It is possible to say that if the canonical commutation relations aren't satisfied, then we cannot "study" the dynamics of a quantum system?
The similarity between Hamiltonian mechanics and quantum mechanics is paramount. On one hand, it is quite clear to me that the canonical relations in classical mechanics just tells us:
Given the pair (roughly, a coordinate chart for the phase space) of positions and momenta $(q^{i},p_{j})$, it was found that the obey:
\begin{cases}{ q^{i},q^{j} }{\mathrm{PB}}={ p{i},p_{j} }{\mathrm{PB}} = 0 \ { q^{i},p{j} }{\mathrm{PB}} = \delta^{i}\hspace{0.3mm}{j}, \tag{1}\end{cases}
for another coordinate chart $(Q^{i}((q^{m},p_{n})),P_{j}(q^{n},p_{m}))$, it is necessary that they "maintain" the form of equations $(1)$, as:
\begin{cases}{ Q^{i},Q^{j} }{\mathrm{PB}}={ P{i},P_{j} }{\mathrm{PB}} = 0 \ { Q^{i},P{j} }{\mathrm{PB}} = \delta^{i}\hspace{0.3mm}{j}. \tag{2}\end{cases}
If we have the "coordinate independece" given by $(2)$, then the same equations of motion, for both the Hamiltonian and for a observable $F$, is guaranteed; i.e.
$$\frac{\partial H}{\partial P} = \dot{Q}, \frac{\partial H}{\partial Q} = -\dot{P} \tag{3}$$
and $$\frac{d f}{d t} = {f,H}_{\mathrm{PB}} \tag{4}$$
are validy just if we have coodinate independence just like $(2)$.
A similar picture is drawn for quantum mechanics of one, finitely many and infinite degrees of freedom (the later is roughly the prototype for Quantum field theory).
My question is: when the text book says "I postulate the comutation relations for the fields" or "the annahilation and creation operators satisfy the commutation relations", can I interpret these steatements as:
"I must to fullfil commutation relations because if not the Heisenberg dynamical equation is useless, then I can't do quantum field theory"?
The question is a bit odd, but I think it makes sense because, often the one starts with a classical field equation, then writes the field in terms of annihilation/creation operators (this is basically a coordinate change); since the one postulated that the fields satisfy the commutation relations the fields satisfy the Heisenberg equation. 101
Then the one searches if the annihilation/creation operators also satisfies the commuation relations, rendering a easier Hamiltionian. Therefore, the whole procedure is somekind of a big "coordinate changing". Then, we define a well-posed quantum framework. 675