A^{\mu}(x) \rightarrow A^{\mu}(x)+ \partial \Lambda(x).$$$$ 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 $$$$\nabla \cdot \boldsymbol{A}=0,$$$$ which is a standard procedure to do in QFT. The second state we want to remove is due to the scalar function $$\Lambda(x)$$, which has the property: $$$$\partial^{\mu} \partial_{\mu} \Lambda(x)=0.$$$$ 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

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