Free real scalar Hamiltonian in oscillator variables ($:H:=\int \frac{d^{3}p}{(2\pi {)}^3}{E}_{\mathbf{p}}{a}_{\mathbf{p}}^{†}{a}_{\mathbf{p}}$)

Insert the real scalar field mode expansion into its quadratic Hamiltonian density and use the Fourier representation of the Dirac delta function. The two-annihilator and two-creator terms cancel by $E_{\mathbf{p}}^2={\mathbf{p}}^2+m^2$. The remaining symmetric expression is $H=\frac{1}{2}\int d^{3}p{E}_{\mathbf{p}}(a^{†}a+a{a}^{†})/(2\pi {)}^3$. Normal ordering removes its constant vacuum energy, yielding the displayed operator. The canonical commutation relations give $[:H:,a_{\mathbf{p}}]=-E_{\mathbf{p}}{a}_{\mathbf{p}}$ and $[:H:,a_{\mathbf{p}}^{†}]=E_{\mathbf{p}}{a}_{\mathbf{p}}^{†}$.