Fock Space Representations for Non-Hermitian Hamiltonians

TL;DR

This paper explores non-Hermitian, PT-symmetric Hamiltonians in Fock space, revealing their relation to magnetic Landau models and harmonic oscillators with imaginary couplings, maintaining real eigenvalues.

Contribution

It provides a new interpretation of non-Hermitian Hamiltonians as Landau models and harmonic oscillators with imaginary couplings, expanding understanding of PT-symmetric quantum systems.

Findings

Non-Hermitian Hamiltonians can be interpreted as Landau models.

PT-symmetric models can have real eigenvalues despite non-Hermiticity.

Multiparticle states are analyzed within this framework.

Abstract

The requirement of Hermiticity of a Quantum Mechanical Hamiltonian, for the description of physical processes with real eigenvalues which has been challenged notably by Carl Bender, is examined for the case of a Fock space Hamilitonian which is bilinear in two creation and destruction operators. An interpretation of this model as a Schr\"odinger operator leads to an identification of the Hermitian form of the Hamiltonian as the Landau model of a charged particle in a plane, interacting with a constant magnetic field at right angles to the plane. When the parameters of the Hamiltonian are suitably adjusted to make it non-Hermitian, the model represents two harmonic oscillators at right angles interacting with a constant magnetic field in the third direction, but with a pure imaginary coupling, and real energy eigenvalues. It is now ${\cal PT}$ symmetric. Multiparticle states are investigated.