  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
(PT) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not
broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian
quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. The crucial
question is whether PT-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 PT
symmetry also possesses a physical symmetry represented by a linear operator
called C. Using 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
positive, and the dynamics is governed by unitary time evolution. After a
review of PT-symmetric quantum mechanics, new results are presented here in
which the C operator is calculated perturbatively in quantum mechanical
theories having several degrees of freedom.
