While Dirac's quantization has been an important inspiration for subsequent work by physicists and mathematicians, there have been certain speculations aimed at settling the initial idea of Dirac's singularity theory from 1931. One such is the two-photon theory. Let us briefly put it in the lens of history in this post.
Since Dirac used a 'singular' (non-local) vector potential, there are some reasonable problems with it. In 1962, Cabibbo and Ferrari proposed (see here) a two-photon theory with two vector potentials instead of one. One vector potential couples to the electric charge, and the other couples to the magnetic charge. In this way, we can avoid the Dirac singularity but at the cost of having two photons, one for each vector potential. However, two photons have not been observed so far.
When we write out symmetric Maxwell's equations with the addition of $\rho_m$, and $J_m$ being charge density and current density respectively, we see that the Maxwell's equations are invariant under interchanging between $E$ and $B$, which the nature implies the symmetry. However, instead of one four-vector potential $A_\mu$, we can write another $B_\mu$, which will be coupled with the magnetic charge in the same manner as $A_\mu$ is coupled with the electron charge. Actually, this is similar to how `t Hooft-Polyakov monopole is defined, except that $B_\mu$ is replaced by a Higgs field, and it becomes more of a topological potential than a gauge potential.
Salam proposed in this work that such a `magnetic' photon will not couple to the leptons but only to hadrons, while the electric photon will interact with both hadrons and leptons. Another variant is a large C-violation in electromagnetism, which is absurd at the experimental level. However, since the model does not require Dirac's singularity now, the electric and magnetic charges decouple, and hence there is no quantization condition. Moreover, the consideration of the mass of the magnetic photon being massless is about the convenience of the symmetry; it can be taken to be massive too due to a spontaneous symmetry breaking (see this). In the simplest speculation, if there are massive magnetic photons, then the quantization conditions will be revised.
While it enjoys the benefits of symmetry, it is not a model of 'quantization', at least the naive model presented here. Moreover, if one cares, the experimental searches for these magnetic photons have only negative results. The model of two potentials has been extended so far to the works here, here, and here.
While it enjoys the benefits of symmetry, it is not a model of 'quantization', at least the naive model presented here. Moreover, if one cares, the experimental searches for these magnetic photons have only negative results. The model of two potentials has been extended so far to the works here, here, and here.
