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abstract: |
    The Hamiltonian $H$ specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that $H$ be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of space-time reflection ($\mathcal{PT}$) symmetry. It is shown that if the $\mathcal{PT}$ symmetry of a Hamiltonian $H$ is not broken, then the spectrum of $H$ is real. Examples of $\mathcal{PT}$-symmetric non-Hermitian quantum-mechanical Hamiltonians are $H=p^2+ix^3$ and $H=p^2-x^4$. The crucial question is whether $\mathcal{P
    T}$-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken $\mathcal{PT}$ symmetry also possesses a physical symmetry represented by a linear operator called $\mathcal{
    C}$. Using $\mathcal{C}$ it is shown how to construct an inner product whose associated norm [*is*]{} positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are 