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I keep reading that the idea of massive photons leads to an explanation of the Meissner effect but I fail to see how photons are involved with the repulsion of fields inside a superconductor.

How can a photon effect the electromagnetic fields inside a superconductor?

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Superconductors don't strictly exclude interior magnetic fields. Over a short distance, called a screening length, external magnetic fields can penetrate into a superconductor. This length can be discussed in terms of an effective photon mass, borrowing concepts from particle physics. The heavier a virtual particle, the shorter the distance it can travel before annihilation. The W and Z bosons are very massive and are carriers of extremely short range forces. The longer-range a force, the lighter its force-carrying boson.

Photons are massless, and therefore may carry forces over infinite distances. If describing superconductivity using photons as force carriers that enforce the electromagnetic field condition inside the superconductor, then their finite action distance requires those particles to have mass. This is not to say that there are literally massive photons inside a superconductor, but such a description is possible.

A more detailed treatment is in J.G. Hey, 2004, section 19.2.

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In fact if you employ Maxwell equations with massive photons, e,g, proca theory. Then electromagnetic interaction inside superconductor would have short range effect. This is hypothesized with London equations. With this treatment superconductivity may normally result from converting electromagnetic energy inside the superconductor so that photons move like a superfluid.

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