Since the electromagnetic field was about the only field known and understood in the early days of quantum physics, 1925 - 30, it was the first to be included in the earliest formulations of quantum physics. Even so, it posed some fairly tricky mathematical problems. And when quantum field theory came along in 1930, quantizing the field itself resulted in some very subtle insights. The first practical applications were to transitions between energy levels in atoms, and in molecules, and soon in nuclei. To obtain a covariant description of electromagnetism we use the vector potential A and the scalar potential φ or A₀. They satisfy B = ∇ × A and E = −∇A₀ − (1/c)∂A/∂t. The basic tool for including electromagnetic interactions into ordinary quantum physics is the minimal electromagnetic substition, adding qA₀ to the Hamiltonian and replacing p by p − qA/c. Since this course is about "subatomic physics," we will consider electromagnetic transitions only in nuclei.  [They tend not to occur in baryons and mesons since the electromagnetic process is very slow compared to strong processes.]

The explanation for why a massless spin-1 particle has only two polarization states, both perpendicular to the particle's momentum, is found as a result of considerations that arise in quantum field theory, so we will just accept it here.   The unphysical state proved an initially baffling problem when a unified theory of weak and electromagnetic interactions was constructed, depending crucially on the Higgs mechanism, as we shall see later.

You can think of the source of the E transitions as being the charge probability distribution of the system, and the source of the M transitions as being the charge probability current distribution of the system... so the moments are polar vectors in one case and axial vectors in the other. Hence the difference in the parity selection rule.  It is less and less likely for the emitted photon to carry away more and more angular momentum, so the fastest transition will, if possible, always be the one where it carries off only its own spin angular momentum.

The lables for single particle states are the traditional spectroscopic labels of j, total angular momentum, and parity.

Even-even nuclei generally have states in order 0+, 2+, 4+, etc., so that almost all transitions are E2.  Even-odd nuclei display spectra of single particle states, while odd-odd nuclei tend to be rare and pathological.

Why does one of these γ transitions take nearly 5 hours, when the others take place in a very, very, very tiny fraction of a second (ns or ps)?

How we verified the existence of color experimentally.  A convenient list of particle accelerators, past and present, can be found here.

Maxwell's equations with monopoles in MKS units.  Here is an article about the history of magnetic monopoles in modern physics.

Magnetic monopoles were studied by Dirac in 1931; their hypothetical existence can be used to explain charge quantization. However, experimentally there is no trace of such particles in our universe. Monopoles do feature in some efforts to extend physics beyond the Standard Model. It's fun thinking about how Maxwell's equations look if monopoles existed in everyday life--- the modifications are simple and straightforward and in fact make the equations much more symmetric, as shown above. Where are they? [Recent biography of Dirac.]

In the Standard Model there is no obvious reason why CP-violating processes should not be observed for strong interactions. The axion boson was first suggested in 1978 independently by Frank Wilczek and Steven Weinberg, as a necessary consequence of the earlier Peccei–Quinn theory, to address the strong CP problem in quantum chromodynamics (QCD). It was suggested that if the free parameter (an angle θ) in the CP-violating term of the QCD Lagrangian density were actually a pseudoscalar field, with a 0⁻ boson called the axion, the suppression of such processes could be understood. Axions have not been detected (they would have properties much like the π⁰ meson but with much, much smaller mass, probably much less than 1 eV), but their properties are so fascinating that there has been a lot of effort to detect them in various ways.

Maxwell's equations with axions!

Shining light through solid walls!  In a strong magnetic field, photons can convert to axions, or axions can convert to photons.

The axion could couple to photons, gluons, leptons and quarks. It is also a candidate for “dark matter,” as we will discuss later.

By the way, since Maxwell's equations are already relativistically correct, it is straightforward to put them in covariant form, without changing much of anything other than notation.

Have you ever heard of the incredible war between proponents of vectors and proponents of quaternions, which had a drastic effect on the history of physics in England in the late 19th Century?

A famous modern application of quaternions is to avoid the notorious problem of gimbal lock.