[Paper Review] PT-symmetric quantum mechanics

Carl M. Bender, Daniel Hook|arXiv (Cornell University)|Dec 28, 2023

Quantum Mechanics and Non-Hermitian Physics82 citations

TL;DR

The paper surveys how non-Hermitian but PT-symmetric Hamiltonians can yield real spectra and unitary quantum theories, introducing the CPT inner product and related constructions.

ABSTRACT

It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.

Motivation & Objective

• Motivate and define PT symmetry as a relaxation of Hermiticity in quantum mechanics.
• Show that PT-symmetric Hamiltonians can have real, bounded spectra and define unitary theories.
• Introduce the C operator and CPT inner product to establish a positive-definite Hilbert space.
• Demonstrate construction methods via complex deformations of Hermitian Hamiltonians.
• Discuss physical realizations and potential research directions in PT-symmetric systems.

Proposed method

• Present PT-symmetric deformations of Hermitian Hamiltonians (e.g., H = 1/2 p^2 + 1/2 x^2 + ε i x).
• Show eigenvalues remain real: E_n = (n + 1/2) (1 + ε^2) for the additive deformation.
• Explain that PT symmetry allows real spectra or complex-conjugate pairs, with exceptional points marking transitions.
• Develop the framework for a CPT-based inner product and the C operator to ensure unitarity.
• Discuss examples and techniques for identifying when eigenstates are PT eigenstates (and real spectra).
• Outline the connection to complex analysis and deformations of oscillator-like systems.

Figure 1: Annual publications referencing $\mathcal{PT}$ symmetry since its inception in 1998 (not including papers on the arXiv). The total number of research publications is growing rapidly and currently exceeds 10,000. Source: Dimensions from Digital Science [ 161 ] .

Experimental results

Research questions

• RQ1 When does a PT-symmetric Hamiltonian have a real and bounded-below spectrum?
• RQ2 How can one construct a consistent inner product (via the C operator) to ensure unitarity in PT-symmetric quantum mechanics?
• RQ3 What are the mechanisms (e.g., exceptional points) that cause PT symmetry to be broken or unbroken in specific models?
• RQ4 How do complex deformations of Hermitian systems relate to PT-symmetric non-Hermitian theories?
• RQ5 What examples illustrate PT transitions in low-dimensional or matrix Hamiltonians and their experimental analogs?

Key findings

• PT-symmetric deformations can yield entirely real spectra despite non-Hermiticity.
• In certain regions (unbroken PT symmetry) eigenvalues are real and eigenfunctions are PT-symmetric.
• Eigenvalues for H = p^2 + x^2 (ix)^ε are real or come in complex-conjugate pairs depending on ε.
• A C operator can be constructed to define a PT-symmetric inner product, enabling unitary evolution.
• PT symmetry provides a broad framework for non-Hermitian quantum theories with potential experimental realizations.
• Exceptional points mark transitions between real spectra and complex-conjugate pair spectra.

Figure 2: Schematic picture of the group space of the homogeneous real Lorentz group. This continuous group consists of four disconnected parts: At the top of the diagram is the proper orthochronous Lorentz group (POLG); the POLG contains the identity element and is a subgroup of the Lorentz group.