 Indefinite Klein-Gordon charge density

For a complex Klein-Gordon field, $j^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })$ is conserved. A mode $\varphi =N{e}^{-iEt+i\mathbf{p}\cdot \mathbf{x}}$ has $j^0=2E|N{|}^2$, so unrestricted positive- and negative-frequency solutions do not have a positive probability norm under this current. In the quantized complex field, the associated conserved quantity is charge: particles and antiparticles contribute with opposite signs while both have positive excitation energy and positive Fock space norm.