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There are very strict limits on the mass of the photon already, so it would only affect our understanding of physics on the largest scales. The cosmologists would have some hard thinking to do, for instance. However, contrary to a comment, it would not affect relativity beyond requiring us to reconsider the usual name for $c$: not "the speed of light" but ...


2

You should be able to show by direct substitution that your proposed wave function $\tilde{\psi}(\mathbf{x},t)$ solves the Schroedinger equation, where the vector potential is included via the minimal coupling prescription $$ \mathbf{p} \to \mathbf{p} - \mathrm{i} q \mathbf{A}(\mathbf{x})$$ (up to a sign convention for $\mathbf{A}$ and some dimensionful ...


1

From right to left: outgoing electron $e^-$ spinor: $\bar{u}(p_2)$ QED vertex: $ie\gamma^\nu$ outgoing photon: $e^*(k_2)$ electron propagator: $\frac{i}{\not{q}-m+i0}$ incoming photon: $e(k_1)$ QED vertex: $ie\gamma^\mu$ incoming electron $e^-$ spinor: $u(p_1)$ There is no photon propagator in this process, and also, only one electron propagator. You ...



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