= Covariant oscillator Hamiltonian of a real scalar field
{title2=$:H:=\int d\Pi_p\,E_p a^\dagger(p)a(p)$}
With $d\Pi_p=d^3p/[(2\pi)^3 2E_p]$, a free <real scalar field> has <mode expansion of a free field> $\phi=\int d\Pi_p(ae^{-ipx}+a^\dagger e^{ipx})$ and <commutator> $[a(p),a^\dagger(q)]=(2\pi)^3 2E_p\delta^3(\mathbf p-\mathbf q)$. Substitution into the quadratic <Hamiltonian density> gives $H=\frac12\int d\Pi_p E_p(a^\dagger a+aa^\dagger)$. The terms with two <annihilation operators> or two <creation operators> cancel by $E_p^2=\mathbf p^2+m^2$. <Normal ordering> removes the constant <vacuum energy>. The <commutators> $[H,a(p)]=-E_pa(p)$ and $[H,a^\dagger(p)]=E_pa^\dagger(p)$ give the <boson> energy without changing under this subtraction. Replacing $a(p)$ by $\sqrt{2E_p}\,c(p)$ recovers the noncovariant normalization $[c(p),c^\dagger(q)]=(2\pi)^3\delta^3(\mathbf p-\mathbf q)$.
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