Making sense of non-Hermitian Hamiltonians

TLDR

The Hamiltonian H governs energy levels and time evolution, and Hermiticity is traditionally required to ensure a real spectrum and unitary evolution. This work proposes a reformulation of quantum mechanics that replaces the Hermiticity axiom with the physically transparent condition of space–time reflection (PT) symmetry. By requiring that a Hamiltonian possess unbroken PT symmetry, the authors demonstrate that its spectrum remains real. The authors provide examples of PT‑symmetric non‑Hermitian Hamiltonians whose energy levels are all real and positive.

Abstract

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!