= Free real scalar Hamiltonian in oscillator variables
{title2=$:H:=\int\frac{d^3p}{(2\pi)^3}E_{\mathbf p}a^\dagger_{\mathbf p}a_{\mathbf p}$}
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 $H=\frac12\int d^3p\,E_{\mathbf p}(a^\dagger a+aa^\dagger)/(2\pi)^3$. <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 $[:H:,a^\dagger_{\mathbf p}]=E_{\mathbf p}a^\dagger_{\mathbf p}$.
Back to article page
