PT-symmetric quantum mechanics

TL;DR

This work surveys PT-symmetric quantum mechanics, showing that non-Hermitian Hamiltonians can yield real spectra and unitary evolution when PT symmetry remains unbroken. It develops the mathematical framework, including the C operator and the CPT inner product, to establish a consistent quantum theory and discusses numerous model systems—from elementary Hamiltonians to higher-order anharmonic oscillators and upside-down potentials. The analysis blends complex-analytic techniques with semiclassical methods (WKB), boundary-condition deformations, and algebraic constructions to reveal when spectra are real, complex, or undergo PT transitions, and it links these findings to observable phenomena in optics and related systems. The work also highlights the physical significance of parity anomalies, quasi-exact solvability, and the broader implications for quantum theory, including potential experimental probes of PT-symmetric spectra and the ongoing debate about the fundamental distinction between PT-symmetric and Hermitian quantum mechanics.

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.

Figures (35)

• 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 pt578.
• 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. The other three components of the homogeneous real Lorentz group are the POLG multiplied by the parity (space-reflection) operator $\mathcal{P}$, the time-reversal operator $\mathcal{T}$, and the spacetime reflection operator $\mathcal{PT}$. If we extend the real Lorentz group to the complex Lorentz group pt599, there are now only two disconnected parts. Continuous paths through complex group space (indicated as vertical and horizontal lines) connect the top and bottom components and the left and right components of the real Lorentz group.
• Figure 3: Lowest seven eigenvalues of $H$ in (\ref{['e2.14']}) plotted for $-1<\varepsilon<0.5$. These eigenvalues are real, positive, discrete, and monotonically increasing for $\varepsilon\geq0$ and they are displayed as solid lines. As $\varepsilon$ decreases below 0, the ground-state energy remains real and becomes singular at $\varepsilon=-1$. The other eigenvalues merge pairwise in a regular and orderly fashion, starting with the highest eigenvalues, and immediately bifurcate sequentially into complex-conjugate pairs (whose real and imaginary parts are indicated by dashed/broken lines). The values of $\varepsilon$ at which pairs of real eigenvalues merge are called exceptional points. These exceptional points are square-root singularities, sometimes referred to as pitchfork bifurcations, in the complex-$\varepsilon$ plane.
• Figure 4: First twenty-one eigenvalues of $H$ in (\ref{['e2.14']}) plotted for $-1<\varepsilon\leq4$. The eigenvalues are all real, positive, discrete, and monotonically increasing for $\varepsilon\geq0$. This plot only shows the eigenvalues when they are real. Note that as $\varepsilon$ decreases below 0 the eigenvalues become degenerate pairwise and enter the complex plane as complex-conjugate pairs, but the ground-state energy continues to remain real and becomes singular as $\varepsilon$ approaches $-1$.
• Figure 5: Schematic picture of the light fiber experiment.