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) $$ \begin{equation} \pi(x) = \frac{\partial {\cal L}}{\partial (\partial_0 Φ(x))}\tag{2} \end{equation} $$ 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 placed mentioned by OP. (OP's issue already arise in QM, and transcribes 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 a 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 exists 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.