Canonical quantization of a complex scalar field

A free complex scalar field has independent particle and antiparticle mode operators. With $d{\Pi }_p=d^3\mathbf{p}/[(2\pi {)}^{3}2E_p]$, the expansion $\varphi =\int d{\Pi }_p(a{e}^{-ipx}+b^{†}{e}^{ipx})$ and its adjoint reproduce equal-time canonical commutation relations when $[a(\mathbf{p}),a^{†}(\mathbf{q})]=[b(\mathbf{p}),b^{†}(\mathbf{q})]=(2\pi {)}^{3}2E_p{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$ and cross commutators vanish. The canonical momenta are $\pi ={\dot{\varphi }}^{†}$, ${\pi }^{†}=\dot{\varphi }$. An overall phase on $b$ can reverse the sign of the antiparticle term without changing the theory.

For the global phase transformation $\delta \varphi =-i\alpha \varphi$, the Noether current is $j^{\mu }=i({\varphi }^{†}{\partial }^{\mu }\varphi -{\partial }^{\mu }{\varphi }^{†}\varphi )$. The Klein-Gordon equation gives ${\partial }_{\mu }{j}^{\mu }=0$. With zero vacuum charge, its normal-ordered charge is $Q=\int d{\Pi }_p(a^{†}a-b^{†}b)$. Thus $[Q,a^{†}]=a^{†}$, $[Q,b^{†}]=-b^{†}$ and $[Q,\varphi ]=-\varphi$. Particle and antiparticle creation carry opposite charges while both increase energy.

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 $\int d{\Pi }_p{E}_p(a^{†}a+b{b}^{†})$. Normal ordering removes the field-independent vacuum energy and gives $H=\int d{\Pi }_p{E}_p(a^{†}a+b^{†}b)$. 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.