The paper claims zero-momentum photon modes generate the QED gauge transformation, but the transformation is forced by choosing the field expression and normalization that produce it.
It is easy to see that these two expressions give different predictions under the local gauge transformation in Eq.( 25)
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A quantum mechanism underlying the gauge symmetry in quantum electrodynamics
The paper claims zero-momentum photon modes generate the QED gauge transformation, but the transformation is forced by choosing the field expression and normalization that produce it.