Complex Extension of Quantum Mechanics

Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry ($\mathcal{P}\mathcal{T}$ symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if $\mathcal{P}\mathcal{T}$ symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry $\mathcal{C}$ of the Hamiltonian. Using $\mathcal{C}$, an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian. Observables exhibit $\mathcal{C}\mathcal{P}\mathcal{T}$ symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.