The Non-Hermitian quantum mechanics and its canonical structure

TL;DR

This paper reformulates non-Hermitian quantum mechanics using Hamilton's canonical equations, resolving traditional issues and clarifying the physical irrelevance of certain complex quantities.

Contribution

It presents a canonical formulation of non-Hermitian quantum mechanics, eliminating previous difficulties and redefining key concepts without approximations.

Findings

Reformulation of non-Hermitian Schrödinger equation as canonical equations

Resolution of issues related to probability amplitudes and operator expectations

Clarification that imaginary parts of eigenenergy and geometric phase are unphysical

Abstract

The non-Hermitian Schr\"odinger equation is re-expressed generally in the form of Hamilton's canonical equation without any approximation. Its quantization called non-Hermitian quantum field theory is discussed. By virtue of the canonical equation, the theory of non-Hermitian quantum mechanics is totally reformulated, including the probability amplitudes of states, the expectations of operators, as well as the expressions of interaction terms. The conventional difficulties in non-Hermitian quantum mechanics are totally overcome by the reformulation. Specifically, the imaginary parts the non-Hermitian eigenenergy and adiabatic geometric phase are actually unphysical, although they are mathematically perfect.