that both inner products, (\[e7\]) and (\[e18\]), remain time independent as the states evolve in time. However, unlike (\[e7\]), the inner product (\[e18\]) is positive definite because $\mathcal{C}$ contributes $-1$ when it acts on states with negative $\mathcal{PT}$ norm. In terms of the $\mathcal{CPT}$ conjugate, the completeness condition (\[e6\]) reads $$\sum_n \phi_n(x)[\mathcal{CPT}\phi_n(y)]=\delta(x-y).
\label{e19}$$ Unlike the inner product of conventional quantum mechanics, the $\mathcal{CPT}$ inner product (\[e19\]) is [*dynamically determined*]{}; it depends implicitly on the Hamiltonian.

The operator $\mathcal{C}$ does not exist as a distinct entity in conventional quantum mechanics. Indeed, if we allow the parameter $\epsilon$ in (\[e1\]) to tend to zero, the operator $\mathcal{C}$ in this limit becomes identical with $\mathcal{P}$. Thus, in this limit the $\mathcal{CPT}$ operator becomes $\mathcal{T}$, which is just complex conjugation. As a consequence, the inner product (\[e18\]) defined with respect to the $\mathcal{CPT}$ conjugation reduces to the complex conjugate inner product of conventional quantum mechanics when $\epsilon\to0$. Similarly, in the $\epsilon\to0$ limit, (\[e19\]) reduces to the usual statement of completeness $\sum_n\phi_n(x)\phi_n^*(y)=\delta(x-y)$.

The $\mathcal{CPT}$ inner-product (\[e18\]) is independent of the choice of integration contour ${\rm C}$ so long as ${\rm C}$ lies inside the asymptotic wedges associated with the boundary conditions for the Sturm-Liouville problem 