the $\mathcal{PT}$ norm of a state, we can think of the $\mathcal{PT}$ norm as the $\mathcal{C}$ “charge” of the state.

The operators $\mathcal{P}$ and $\mathcal{C}$ are distinct square roots of the unity operator $\delta(x-y)$. That is, $\mathcal{P}^2=\mathcal{C}^2=1$, but $\mathcal{P}\neq\mathcal{C}$. Indeed, $\mathcal{P}$ is real, while $\mathcal{C}$ is complex. Note that the parity operator in coordinate space is explicitly real $\mathcal{P}(x,y)=\delta(x+y)$; the operator $\mathcal{C}(x,y)$ is complex because it is a sum of products of complex functions, as we see in (\[e14\]). The complexity of the $\mathcal{C}$ operator can be seen explicitly in perturbative calculations of $\mathcal{C}(x,y)$ [@rr11]. We show how to perform these perturbative calculations in Secs. \[s4\] and \[s5\]. Furthermore, these two operators do not commute; in the position representation $$(\mathcal{CP})(x,y)=\sum_n\phi_n(x)\phi_n(-y)~~{\rm but}~~(\mathcal{PC})(x,y)=
\sum_n\phi_n(-x)\phi_n(y),
\label{e17}$$ which shows that $\mathcal{CP}=(\mathcal{PC})^*$. However, $\mathcal{C}$ [*does*]{} commute with $\mathcal{PT}$.

Finally, having obtained the operator $\mathcal{C}$ we define a new inner product structure having [*positive definite*]{} signature by $$\langle f|g\rangle\equiv\int_{\rm{C}} dx\,[\mathcal{CPT}f(x)]g(x).
\label{e18}$$ This inner product is phase independent and conserved in time like the $\mathcal
{PT}$ inner product (\[e7\]). This is because the time evolution operator, just as in ordinary quantum mechanics, is $e^{iHt}$. The fact that $H$ commutes with the $\mathcal{PT}$ and the $\mathcal{CPT}$ operators implies 