# How Can Maxwell's Equations Describe Both Photons and Electrons/Protons?

As an analogue to an existing question about how Maxwell's equations and photons relate [1], I'm curious how Maxwell's equations relate to charged particles, e.g. electrons and protons? That is, how does a single system of equations manage to describe the behavior of both charged matter (such as electrons and protons) and the propagation of photons. Particularly when the motivation and derivations I've seen all focused on the charged matter aspect exclusively.

I understand that both photons and electrons are (quantum mechanical) particles, while Maxwell's equations are about fields and continuous current/charge densities.

The kind of answer I'm looking for is, for example, if Maxwell's equations happen to be a reasonable approximation to 2 other sets of equations, one for photons and one for electrons.

• It doesn't. Maxwell's equations don't describe charged particles at all. Commented Dec 16, 2021 at 14:06
• Dirac equation is what describes electrons (fermions in general) Commented Dec 16, 2021 at 14:37
• Are you familiar with the Lorentz force law?
– J.G.
Commented Dec 16, 2021 at 14:38
• @Quantumwhisp -- The divergence of the electric field is a charge density. That certainly seems like Maxwell's equations are describing the behavior of charged particles, or at least the electromagnetic forces that create and exert on each other. Commented Dec 16, 2021 at 16:44
• @cjordan1 Can you derive the Lorentz-Force from maxwell's equations? Commented Dec 16, 2021 at 17:10

The following is not well-known, but (modified) Maxwell equations can indeed describe both electromagnetic field and electrons.

@Quantumwhisp commented: "Maxwell's equations don't describe charged particles at all", and then asked: "Can you derive the Lorentz-Force from maxwell's equations?"

I am not saying these comments are unreasonable, but, surprisingly, Dirac did derive the Lorentz force from Maxwell equations (Proc. Roy. Soc. London A 209, 291 (1951)).

I summarized Dirac's derivation elsewhere as follows.

Dirac considers the following conditions of stationary action for the free electromagnetic field Lagrangian subject to the constraint $$A_\mu A^\mu=k^2$$: $$$$\label{eq:pr1} \Box A_\mu-A^\nu_{,\nu\mu}=\lambda A_\mu,$$$$ where $$A^\mu$$ is the potential of the electromagnetic field, and $$\lambda$$ is a Lagrange multiplier. The constraint represents a nonlinear gauge condition. One can assume that the conserved current in the right-hand side of the equation is created by particles of mass $$m$$, charge $$e$$, and momentum (not generalized momentum!) $$p^\mu=\zeta A^\mu$$, where $$\zeta$$ is a constant. If these particles move in accordance with the Lorentz equations $$$$\label{eq:pr2} \frac{dp^\mu}{d\tau}=\frac{e}{m}F^{\mu\nu}p_\nu,$$$$ where $$F^{\mu\nu}=A^{\nu,\mu}-A^{\mu,\nu}$$ is the electromagnetic field, and $$\tau$$ is the proper time of the particle ($$(d\tau)^2=dx^\mu dx_\mu$$), then $$$$\label{eq:pr3} \frac{dp^\mu}{d\tau}=p^{\mu,\nu}\frac{dx_\nu}{d\tau}=\frac{1}{m}p_\nu p^{\mu,\nu}=\frac{\zeta^2}{m}A_\nu A^{\mu,\nu}.$$$$ Due to the constraint, $$A_\nu A^{\nu,\mu}=0$$, so $$$$\label{eq:pr4} A_\nu A^{\mu,\nu}=-A_\nu F^{\mu\nu}=-\frac{1}{\zeta}F^{\mu\nu}p_\nu.$$$$ Therefore, the last three equations are consistent if $$\zeta=-e$$, and then $$p_\mu p^\mu=m^2$$ implies $$k^2=\frac{m^2}{e^2}$$ (so far the discussion is limited to the case $$-e A^0=p^0>0$$).

Thus, the first equation with the gauge condition $$$$\label{eq:pr5} A_\mu A^\mu=\frac{m^2}{e^2}$$$$ describes both independent dynamics of electromagnetic field and consistent motion of charged particles in accordance with the Lorentz equations. The words "independent dynamics" mean the following: if values of the spatial components $$A^i$$ of the potential ($$i=1,2,3$$) and their first derivatives with respect to $$x^0$$, $$\dot{A}^i$$, are known in the entire space at some moment in time ($$x^0=const$$), then $$A^0$$, $$\dot{A}^0$$ may be eliminated using the gauge condition, $$\lambda$$ may be eliminated using the first equation for $$\mu=0$$ (the equation does not contain second derivatives with respect to $$x^0$$ for $$\mu=0$$), and the second derivatives with respect to $$x^0$$, $$\ddot{A}^i$$, may be determined from the first equation for $$\mu=1,2,3$$.

However, the above is about classical electrodynamics. What about quantum theory? It turns out that modified Maxwell equations can be equivalent to the Klein-Gordon-Maxwell electrodynamics or (with some caveats) to the Dirac-Maxwell electrodynamics (see my article Eur. Phys. J. C (2013) 73:2371 at https://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4 ).

That is, how does a single system of equations manage to describe the behavior of both charged matter (such as electrons and protons) and the propagation of photons.

Photons are quantum-mechanical, and Maxwell's equations are classical, so they don't describe photons. They do describe electromagnetic fields and waves.

Maxwell's equations don't directly predict everything about charged matter. However, if you have some externally imposed picture that provides at least some constraint on what your charged matter is like, then Maxwell's equations do provide quite a bit of predictive value. This is partly because Maxwell's imply conservation of energy, momentum, and charge, and in many cases those conservation laws are enough to predict what you want to know.

Maxwell equations alone do not determine the behavior of the electromagnetic field and charged particles. Rather, these equations describe:

• Behavior of free EM field
• Coupling of this field to charges and currents

Mathematically Maxwell equations are incomplete and need to be supported by the material equations, describing the response of the particles to the electromagnetic field. These can take form of simple phenomenological laws, such as the Ohm's law, $$\mathbf{j}=\hat{\sigma}\mathbf{E},$$ or appear as solutions of Schrödinger equation, and other various relativistic and non-relativistic models.