Complex Extension of Quantum Mechanics

Abstract

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 ([Formula presented] symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if [Formula presented] symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry [Formula presented] of the Hamiltonian. Using [Formula presented], an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any [Formula presented]-symmetric Hamiltonian. Observables exhibit [Formula presented] 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. © 2002 The American Physical Society.