Must a Hamiltonian be Hermitian?

TL;DR

This paper explores PT-symmetric non-Hermitian Hamiltonians in quantum mechanics, showing they can produce unitary evolution and consistent physical theories, expanding the scope beyond traditional Hermitian Hamiltonians.

Contribution

It demonstrates that PT-symmetric Hamiltonians can define a positive definite inner product, enabling unitary evolution without Hermiticity, thus broadening the class of physically acceptable quantum theories.

Findings

PT-symmetric Hamiltonians can have real spectra

A positive definite inner product can be constructed for PT-symmetric systems

Quantum dynamics remain unitary under PT symmetry

Abstract

A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite. This construction is general and works for any PT-symmetric Hamiltonian. The dynamics is governed by unitary time evolution. This formulation does not conflict with the requirements of conventional quantum mechanics. There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.