Does the Heisenberg equation for fields and canonical momentums hold as well for the hamiltonian density operator instead of the Hamiltonian operator?

In quantum field theory, with the field $\phi$ and the momentum $\pi$ being operators, their time evolution is governed (in the Heisenberg-picture) by the Heisenberg equation:

$$\begin{align} \dot{\phi} = \frac{i}{\hbar}[ \hat{H}, \phi] \\ \dot{\pi} = \frac{i}{\hbar}[ \hat{H}, \pi]. \\ \end{align}$$

Now, in case the Hamiltonian operator $\hat{H}=\int d^3x ~\hat{\cal H}$ can be written as an integral over the hamiltonian density $\hat{\cal H}$, and the fields and the momenta commute at non-equal positions, do the same equations hold as well with the Hamiltonian operator being replaced by it's density? What would the caveats be?

$$\begin{align} \dot{\phi} = \frac{i}{\hbar}[ \hat{\cal H}, \phi] \\ \dot{\pi} = \frac{i}{\hbar}[ \hat{\cal H}, \pi]. \\ \end{align}$$

Answer

You have $\hat{H} = \int d^3x \hat{\tilde{H}}(x)$. That implies that canonical Relations will be slightly altered.

For a Quantum field Operator $\hat{\phi}(x',t)$ distributed over space $x'$ and time $t$, you will have a relation like the following:

$[\hat{\tilde{H}}(x),\hat{\phi}(x',t)] = \frac {\partial}{\partial t} \hat{\phi(x',t)} \delta(x-x')$.

The Delta function factor ensures not only the commutation of Operators for nonequal space Points; also that after Integration over space, the ordinary commutation Relations are obtained