Complex Scalar Field

A complex scalar describes spin-zero particles carrying a conserved global charge, together with antiparticles carrying the opposite charge. It is the natural free starting point for charged bosons, scalar electrodynamics, and Bose fields before interactions are introduced.

A complex field and its charge

In flat spacetime, let $m>0$ and

$$\mathcal L=\partial_\mu\phi^*\partial^\mu\phi-m^2\phi^*\phi.$$

Writing $\phi=(\phi_1+i\phi_2)/\sqrt2$ gives two real scalar fields with identical masses. The phase transformation $\phi\mapsto e^{i\alpha}\phi$ rotates that pair. To fix the charge sign, use $D_\mu=\partial_\mu-iqA_\mu$ for an external source and define the current by $\delta\mathcal L/\delta A_\mu=qj^\mu$ at $A=0$. Then

$$j^\mu=i\bigl(\phi^*\partial^\mu\phi -\phi\partial^\mu\phi^*\bigr), \qquad \partial_\mu j^\mu=0.$$

The conservation law follows directly from the Klein–Gordon equations for $\phi$ and $\phi^*$: the two mass terms cancel. The global charge is $Q=\int\mathrm d^{d-1}x\,j^0$.

Why two oscillator species are needed

The field equation is the same as for a real scalar, but Hermiticity no longer relates the positive- and negative-frequency coefficients. Thus $\phi$ contains a particle annihilator and an antiparticle creator, with independent oscillator families. After normal ordering,

$$H=\int\mathrm d\Pi_{\mathbf p}\,E_{\mathbf p} \bigl(a^\dagger_{\mathbf p}a_{\mathbf p} +b^\dagger_{\mathbf p}b_{\mathbf p}\bigr), \qquad Q=\int\mathrm d\Pi_{\mathbf p} \bigl(a^\dagger_{\mathbf p}a_{\mathbf p} -b^\dagger_{\mathbf p}b_{\mathbf p}\bigr).$$

Here $\mathrm d\Pi_{\mathbf p}=\mathrm d^{d-1}p/[(2\pi)^{d-1}2E_{\mathbf p}]$, with $[a_{\mathbf p},a^\dagger_{\mathbf q}]=(2\pi)^{d-1}2E_{\mathbf p}\delta^{(d-1)}(\mathbf p-\mathbf q)$ and the same relation for $b$. Energy counts both species positively; charge counts their difference. A neutral particle–antiparticle pair can therefore have positive energy and zero charge. The relation between continuous symmetry and this charge is treated in Srednicki 2007, §22.

Interactions and physical limits

A potential depending only on $\phi^*\phi$ preserves the global $U(1)$ charge. It need not preserve the total number of particles plus antiparticles: sufficiently energetic processes can create pairs. The exact conservation law concerns the signed difference.

Coupling to a dynamical Maxwell field produces scalar electrodynamics. A global phase rotation and a local gauge redundancy then play different roles; the free global model by itself has neither a Coulomb field nor gauge constraints. Likewise, introducing a chemical potential requires specifying the ensemble and checking stability: for a free gas the approach to $\lvert\mu\rvert=m$ is physically consequential.

Develop the calculation

Complex Scalars and Conserved Charge gives the normalized mode expansion and charge operator. Maxwell Theory supplies the gauge field; Bose–Hubbard Model gives a nonrelativistic lattice example with exact number conservation.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§22 and 61. DOI.