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 positive kinetic coefficient $\zeta$ in ${\mathcal{L}}_0=\zeta ({\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi )$, the canonical momentum is ${\pi }_{\varphi }=\zeta {\dot{\varphi }}^{\ast }$. Positive-energy oscillator modes with unit oscillator commutators have amplitude $1/\sqrt{2\zeta E_{\mathbf{p}}}$. The mixed time-ordered propagator is $C={\zeta }^{-1}{D}_F$, with $D_F$ the ordinary scalar propagator. This also follows by inverting the quadratic form in the path integral. Rescaling $\varphi$ changes interaction coefficients as well as external correlation-function factors.

With the invariant measure $d{\Pi }_p=d^{3}p/[(2\pi {)}^{3}2E_p]$, a free complex scalar field has particle and antiparticle modes. At any fixed time the displayed inverse and $b^{†}(p)=\int d^{3}x{e}^{-ip\cdot x}(E_p\varphi -i\dot{\varphi })$ project onto its positive- and negative-frequency components. The equal-time canonical commutation relations then give $[a(p),a^{†}(q)]=[b(p),b^{†}(q)]=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$; cross terms vanish because their coefficient contains $E_p-E_q$ on the momentum delta function.

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.

Substituting the free complex scalar field modes into $i\int ({\varphi }^{†}\dot{\varphi }-{\dot{\varphi }}^{†}\varphi )d^{3}x$ gives the formally ordered expression $Q_{\mathrm{bare}}=\int d{\Pi }_p(a^{†}a-b{b}^{†})$. Its divergent vacuum constant is removed by normal ordering, equivalently by choosing $Q|0⟩=0$. The resulting complex scalar charge operator has $[Q,a^{†}]=a^{†}$ and $[Q,b^{†}]=-b^{†}$. Subtracting the constant changes neither its action on fields nor current conservation, but it is essential when identifying the vacuum charge and the charges of one-particle states.

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.