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The Hamiltonian H specifies the energy levels and the time evolution of aquantum theory. It is an axiom of quantum mechanics that H be Hermitian becauseHermiticity guarantees that the energy spectrum is real and that the timeevolution is unitary (probability preserving). This paper investigates analternative way to construct quantum theories in which the conventionalrequirement of Hermiticity (combined transpose and complex conjugate) isreplaced 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 notbroken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitianquantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. The crucialquestion is whether PT-symmetric Hamiltonians specify physically acceptablequantum theories in which the norms of states are positive and the timeevolution is unitary. The answer is that a Hamiltonian that has an unbroken PTsymmetry also possesses a physical symmetry represented by a linear operatorcalled C. Using C it is shown how to construct an inner product whoseassociated norm is positive definite. The result is a new class of fullyconsistent complex quantum theories. Observables are defined, probabilities arepositive, and the dynamics is governed by unitary time evolution. After areview of PT-symmetric quantum mechanics, new results are presented here inwhich the C operator is calculated perturbatively in quantum mechanicaltheories having several degrees of freedom

The Script C operator in Script PScript T-symmetric quantum theories