Quantizing complex scalar fields
# Scalar QED
Definition: Scalar QED
Scalar QED consists of a massless vector and a complex scalar field.
The lagrangian for scalar QED is given by $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + |D_\mu\phi|^2 - m^2|\phi|^2$$ with the covariant derivative $$D_\mu\phi = \partial_\mu\phi + ieA_\mu\phi$$ and $$F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu.$$
The scalar field $\phi$ needs to be complex! This is because
- $A_\mu$ transforms under gauge transformation
- Can therefore only couple to fields that account for that transformation.
- Need two d.o.f. to have nontrivial gauge transformation. In the complex case this is $\phi\to e^{-i\alpha}\phi$
E.o.M: $$(\Box + m^2)\phi = i(-eA_\mu)\partial_\mu\phi + i\partial_\mu(-eA_\mu\phi)+(-eA_\mu)^2\phi$$ $$(\Box +m^2)\phi^*=i(eA_\mu)\partial_\mu\phi^*+i\partial_\mu(eA_\mu\phi^*)+(eA_\mu)^2\phi^**$$
- The two d.o.f. have the same equation, but the coupling charge $e\to -e$: $\phi^*$ is the antiparticle to $\phi$
The quantization for the real scalar field was
$$\phi(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}} (a_p e^{-ipx} + a^\dagger_p e^{ipx})$$
For the field to be complex $\phi\neq\phi^*$ we have to introduce two sets of creation and annihilation operators $a_p$ and $b_p$ leading to
Quantization of complex scalar field
The complex scalar field $\phi(x)$ is quantized as $$\phi(x)=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_pe^{-ipx}+b^\dagger_pe^{ipx})$$ and the complex conjugate follows as $$\phi^*(x)=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a^\dagger_pe^{ipx}+b_pe^{-ipx})$$
$b_p$ annihilates particles of opposite charge and same mass compared to $a_p$: Antiparticle
Matter coupling to spin-1 $\rightarrow$ Phase rotation symmetry (to have gauge symmetry) $\rightarrow$ charge $\rightarrow$ complex fields $\rightarrow$ antiparticles