Connection between definitions of "conjugate momentum density" as "generator of displacement of the field" and as "Lagrangian partial derivative"

Question

I am reading Jakob Schwichtenberg Physics from Symmetry where in 5.2 conjugate momentum density $\pi(x)$ is defined as generator of displacement of the field itself (1): $$ \pi(x) = −i\hbar\frac{\partial}{\partial Φ(x)}\tag{1} $$

From that definition it follows that $[Φ(x), π(y)] = i\hbar δ(x − y).$

So far so good, but in 9.1 we define (2) $$ \begin{equation} \pi(x) = \frac{\partial {L}}{\partial (\partial_0 Φ(x))}\tag{2} \end{equation} $$ and still use $[Φ(x), π(y)] = i\hbar δ(x − y)$ that was derived from definition (1)

So my two questions

1. What is the connection between two distinct definitions of $\pi(x)$?

2. Why we can still use $[Φ(x), π(y)] = i\hbar δ(x − y)$ for the definition (2)?

Answer 1

Ref. 1 is possibly a bit skimpy on details at the 2 places mentioned by OP. (OP's issue already arises in QM and is transcribed in a natural way to QFT.) Let us stress the following facts:

• The momentum definition $(2)$ is valid within the context of a classical Lagrangian formulation.

• It is then implicitly understood that we next Legendre transform to a classical Hamiltonian formulation with fundamental Poisson brackets $$\{\Phi(x),\Phi(y)\}~=~ 0, \qquad \{\Phi(x),\pi(y)\}~=~ \delta(x-y), \qquad \{\pi(x),\pi(y)\}~=~0,$$ which we, in turn, quantize to a quantum mechanical formulation with a CCR that OP mentions.

• The momentum definition $(1)$ is the Schrödinger position representation of this CCR. (There exist other representations of the CCR, such as, e.g., the Schrödinger momentum representation, cf., e.g. this Phys.SE post.)

• Moreover, the CCR implies that the momentum operator is the generator of position translations.

References:

1. J. Schwichtenberg, Physics from Symmetry, 2nd edition, 2018.