PT Symmetry

The following sections are included:

• Dedication
• Preface
• About the Authors
• Acknowledgment
• Contents

This chapter introduces the basic ideas of $\mathcal{P}\mathcal{T}$-symmetric systems. It begins with a brief discussion of closed (isolated) and open (non-isolated) systems and explains that $\mathcal{P}\mathcal{T}$-symmetric systems are physical configurations that may be viewed as intermediate between open and closed systems. The chapter then presents elementary examples of quantum-mechanical and classical $\mathcal{P}\mathcal{T}$-symmetric systems. It demonstrates that the Hamiltonians that describe $\mathcal{P}\mathcal{T}$-symmetric systems are complex extensions (deformations) of conventional real Hamiltonians. Finally, this chapter shows that real systems that are unstable may become stable in the more general complex setting. Thus, by deforming real systems into the complex domain one may be able to tame or even eliminate instabilities.

Perhaps the most remarkable feature of a $\mathcal{P}\mathcal{T}$-symmetric quantummechanical Hamiltonian is that its eigenvalues can be all real even if the Hamiltonian is not Hermitian. The purpose of this chapter is to explain the mathematics (complex asymptotic methods, WKB approximation, Stokes sectors) needed to define and solve the eigenvalue problems associated with $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians. We apply these techniques to a wide variety of examples that illustrate the parametric deformation (analytic continuation) of differential-equation eigenvalue problems…

We have seen in Chap. 2 that a $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian can have an unbroken region of $\mathcal{P}\mathcal{T}$ symmetry in which its eigenvalues are all real and positive even though the Hamiltonian is not Hermitian. This raises an obvious question: Can a non-Hermitian Hamiltonian such as Ĥ in (2.7) define an actual physical theory of quantum mechanics or is the reality and positivity of its spectrum just a mathematical curiosity having no important physical consequences? A Hamiltonian that defines a physical quantum theory must have a real energy spectrum because an energy measurement must return a real number. (The energy spectrum must also be bounded below because a physical theory must have a stable ground state.) Second, the theory must possess a Hilbert space of state vectors that is endowed with an inner product having a positive norm; this is because the norm of a state is interpreted as a probability, and a probability is a positive number. Third, the Hamiltonian of the theory must generate unitary time evolution because probability must be preserved in time…

The requirement that a physical system be $\mathcal{P}\mathcal{T}$ symmetric (that is, invariant under combined space and time reection) can be imposed on classical as well as on quantum systems. Elementary examples of classical $\mathcal{P}\mathcal{T}$-symmetric systems, such as coupled linear oscillators with balanced gain and loss, are used in Chap. 1 to illustrate the basic concepts of $\mathcal{P}\mathcal{T}$ symmetry. Classical $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians arise as the classical limit (ħ → 0) of quantum-mechanical $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians. We saw in Chap. 1 that it is useful to study such classical systems because closed classical trajectories (see Figs. 1.14, 1.15, and 2.12) indicate that the corresponding quantum system has an unbroken $\mathcal{P}\mathcal{T}$ symmetry (with a real spectrum), and open classical trajectories (see Figs. 1.18 and 2.14) indicate that the corresponding quantum system has a broken $\mathcal{P}\mathcal{T}$ symmetry (with a partially or entirely complex spectrum)…

Two profound advances in the early 20th century transformed classical physics into what we view today as modern physics. The first was the discovery of quantum mechanics, a theory that describes the properties of matter. The second was the discovery of relativity, a theory that describes the geometry of space and time. When these two theories are combined, the resulting relativistic quantum theory is called quantum field theory. Quantum field theory describes the properties and interactions of elementary particles, which are the fundamental constituents of the natural world…

The purpose of this chapter is to study the $\mathcal{P}\mathcal{T}$-symmetric eigenvalue problem for the Hamiltonian $H=p^2+x^2{\left(ix\right)}^{\epsilon }$ at a rigorous level. We start with the $\mathcal{P}\mathcal{T}$-symmetric cubic oscillator $\left(\epsilon =1\right)$ that, as explained in the preface of this book, brought the issue of spectral reality to the attention of Bender and Boettcher [Bender and Boettcher (1998b)] and initiated the current upsurge of interest in the subject of $\mathcal{P}\mathcal{T}$ symmetry. The Hamiltonian for this oscillator is

In attempting to understand an emergent area of physics, it is extremely helpful to study elementary solvable models. $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics is a complex generalization of conventional quantum mechanics, and some of its features are unexpected and nonintuitive. By examining the solutions to exactly solvable $\mathcal{P}\mathcal{T}$-symmetric quantum-mechanical models, we can develop an understanding of the nature of $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics. Exactly solvable $\mathcal{P}\mathcal{T}$-symmetric quantum-mechanical models are usually constructed as generalizations of exactly solvable models in conventional quantum mechanics. This section explores the properties of some carefully selected $\mathcal{P}\mathcal{T}$-symmetric potentials.

The key objective of PTQM is the development of modern physics using Hamiltonians that are not Hermitian, but have certain characteristics of symmetry (like $\mathcal{P}\mathcal{T}$ symmetry). Usually, $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians can be naturally considered as Hermitian operators with respect to an indefinite inner product ($\mathcal{P}\mathcal{T}$ inner product). This opens up a promising way to reach deeper insights into the structural subtleties of the $\mathcal{P}\mathcal{T}$-symmetric quantum theory with the use of the Kreĭn-space methods.

We begin with an elementary presentation of the Kreĭn-space theory focusing on those aspects that may be more appealing to PTQM-related studies (the mathematical background of $\mathcal{P}\mathcal{T}$ and $\mathcal{C}$-symmetries, the meaning of the $\mathcal{C}\mathcal{P}\mathcal{T}$ inner product). The final part of the chapter deals with more advanced results and open problems.

Integrable models occupy a special place in the set of classical as well as quantum systems. Classically they are characterized by the special feature that they possess enough conserved quantities that allow one to construct exact analytical solutions to their equations of motion. In quantum fieldtheoretic settings, integrability is usually identified with the factoring of multiparticle scattering amplitudes into a product of sequential two-particle amplitudes that can be determined to all orders in perturbation theory. Integrable systems can often be used as cornerstones or benchmarks around which new, often more realistic systems can be constructed in a perturbative or deformed fashion…

The ideas of $\mathcal{P}\mathcal{T}$ symmetry were originally introduced in the context of quantum mechanics, but in recent years they have led to rapid developments in the apparently unconnected field of classical optics, starting with the pioneering papers [El-Ganainy et al. (2007); Musslimani et al. (2008a); Makris et al. (2008); Klaiman et al. (2008); Guo et al. (2009); Makris et al. (2010); Rüter et al. (2010); Ramezani et al. (2010)]. The reason for this surprising connection is that when one makes a particular approximation, the paraxial approximation, for the equation of propagation of an electromagnetic wave, the resulting equation is formally identical to the Schrödinger equation, but with completely different interpretations for the symbols appearing therein. In particular, the role of the quantum-mechanical potential is taken by the refractive index. The complex potentials appearing in $\mathcal{P}\mathcal{T}$ quantum mechanics thus correspond to a complex refractive index, whose imaginary part corresponds to loss or gain…

The following sections are included:

• Bibliography
• Index