Canonical equal time commutation relations in QED I understand that to quantize the classical electromagnetic field one needs to impose commutation relations and express the field in terms of creation and annihilation operators. I notice that the canonical equal-time commutation rule comes in to use here. How do you derive this? What is it meant to represent?
 
Also, how do you prove the following commutation relation? 

 A: In classical Hamiltonian mechanics one has the generalized coordinates $q^{i}(t)$ and momenta $p_{i}(t)$. The Poisson bracket is defined as,
$$
[F,G]_{PB}=\frac{\partial F}{\partial q^{k}}\frac{\partial G}{\partial p_{k}}-\frac{\partial F}{\partial p_{k}}\frac{\partial G}{\partial q^{k}}\ .
$$ 
Using the q's and p's in place of $F$ and $G$ one has the fundamental PBs,
$$
[q^{i}(t),p_{j}(t)]_{PB}=\delta^{i}_{k}\delta^{k}_{j}=\delta^{i}_{j} 
$$
$$
[q^{i}(t),q^{j}(t)]_{PB}=0
$$
$$
[p_{i}(t),p_{j}(t)]_{PB}=0 \ .
$$
In order to set up a quantum theory, Dirac says that the phase space functions $F$ and $G$ are changed to operators $\hat{F}$ and $\hat{G}$ and the PB becomes the commutator,
$$
[\hat{F},\hat{G}]_{-}=\hat{F}\hat{G}-\hat{G}\hat{F}=i\widehat{[F,G]_{PB}} \ .
$$
So, going over to the quantum theory in this way, the fundamental PBs become commutators,
$$
[\hat{q}^{i}(t),\hat{p}_{j}(t)]_{-}=i\delta^{i}_{j} 
$$
$$
[\hat{q}^{i}(t),\hat{q}^{j}(t)]_{-}=0
$$
$$
[\hat{p}_{i}(t),\hat{p}_{j}(t)]_{-}=0 \ .
$$
These three commutators are the ones in the question. This is because the generalized coordinates for the classical electromagnetic field are,
$$
q^{i}(t)\rightarrow q^{(\mu,x)}(t) \rightarrow A_{\mu}(t,x) \ .
$$
The generalized momenta $p_{i}(t)$ will be the values of a vector field $\pi^{\mu}(t,x)$ and so the first fundamental commutator becomes,
$$
[\hat{A}_{\mu}(t,x),\hat{\pi}^{\nu}(t,y)]_{-}=i\delta^{\nu}_{\mu}\delta(x-y)
$$
and this is the first commutator in the question. The other two commutators in the question follow from the remaining two commutators for the q's and p's with a bit of raising/lowering indices of four-vectors.
