A confusing point of the Hamiltonian for a particle interacting with electromagnetic fields In non-relativistic quantum theory the Hamiltonian for a particle interacting with electromagnetic fields is
$$H=\frac{(\mathbf{p}-\mathbf{A}*e/c)^2}{2m}+e\phi+\int\,d^3x \frac{\mathbf{E^2}+\mathbf{B^2}}{8\pi}.\tag{1}$$  
According to Hamiltonian's equations, 
$$\dot{r_i}=\frac{\partial H}{\partial p_i}, \tag{2}$$  $$\dot{p_i}=-\frac{\partial H}{\partial r_i}.\tag{3}$$
They certainly can not produce the equations of motion of the particle as well as the electromagnetic fields. Where am I wrong? What are the coordinates and the canonical momentum for the fields? 
 A: I) The Hamiltonian for point charges and EM fields can certainly produce the EOMs of the particle(s) as well as the EM fields. 
A full explanation is quite a long story. For pedagogical reasons, to see how this works, it is best to: 


*

*Firstly, understand the corresponding Lagrangian formulation.

*Secondly, understand how the Hamiltonian formulations work for point charges and EM fields separately, see e.g. this & this Phys.SE posts. 

*Thirdly, try to construct a Hamiltonian formulation for both point charges and EM fields together.
II) One correction: OP's Hamiltonian (1) yields the correct total energy, but OP asks how to produce Maxwell's equations. For the latter purpose, OP's Hamiltonian (1) is missing a Lagrange multiplier term that imposes Gauss' law.
III) Concretely, the minimal phase space is as follows: 


*

*Particle position ${\bf r}(t)$ and particle momentum ${\bf p}(t)$:
$$\{r^k(t), p_{\ell}(t)\}= \delta_{\ell}^k.$$

*(Minus$^1$) the electric field ${\bf E}(x)$ is the canonical conjugate variable to the magnetic gauge potential ${\bf A}(x)$: $$\{A_i({\bf x},t), E^j({\bf x}^{\prime},t)\}~=~ -\delta_i^j~\delta^3({\bf x}\!-\!{\bf x}^{\prime}).$$ 

*Lagrange multiplier $A^0(x)\equiv \phi(x)$.
IV) The equations come about as follows:


*

*The magnetic field ${\bf B}\equiv{\bf \nabla}\times {\bf A}$ is defined as the curl of the magnetic gauge potential ${\bf A}$.

*The Hamilton's equations for ${\bf r}$ and ${\bf p}$ yield (i) the
Newton's 2nd law with a Lorentz force, and (ii) the relation between velocity $\dot{\bf r}$ and momentum ${\bf p}$.

*The Hamilton's equations for ${\bf A}$ and ${\bf E}$ yield (i) the
Maxwell–Ampere's law, and (ii) the relation between the electric field ${\bf E}$ and the gauge potential $A_{\mu}$.

*The Lagrange multiplier $A^0\equiv\phi$ imposes Gauss' law.

*The source-free Maxwell equations follows from the existence of the gauge potential $A_{\mu}$.
--
$^1$ We use $(-,+,+,+)$ Minkowski sign convention with $c=1$.
A: Remove the second term in the hamiltonian altogether. In the first term only consider the field of other particles. 
A: Normally the EMF Hamiltonian must be written in terms of $Q$s and $P$s of the electromagnetic field which is done by representing it via harmonic oscillators. You may find decomposition of EMF into a set of harmonic oscillators in many textbooks.
