I've read and heard that quantum electrodynamics is more fundamental than maxwells equations. How do you go from quantum electrodynamics to Maxwell's equations?
Electromagnetism – How to Transition from Quantum Electrodynamics to Maxwell’s Equations
electromagnetismmathematical physicsmaxwell-equationsquantum-electrodynamicsspecial-relativity
Related Solutions
For what it's worth, throughout my career as a professor of theoretical physics specializing in high energy physics and the foundations of Lorentz covariant quantum theory, I never heard of Harmuth or his modification of Maxwell's equations until reading this question and the indicated foreword to his book. Now it's hard to keep track of everything going on in science, even within ones specialty. But I think it's safe to say that Harmuth's theory is not accepted by the physics community as an improvement on Maxwell's equations. There have been proposed modifications that have received extensive attention and study, such as Born-Infeld electrodynamics, and ever since the development of the Electro-Weak unified theory our understanding of the status of the EM field has been altered. But no proposals for modifying Maxwell's equations, per se, have been accepted as established improvements.
Maxwellian electrodynamics fails when quantum mechanical phenomena are involved, in the same way that Newtonian mechanics needs to be replaced in that regime by quantum mechanics. Maxwell's equations don't really "fail", as there is still an equivalent version in QM, it's just the mechanics itself that changes.
Let me elaborate on that one for a bit. In Newtonian mechanics, you had a time-dependent position and momentum, $x(t)$ and $p(t)$ for your particle. In quantum mechanics, the dynamical state is transferred to the quantum state $\psi$, whose closest classical analogue is a probability density in phase space in Liouvillian mechanics. There are two different "pictures" in quantum mechanics, which are exactly equivalent.
In the Schrödinger picture, the dynamical evolution is encoded in the quantum state $\psi$, which evolves in time according to the Schrödinger equation. The position and momentum are replaced by static operators $\hat x$ and $\hat p$ which act on $\psi$; this action can be used to find the expected value, and other statistics, of any measurement of position or momentum.
In the Heisenberg picture, the quantum state is fixed, and it is the operators of all the dynamical variables, including position and momentum, that evolve in time, via the Heisenberg equation.
In the simplest version of quantum electrodynamics, and in particular when no relativistic phenomena are involved, Maxwell's equations continue to hold: they are exactly the Heisenberg equations on the electric and magnetic fields, which are now operators on the system's state $\psi$. Thus, you're formally still "using" the Maxwell equations, but this is rather misleading as the mechanics around it is completely different. (Also, you tend to work on the Schrödinger picture, but that's beside the point.)
This regime is used to describe experiments that require the field itself to be quantized, such as Hong-Ou-Mandel interferometry or experiments where the field is measurably entangled with matter. There is also a large gray area of experiments which are usefully described with this formalism but do not actually require a quantized EM field, such as the examples mentioned by Anna. (Thus, for example, black-body radiation can be explained equally well with discrete energy levels on the emitters rather than the radiation.)
This regime was, until recently, pretty much confined to optical physics, so it wasn't really something an electrical engineer would need to worry about. That has begun to change with the introduction of circuit QED, which is the study of superconducting circuits which exhibit quantum behaviour. This is an exciting new research field and it's one of our best bets for building a quantum computer (or, depending on who you ask, the model used by the one quantum computer that's already built. ish.), so it's something to look at if you're looking at career options ;).
The really crazy stuff comes in when you push electrodynamics into regimes which are both quantum and relativistic (where "relativistic" means that the frequency $\nu$ of the EM radiation is bigger than $c^2/h$ times the mass of all relevant material particles). Here quantum mechanics also changes, and becomes what's known as quantum field theory, and this introduces a number of different phenomena. In particular, the number of particles may change over time, so you can put a photon in a box and come back to find an electron and a positron (which wouldn't happen in classical EM).
Again, here the problem is not EM itself, but rather the mechanics around it. QFT is built around a concept called the action, which completely determines the dynamics. You can also build classical mechanics around the action, and the action for quantum electrodynamics is formally identical to that of classical electrodynamics.
This regime includes pair creation and annihilation phenomena, and also things like photon-photon scattering, which do seem at odds with classical EM. For example, you can produce two gamma-ray beams and make them intersect, and they will scatter off each other slightly. This is inconsistent with the superposition principle of classical EM, as it breaks linearity, so you could say that the Maxwell equations have failed - but, as I pointed out, it's a bit more subtle than that.
Best Answer
Disclaimer: This is answer is given from a mathematical physics point of view, and it is a little bit technical. Any comment or additional answer from other points of view is welcome.
The classical limit of quantum theories and quantum field theories is not straightforward. It is now a very active research topic in mathematical physics and analysis.
The idea is simple: by its own construction, quantum mechanics should reduce to classical mechanics in the limit $\hslash\to 0$. I don't think it is necessary to go into details, however for QM this procedure is now well understood and rigorous form a mathematical standpoint.
For QFTs, such as QED, the situation is similar, although more complicated, and it can be mathematically handled only in few situations. Although it has not been proved yet, I think it is possible to prove convergence to classical dynamics for a (simple) model of QED, describing rigid charges interacting with the quantized EM field.
The Hilbert space is $\mathscr{H}=L^2(\mathbb{R}^{3})\otimes \Gamma_s(\mathbb{C}^2\otimes L^2(\mathbb{R}^3))$ ($\Gamma_s$ is the symmetric Fock space). The Hamiltonian describes an extended charge (with charge/mass ratio $1$) coupled with a quantized EM field in the Coulomb gauge: \begin{equation*} \hat{H}=(\hat{p} - c^{-1} \hat{A}(\hat{x}))^2+\sum_{\lambda=1,2}\hslash\int dk\;\omega(k)a^*(k,\lambda)a(k,\lambda)\; , \end{equation*} where $\hat{p}=-i\sqrt{\hslash}\nabla$ and $\hat{x}=i\sqrt{\hslash}x$ are the momentum and position operators of the particle, $a^{\#}(k, \lambda)$ are the annihilation/creation operators of the EM field (in the two polarizations) and $\hat{A}(x)$ is the quantized vector potential \begin{equation*} \hat{A}(x)=\sum_{\lambda=1,2}\int \frac{dk}{(2\pi)^{-3/2}}\;c\sqrt{\hslash/2\lvert k\rvert}\;e_\lambda(k)\chi(k)(a(k,\lambda)e^{ik\cdot x}+a^*(k,\lambda)e^{-ik\cdot x})\; ; \end{equation*} with $e_\lambda(k)$ orthonormal vectors such that $k\cdot e_\lambda(k)=0$ (they implement the Coulomb gauge) and $\chi$ is the Fourier transform of the charge distribution of the particle. The magnetic field operator is $\hat{B}(x)=\nabla\times \hat{A}(x)$ and the (perpendicular) electric field is $$\hat{E}(x)=\sum_{\lambda=1,2}\int \frac{dk}{(2\pi)^{-3/2}}\;\sqrt{\hslash\lvert k\rvert/2}\;e_\lambda(k)\chi(k)i(a(k,\lambda)e^{ik\cdot x}-a^*(k,\lambda)e^{-ik\cdot x})\; $$
$\hat{H}$ is a self adjoint operator on $\mathscr{H}$, if $\chi(k)/\sqrt{\lvert k\rvert}\in L^2(\mathbb{R}^3)$, so there is a well defined quantum dynamics $U(t)=e^{-it\hat{H}/\hslash}$. Consider now the $\hslash$-dependent coherent states \begin{equation*} \lvert C_\hslash(\xi,\pi,\alpha_1,\alpha_2)\rangle=\exp\Bigl(i\hslash^{-1/2}(\pi\cdot x+i\xi \cdot\nabla)\Bigr)\otimes\exp\Bigl(\hslash^{-1/2}\sum_{\lambda=1,2}(a^*_\lambda(\alpha_\lambda)-a_\lambda(\bar{\alpha}_\lambda))\Bigr)\Omega\; , \end{equation*} where $\Omega=\Omega_1\otimes\Omega_2$ with $\Omega_1\in C_0^\infty(\mathbb{R}^3)$ (or in general regular enough, and with norm one) and $\Omega_2$ the Fock space vacuum.
What it should be at least possible to prove is that ($\alpha_\lambda$ is the classical correspondent of $\sqrt{\hslash}a_\lambda$, and it appears inside $E(t,x)$ and $B(t,x)$ below): \begin{gather*} \lim_{\hslash\to 0}\langle C_\hslash(\xi,\pi,\alpha_1,\alpha_2),U^*(t)\hat{p}U(t)C_\hslash(\xi,\pi,\alpha_1,\alpha_2)\rangle=\pi(t)\\ \lim_{\hslash\to 0}\langle C_\hslash(\xi,\pi,\alpha_1,\alpha_2),U^*(t)\hat{x}U(t)C_\hslash(\xi,\pi,\alpha_1,\alpha_2)\rangle=\xi(t)\\ \lim_{\hslash\to 0}\langle C_\hslash(\xi,\pi,\alpha_1,\alpha_2),U^*(t)\hat{E}(x)U(t)C_\hslash(\xi,\pi,\alpha_1,\alpha_2)\rangle=E(t,x)\\ \lim_{\hslash\to 0}\langle C_\hslash(\xi,\pi,\alpha_1,\alpha_2),U^*(t)\hat{B}(x)U(t)C_\hslash(\xi,\pi,\alpha_1,\alpha_2)\rangle=B(t,x)\; ; \end{gather*} where $(\pi(t),\xi(t),E(t,x),B(t,x))$ is the solution of the classical equation of motion of a rigid charge coupled to the electromagnetic field: \begin{equation*} % \left\{ \begin{aligned} &\left\{\begin{aligned} \partial_t &B + \nabla\times E=0\\ \partial_t &E - \nabla\times B=-j \end{aligned}\right. \mspace{20mu} \left\{\begin{aligned} \nabla\cdot &E=\rho\\ \nabla\cdot &B=0 \end{aligned}\right.\\ &\left\{\begin{aligned} \dot{\xi}&= 2\pi\\ \dot{\pi}&= \frac{1}{2}[(\check{\chi}*E)(\xi)+2\pi\times(\check{\chi}*B)(\xi)] \end{aligned}\right. \end{aligned} % \right. \end{equation*} with $j=2\pi\check{\chi}(\xi-x)$, and $\rho=\check{\chi}(\xi-x)$ (charge density and current).
To sum up: the time evolved quantum observables averaged over $\hslash$-dependent coherent states converge in the limit $\hslash\to 0$ to the corresponding classical quantities, evolved by the classical dynamics.
Hoping this is not too technical, this picture gives a precise idea of the correspondence between the classical and quantum dynamics for an EM field coupled to a charge with extended distribution (point charges cannot be treated mathematically on a completely rigorous level both classically and quantum mechanically).