In this answer my2cts says "The electromagnetic field is to photons what the Schrödinger or Klein-Gordon wave function is to electrons." Could someone expand on this further? Is this just a descriptive analogy, or is there some kind of mathematical equivalence between the field description of e.g. photons and the vector spaces & operators description of particles?
Overall I am confused at how exactly to decompose the statement and whether there is some deeper understanding to be gained from it.
There is some truth to the statement, but it is rather unhelpful. The electromagnetic four-potential $A^{\mu}$ is a field. That is to say, it is a four-vector assigned to every point in space and time. A free electromagnetic field obeys Maxwell's equations: $$\Box A^{\mu}=0$$
Schrödinger's equation $\hat{H}|\Psi(t)\rangle=i\hbar\frac{d}{dt}|\Psi(t)\rangle$ describes the time evolution of the state. In position space, we can identify the wave function $\Psi(\textbf{r}_1,\textbf{r}_2, ...,\textbf{r}_N, t)=\langle\textbf{r}_1,\textbf{r}_2, ...,\textbf{r}_N|\Psi(t)\rangle$ which obeys the position space Schrödinger equation: $$\sum_{i=1}^N\left(-\frac{\hbar^2}{2m_i}\nabla_i\right)\Psi+V(\textbf{r}_1,\textbf{r}_2, ...,\textbf{r}_N, t)\Psi=i\hbar\frac{\partial}{\partial t}\Psi$$ The wave function $\Psi(\textbf{r}_1,\textbf{r}_2, ...,\textbf{r}_N, t)$ is not a field*. It has a value not for every point in our real space and time $(\textbf{r},t)$, but for every point in configuration space and time ($\textbf{r}_1,\textbf{r}_2, ...,\textbf{r}_N,t)$. It is especially tempting to conflate the two in the case of one particle, but they are not the same.
Things are very different when we apply quantum field theory. In QFT, a free spin-0 field and a free spin-1/2 respectively obey the Klein-Gordon equation: $$(\Box+m^2)\phi=0 $$ and the Dirac equation: $$ (i\not \partial-m)\psi = 0$$ In this case, $\phi$ and $\psi$ are fields like $A^{\mu}$. In the same way that photons are quanta of $A^{\mu}$, electrons are quanta of $\psi$ and Higgs bosons are quanta of $\phi$. As we have established, a field is not the same as a wave function. Rather, fields are operators which act on the state. I guess the main takeaway is this:
In non-relativistic quantum mechanics: There is a state $|\Psi(t)\rangle$. Physical quantities such as the position of a particle $\textbf{r}_i$ are operators which act on $|\Psi(t)\rangle$. The wave function $\Psi(\textbf{r}_1,\textbf{r}_2, ...,\textbf{r}_N, t)$ is a way of representing the state in position space.
In quantum field theory: There is a state $|\Psi\rangle$**. Fields such as $\phi$, $\psi$ and $A^\mu$ are operators which act on $|\Psi\rangle$. These fields give rise to particles such as the Higgs boson, electron and photon. There is no wave function$\text{***}$.
I agree with Ghoster that the correct statement would be "The quantized electromagnetic field is to photons what the quantized Dirac field is to electrons."
$\text{*}$ To make matters worse, there is such a thing as a Schrödinger field, which obeys the same differential equation as the single-particle wave function. Still, they are two different things.
$\text{**}$ It is often useful in QFT to use the Heisenberg picture in which the state remains constant and the operators (fields) change with time.
$\text{***}$ There is something called the wave functional $\Psi[\phi]$ which is essentially the same thing but instead of being a function of the position configurations of a system it is a function of the field configurations.
The statement is based on the fact that all free matter in special relativity obeys the Klein Gordon equation. The latter expresses the Einstein energy momentum expression, which is the cornerstone of special relativity. The wave equation of the electromagnetic four-potential is exactly the massless Klein Gordon equation of a Lorentz vector potential. To avoid theoretical hairsplitting, let's discuss the dual slit experiment. Photons, electrons and even molecules all exhibit the same experimental behaviour as described here. The probability of finding a particle on the screen is proportional to the square of the EM potential field/wave function (use Schrodinger, Dirac or Klein-Gordon as you prefer).
Of course, when many particles must be described and Fermi, Bose and Coulomb correlations as well as interactions come into play things are different, let alone when condensed matter and molecules are involved. My statement was not intended for such cases.
