Equal time commutation relations in canonical quantization of relativistic free fields

As a matter of fact one could also discuss commutation relations at different time: $$[\phi(x),\phi(y)] = i E(x,y) I\quad (1)$$ for free fields $E$ is the so-called causal propagator or advanced minus retarded fundamental solution that depends on the free field equation satisfied by $\phi$.

The point is that, passing to considering interacting fields, at least formally, equal time commutation relations remain unchanged with respect to the free case, whereas the corresponding of (1) changes into an, in practice, unknown form as they include the full dynamics.

Actually even this idea does not work completely, as interacting fields $\Phi$ are affected by a renormalization constant $Z^{1/2}$: $$\Phi (t,\vec x) \to Z^{1/2} \phi(t, \vec x)\quad \mbox{in weak sense as } t \to \pm \infty$$ and, dealing with naively with renormalization procedure, it arises $Z=0$. So canonical commutation relations seem to be untenable for fields $\Phi(x), \partial_t \Phi(y) = \Pi(y)$. However all that sounds a bit academic as the renormalization procedure, in a sense, solves the problem.

I would like to stress that the fact that commutation relations are taken at equal time is not in contradiction with relativistic invariance: Covariance (i.e., the use of tensors and taking space and time on the same footing) is just one way to make explicit relativistic invariance, but it is by no means the unique one!

Hamiltonian formalism is not covariant, though it is relativistically invariant: all equations (including CCR) take the same form in every inertial reference frame.

Surely, the field and the momentum fail to commute when examined at the same instance, by extension of the logic of quantum mechanics. However, there is no reason why the field and momentum shouldn't commute when they are well-separated in time. Very loosely speaking, they just don't really have anything to do with each other. The logical conclusion is that one should use equal-time commutation relations that denote that the quantities do not commute at equal times but do commute when well-separated in time.

Let me sketch an answer elaborated in the reference A First Book of Quantum Field Theory by Lahiri and Pal (Second edition, Page $30$).

According to this reference above, the commutator $$[\phi(t,\textbf{x}),\pi(t,\textbf{y})]=i\delta^{(3)}(\textbf{x}-\textbf{y})\tag{1}$$ is actually covariant though the covariance is not manifest!

If two spacetime points coincide in one inertial frame, they will also coincide in different inertial frame. Therefore, at least for the special case when $x=y$ (i.e., $x^0=y^0=t$ and $\textbf{x}=\textbf{y}$), the commutation relation $(1)$ is satisfied in the same form in a different inertial frame.

Now, let us ask whether the relation $(1)$, in general, transforms covariantly under Lorentz transformation. The first thing to note is that $\phi$ is a scalar, and $\pi=\partial_0\phi$. Therefore, the Lorentz transformation property of the LHS comes from $\partial_0$ which makes it clear that LHS must transform like the time component of a four-vector. Now, since $$\int dt\int d^3\textbf{y}\delta^{(3)}(\textbf{x}-\textbf{y})=\int dt\tag{2}$$ and $d^4y=dtd^3\textbf{y}$ is Lorentz invariant, $\delta^{(3)}(\textbf{x}-\textbf{y})$ must also transform like the time component of a four-vector. Therefore, $(1)$ is covariant.