Complex extension of quantum mechanics

Abstract:
This talk examines Hamiltonians H that are not Hermitian but do exhibit space-time reflection (PT) symmetry. If the (PT) symmetry of H is not spontaneously broken, then the spectrum of H is entirely real and positive. Examples of PT-symmetric non-Hermitian Hamiltonians are H = p² + ix³ and H = p² - x⁴. The apparent shortcoming of quantum theories arising from PT-symmetric Hamiltonians is that the PT norm is not positive definite. This suggests that it may be difficult to develop a quantum theory based on such Hamiltonians. In this talk it is shown that these difficulties can be overcome by introducing a previously unnoticed underlying physical symmetry C of Hamiltonians having an unbroken PT symmetry. 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 in these theories exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.