of an indefinite norm by finding a physical interpretation for the negative norm states. We claim that in [*any*]{} theory having an unbroken $\mathcal{PT}$ symmetry there exists a symmetry of the Hamiltonian connected with the fact that there are equal numbers of positive-norm and negative-norm states. To describe this symmetry we construct a linear operator denoted by $\mathcal{C}$ and represented in position space as a sum over the energy eigenstates of the Hamiltonian [@rr7]: $$\mathcal{C}(x,y)=\sum_n\phi_n(x)\phi_n(y).
\label{e14}$$

The properties of this new operator $\mathcal{C}$ closely resemble those of the charge conjugation operator in quantum field theory. For example, we can use (\[e6\]) – (\[e8\]) to verify that the square of $\mathcal{C}$ is unity ($\mathcal{C}^2=1$): $$\int dy\,\mathcal{C}(x,y)\mathcal{C}(y,z)=\delta(x-z).
\label{e15}$$ Thus, the eigenvalues of $\mathcal{C}$ are $\pm1$. Also, $\mathcal{C}$ commutes with the Hamiltonian $H$. Therefore, since $\mathcal{C}$ is linear, the eigenstates of $H$ have definite values of $\mathcal{C}$. Specifically, if the energy eigenstates satisfy (\[e8\]), then we have $\mathcal{C}\phi_n=(-1)^n
\phi_n$ because $$\mathcal{C}\phi_n(x)=\int dy\,\mathcal{C}(x,y)\phi_n(y)=\sum_m\phi_m(x)\int dy\,
\phi_m(y)\phi_n(y).
\label{e16}$$ We then use $\int dy\,\phi_m(y)\phi_n(y)=(\phi_m,\phi_n)$ according to our convention. We conclude that $\mathcal{C}$ is the operator observable that represents the measurement of the signature of the $\mathcal{PT}$ norm of a state. Note that since the $\mathcal{C}$ operator measures 