symmetry by $\mathcal{C}$ because the properties of this operator resemble those of the charge conjugation operator in particle physics. This allows us to introduce an inner product structure associated with $\mathcal{CPT}$ conjugation for which the norms of quantum states are positive definite. Because of this we can say that $\mathcal{PT}$ symmetry is an alternative to conventional Hermiticity; it introduces the new concept of a [*dynamically determined*]{} inner product (one that is defined by the Hamiltonian itself). Consequently, we can extend the Hamiltonian and its eigenstates into the complex domain so that the associated eigenvalues are real and the underlying dynamics is unitary. This shows that $\mathcal{PT}$-symmetric Hamiltonians are Hermitian in an extended (non-Dirac) sense.

This paper is organized as follows. In Sec. \[s2\] we give a general discussion of the $\mathcal{C}$ operator and in Sec. \[s3\] we present a simple $2\times2$ matrix example of this operator. In Sec. \[s4\] we show how to calculate $\mathcal{C}$ using perturbation theory for the cubic Hamiltonian $H={\mbox{$\textstyle\frac{1}{2}$}}p^2+{\mbox{$\textstyle\frac{1}{2}$}}\mu^2x^2+i\epsilon x^3$. In Secs. \[s5\] and \[s6\] we calculate $\mathcal{C}$ for quantum mechanical Hamiltonians having two and three degrees of freedom. This is the principal new result in this paper. In Secs. \[s7\] and \[s8\] we consider possible 