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

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 spacetime reflection symmetry. It is shown that if the symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of -symmetric non-Hermitian quantum mechanical Hamiltonians are H = p2 + ix3 and H = p2 − x4. The crucial question is whether -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 symmetry also possesses a physical symmetry represented by a linear operator called . Using 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 -symmetric quantum mechanics, new results are presented here in which the operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.