A massless spin $1$ particle has 2 degrees of freedom. However, we usually describe it using four-vectors, which have four components. Hence, somehow we must get rid of the superfluous degrees of freedom. This job is done by the Maxwell equations. To quote from Gilmore's "Lie Groups, Physics, and Geometry"
“In some sense, Maxwell's equations were a historical accident. Had the discovery of quantum
mechanics preceded the unification of electricity and magnetism, Maxwell's equations might not have
loomed so large in the history physics. … Since the quantum description has only two independent
components associated with each four momentum, there are four dimensions worth of linear
combinations of the classical field components that do not describe physically allowed states, for each
four momentum. Some mechanism must be derived for annihilating these superpositions. This
mechanism is the set of equations discovered by Maxwell. In this sense, Maxwell's equations are an
expression of our ignorance.”
An analogous argument can be made for the Dirac equation. For example at page 444 in Spin in Particle Physics by Elliot Leader he writes
At least for me this seems like a quite unconventional way to interpret the famous EOM's of QFT and I'm having problems to understand it fully. (I've read the accompanying sections in boths books, but wasn't fully able to grasp what they mean.) Therefore my questions:
- How does this interpretation of the equations of motion come about? (If it is correct, after all)
- Is there a textbook or paper that discusses this pedagogically?