 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.