= Normal-ordered Hamiltonian of a free complex scalar field
{title2=$H=\int d\Pi_p E_p(a^\dagger a+b^\dagger b)$}
Spatial integration of the free <Hamiltonian density> cancels particle-<antiparticle> pair terms through $E_p^2-\mathbf p^2-m^2=0$. The raw <Hamiltonian> is $\int d\Pi_p E_p(a^\dagger a+bb^\dagger)$. <Normal ordering> removes the field-independent <vacuum energy> and gives $H=\int d\Pi_p E_p(a^\dagger a+b^\dagger b)$. Both <creation operators> increase the energy by $E_p$, so a negative-frequency field mode represents a positive-energy <antiparticle>, not a negative-energy state.
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