= 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 $\phi=\int d\Pi_p(ae^{-ipx}+b^\dagger e^{ipx})$ and its adjoint reproduce equal-time <canonical commutation relations> when $[a(\mathbf p),a^\dagger(\mathbf q)]=[b(\mathbf p),b^\dagger(\mathbf q)]=(2\pi)^3 2E_p\delta^{(3)}(\mathbf p-\mathbf q)$ and cross commutators vanish. The <canonical momenta> are $\pi=\dot\phi^\dagger$, $\pi^\dagger=\dot\phi$. An overall phase on $b$ can reverse the sign of the <antiparticle> term without changing the theory.
Back to article page
