= Solution
Define $E_{\mathbf p}=\sqrt{\kappa^2|\mathbf p|^4+m^2}$ and $\int_{\mathbf p}=\int d^3p/(2\pi)^3$. The Hermitian <quantum field> has the <mode expansion of a free field>
$$
\boxed{\phi(t,\mathbf x)=\int_{\mathbf p}\frac1{\sqrt{2E_{\mathbf p}}}
\left[a_{\mathbf p}e^{-iE_{\mathbf p}t+i\mathbf p\cdot\mathbf x}
+a_{\mathbf p}^\dagger e^{iE_{\mathbf p}t-i\mathbf p\cdot\mathbf x}\right],}
$$
where the <annihilation operators> and <creation operators> obey the bosonic <canonical commutation relations>
$$
\boxed{[a_{\mathbf p},a_{\mathbf q}^\dagger]=(2\pi)^3\delta^{(3)}(\mathbf p-\mathbf q),\qquad
[a_{\mathbf p},a_{\mathbf q}]=[a_{\mathbf p}^\dagger,a_{\mathbf q}^\dagger]=0.}
$$
Because $\pi=\dot\phi$, its <mode expansion of a free field> is
$$
\pi(t,\mathbf x)=-i\int_{\mathbf p}\sqrt{\frac{E_{\mathbf p}}2}
\left[a_{\mathbf p}e^{-iE_{\mathbf p}t+i\mathbf p\cdot\mathbf x}
-a_{\mathbf p}^\dagger e^{iE_{\mathbf p}t-i\mathbf p\cdot\mathbf x}\right].
$$
The two nonzero cross <commutators> then give
$$
[\phi(t,\mathbf x),\pi(t,\mathbf y)]
=\frac i2\int_{\mathbf p}\left[e^{i\mathbf p\cdot(\mathbf x-\mathbf y)}+e^{-i\mathbf p\cdot(\mathbf x-\mathbf y)}\right]
=i\delta^{(3)}(\mathbf x-\mathbf y),
$$
and the equal-time field-field and momentum-momentum <commutators> vanish. This verifies the factor $1/\sqrt{2E_{\mathbf p}}$ without requiring <Lorentz invariance>.
Substitution into the <Hamiltonian>, followed by <normal ordering>, yields
$$
\boxed{{:H:}=\int_{\mathbf p}E_{\mathbf p}a_{\mathbf p}^\dagger a_{\mathbf p}.}
$$
Consequently $[{:H:},a_{\mathbf p}^\dagger]=E_{\mathbf p}a_{\mathbf p}^\dagger$: one application of a <creation operator> adds precisely the required <quasiparticle> <energy>. Before <normal ordering> there is the usual constant zero-point <vacuum energy>. If $m=0$, the exactly zero-<momentum> mode has $E_{\mathbf0}=0$ and is a free canonical coordinate, not an oscillator with a normalizable oscillator <vacuum state>; in a finite box handle that mode separately or introduce a small positive <mass>. The formula applies directly to all modes with $E_{\mathbf p}>0$.
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