Automatic Hermiticity

TL;DR

This paper demonstrates that hermiticity can emerge dynamically in non-hermitian Hamiltonians through long-term evolution, challenging the notion that fundamental Hamiltonians must be hermitian.

Contribution

It introduces a mechanism where hermiticity arises automatically from non-hermitian Hamiltonians via state evolution and inner product adjustments.

Findings

Hermiticity emerges automatically after long time evolution.

A conserved probability current density can be constructed locally.

Misestimation of past states can occur when extrapolating with hermitian Hamiltonians.

Abstract

We study a diagonalizable Hamiltonian that is not at first hermitian. Requirement that a measurement shall not change one Hamiltonian eigenstate into another one with a different eigenvalue imposes that an inner product must be defined so as to make the Hamiltonian normal with regard to it. After a long time development with the non-hermitian Hamiltonian, only a subspace of possible states will effectively survive. On this subspace the effect of the anti-hermitian part of the Hamiltonian is suppressed, and the Hamiltonian becomes hermitian. Thus hermiticity emerges automatically, and we have no reason to maintain that at the fundamental level the Hamiltonian should be hermitian. If the Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions. We also point out a possible misestimation of a past state by extrapolating back in time with the hermitian Hamiltonian. It is a seeming past state, not a true one.