The reason Maxwell theory cannot describe the photon is that the radiation
phenomena is mutual energy phenomena.
It is not a self-energy phenomena.
traditional solutions of electromagnetic fields make mistakes here. Mutual energy phenomena includes, mutual energy theorem, mutual energy flow flow theorem, mutual energy principle. All this relates mutual inductance. Self-energy phenomena includes self-energy flow (Poynting vector energy flow), self-energy principle(self-energy flow do not carry energy). All this relates to the self-inductance.
Energy conservation law

How to describe mutual energy radiation phenomena? Maxwell equations
4 formula should be add another formula which is the energy conservation
law. Assume there are $N$ current sources:$\boldsymbol{J}_{i}$,
$i=1,...N$. The corresponding fields are $\xi_{i}=[\boldsymbol{E}_{i},\boldsymbol{H}_{i}]$,
One current $\boldsymbol{J}_{i}$ will offer another current $\boldsymbol{J}_{j}$
some power,
\begin{equation}
P_{ij}=\iiint_{V}(\boldsymbol{J}_{j}\cdot\boldsymbol{E}_{i})dV\label{eq:1}
\end{equation}
The above is the power current $\boldsymbol{J}_{i}$ lost. This power
is received by the current $\boldsymbol{J}_{j}$.
\begin{equation}
P_{ji}=\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j})dV\label{eq:2}
\end{equation}
is the power current $\boldsymbol{J}_{j}$ give to the current $\boldsymbol{J}_{i}$.
When $\boldsymbol{J}_{i}$ lost some energy this energy, it will be
received by current $\boldsymbol{J}_{j}$. Hence, the total energy
will not change, that means,
\begin{equation}
\intop_{t=-\infty}^{\infty}(P_{ji}+P_{ji})dt=0\label{eq:3}
\end{equation}
Consider all $N$ current sources, there is,
\begin{equation}
\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\intop_{t=-\infty}^{\infty}dt\iiint_{V}(\boldsymbol{J}_{j}\cdot\boldsymbol{E}_{i})dV=0\label{eq:4}
\end{equation}
This formula is self-explanatory (introduced by Shuang-ren Zhao).
It should add to the Maxwell equations.
Mutual energy principle
Another formula which should
also add to Maxwell equation which is the mutual energy principle
(introduced by Shuang-ren Zhao)
\begin{equation}
-\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\iint_{\Gamma}(\boldsymbol{E}_{i}\times\boldsymbol{H}_{j})\cdot\hat{n}d\Gamma=\sum_{i=1}^{N}\sum_{J=1,j\neq i}^{N}\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j}+\frac{\partial}{\partial t}(\boldsymbol{E}_{i}\cdot\boldsymbol{D}_{j}+\boldsymbol{H}_{i}\cdot\boldsymbol{B}_{j}))dV\label{eq:5}
\end{equation}
The mutual energy principle can be derived from Maxwell equations
by adding some conditions. The conditions are the Maxwell equation
must established as pairs. In each pair there are solution for transmitting
antenna and receiving antenna. Or pair for emitter and absorber. This
means assume the receiving antenna and absorber also radiate waves.
This also means assume the radiation is a mutual energy phenomena,
that receiving antenna and absorber must also join to the radiation
theory.
$R$ is the set of the solution of retarded wave. $A$ is the set of the solution of the advanced wave. $R\cup A$ is the set of the solutions of Maxwell's equations. $R\cap A$ is the set of the solution the mutual energy principle. $R\cap A$ is the solutions of physics. $R\cap A$ is possible the solution of physics, but is also possible a invalid solutions.

We also can build the electromagnetic field theory by adding the above
mutual energy principle formula, then the above descriptions can be
derived from the mutual energy principle.
Assume both energy conservation law and the mutual energy principle
are accept by us as two new axioms. From these two laws we can prove
that,
\begin{equation}
\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\intop_{t=-\infty}^{\infty}dt\iint_{\Gamma}(\boldsymbol{E}_{i}\times\boldsymbol{H}_{j})\cdot\hat{n}d\Gamma=0\label{eq:6}
\end{equation}
$\Gamma$ is the boundary of the volume $V$. It can be chosen as
big sphere with radius as infinity. This means there should no mutual
energy flow go to the outside of our universe. This is clear a correct
theorem. In order the above formula as 0, the two electromagnetic
fields $\xi_{i}=[\boldsymbol{E}_{i},\boldsymbol{H}_{i}]$ and $\xi_{j}=[\boldsymbol{E}_{j},\boldsymbol{H}_{j}]$
must be one is retarded wave and another is advanced wave. The retarded
wave reaches the surface at a future time. The advanced wave reaches
the surface at a past time. The electromagnetic fields will not nonzero
in the same time in the surface, hence the surface integral will be
0.
We can assume the current sends the retarded wave as transmitting
antenna or emitter. The current sends advanced wave as receiving antenna
or absorber. Hence, transmitting antenna and emitter must radiate
the retarded wave. The receiving antenna and the absorber must radiate
advanced wave.
about the advanced wave, Wheeler and Feynman have the absorber theory.
John Cramer has the transactional interpretation of the quantum mechanics.
The mutual energy flow theorem, photon is the mutual energy flow
From the mutual energy principle the mutual energy flow theorem can
be derived. Assume $N=2$, the mutual energy flow theorem is,
\begin{equation}
-\intop_{t=-\infty}^{\infty}dt\iiint_{V_{1}}(\boldsymbol{J}_{1}\cdot\boldsymbol{E}_{2})dV=(\xi_{1},\xi_{2})=\intop_{t=-\infty}^{\infty}dt\iiint_{V_{2}}(\boldsymbol{J}_{2}\cdot\boldsymbol{E}_{1})dV\label{eq:7}
\end{equation}
The flowing is the mutual energy flow:
\begin{equation}
(\xi_{1},\xi_{2})=\intop_{t=-\infty}^{\infty}dt\iint_{\Gamma}(\boldsymbol{E}_{1}\times\boldsymbol{H}_{2}+\boldsymbol{E}_{2}\times\boldsymbol{H}_{1})\cdot\hat{n}d\Gamma\label{eq:8}
\end{equation}
$\Gamma$ is any surface which separates the volume $V_{1}$ and $V_{2}$. See the following figure for the shape of the mutual energy flow.
$\iint_{\Gamma}(\boldsymbol{E}_{1}\times\boldsymbol{H}_{2}+\boldsymbol{E}_{2}\times\boldsymbol{H}_{1})\cdot\hat{n}d\Gamma$ is the mutual energy flow. The mutual energy flow is defined in contrast
to the self-energy flow: $\iint_{\Gamma}(\boldsymbol{E}_{1}\times\boldsymbol{H}_{1})\cdot\hat{n}d\Gamma$
$\iint_{\Gamma}(\boldsymbol{E}_{2}\times\boldsymbol{H}_{2})\cdot\hat{n}d\Gamma$.
$(\xi_{1},\xi_{2})$ is the mutual energy go through the surface $\Gamma$
through the mutual energy flow. Mutual energy flow do not decrease
like wave. The amplitude of the wave will decrease when it propagates.
The mutual energy flow will not decrease when it propagates. The mutual
energy flow is very thin when it is radiated or received. The mutual
energy will be thick between it its source and sink. Hence, the mutual
energy flow looks very like a photon. We can say the photon actually
is the mutual energy flow.
According these theory, the self-energy flow or self-energy radiation
do not transfer energy in space. Self-energy flow is the normal wave
solution of Maxwell equations (the Maxwell equation only for one current
source). This wave are canceled by time-reversal waves. there are
two kinds of time-reversal waves corresponding to retarded wave and
the advanced wave.
The following is a figure of photon. The emitter sends the retarded wave, the absorber sends the advanced wave. The retarded wave and the advanced wave either are reactive wave or they collapse back. The mutual energy flow bring the photon energy from the emitter to the absorber.

Wave collapse
In quantum mechanics waves collapse, this can be shown as,

In the mutual energy theory, the wave collapse actually is done by a wave backward-collapse process and a mutual energy flow process:

Summary, (1) photon is not a wave, but it is the mutual energy flow.
The mutual energy flow is built by the retarded wave sends from an
emitter and the advanced wave sends from an absorber. (2) there are
4 waves, retarded wave, advanced wave and two time-reversal waves.
All 4 waves cancel each other. However, the mutual energy flow survive.
(3) Wave collapse can be described by the two phenomena: The energy
is transferred through the mutual energy. The retarded wave and the
advanced wave is canceled by the time-reversal waves.
If this theory is interesting, the details can be google searched
by the keyword mutual energy principle'' or
mutual energy flow'',
``self-energy principle''.