= Solution
For a free real <scalar field> in four spacetime dimensions, $\mathcal L=(\partial\phi)^2/2-m^2\phi^2/2$ and $\pi=\dot\phi$. Canonical quantization promotes these canonical fields to Hermitian operator-valued distributions with equal-time <canonical commutation relations>
$$
[\phi(t,\mathbf x),\pi(t,\mathbf y)]=i\delta^{(3)}(\mathbf x-\mathbf y),\qquad[\phi(t,\mathbf x),\phi(t,\mathbf y)]=[\pi(t,\mathbf x),\pi(t,\mathbf y)]=0.
$$
Expand the solution of the <Klein-Gordon equation> into positive- and negative-frequency modes:
$$
\phi(x)=\int d\Pi_k\,[a(\mathbf k)e^{-ikx}+a^\dagger(\mathbf k)e^{ikx}],\quad d\Pi_k=\frac{d^3k}{(2\pi)^3 2\omega_k},\quad\omega_k=\sqrt{\mathbf k^2+m^2}.
$$
Hermiticity pairs the two coefficients as adjoints. The field commutators are equivalent to
$$
[a(\mathbf k),a^\dagger(\mathbf q)]=(2\pi)^3 2\omega_k\delta^{(3)}(\mathbf k-\mathbf q),\qquad[a,a]=[a^\dagger,a^\dagger]=0.
$$
For example, inserting the expansion in $[\phi,\dot\phi]$ yields two equal Fourier integrals, each with coefficient $i/2$, and hence $i\delta^{(3)}$. The vacuum obeys $a(\mathbf k)|0\rangle=0$, and repeated <creation operators> generate the symmetric bosonic <Fock space>. <Normal ordering> of the quadratic <Hamiltonian> removes its vacuum constant and gives $H=\int d\Pi_k\,\omega_k a^\dagger a$; the same computation for the spatial <momentum> gives the stipulated <four-momentum> $P^\mu=\int d\Pi_k\,k^\mu a^\dagger a$.
The oscillator algebra now implies
$$
[P^\mu,a(\mathbf k)]=-k^\mu a(\mathbf k),\qquad[P^\mu,a^\dagger(\mathbf k)]=k^\mu a^\dagger(\mathbf k).
$$
Consequently
$$
\begin{aligned}
[P^\mu,\phi(x)]&=\int d\Pi_k\,[-k^\mu a(\mathbf k)e^{-ikx}+k^\mu a^\dagger(\mathbf k)e^{ikx}]\\
&=\boxed{-i\partial^\mu\phi(x)}.
\end{aligned}
$$
Indeed, $\partial^\mu e^{\mp ikx}=\mp ik^\mu e^{\mp ikx}$. For $\mu=0$, the result is $[H,\phi]=-i\dot\phi$, which is the <Heisenberg equation of motion> with the correct sign. As usual, the field identities are interpreted after smearing the operator-valued distributions, or on a suitable common finite-particle domain.
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