PT symmetry was initially studied as a specific system in non-Hermitian quantum mechanics, where Hamiltonians are not

Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...

. In 1998, physicist Carl Bender and former graduate student Stefan Boettcher published in ''

Physical Review Letters ''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the ''Journa ...

'' a paper in

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

, "Real Spectra in non-Hermitian Hamiltonians Having PT Symmetry." In this paper, the authors found non-Hermitian Hamiltonians endowed with an unbroken PT symmetry (invariance with respect to the simultaneous action of the parity-inversion and time reversal symmetry operators) also may possess a real spectrum. Under a correctly-defined

inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

, a PT-symmetric Hamiltonian's

eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...

s have positive norms and exhibit unitary

time evolution Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...

, requirements for quantum theories. Bender won the 2017

Dannie Heineman Prize for Mathematical Physics Dannie Heineman Prize for Mathematical Physics is an award given each year since 1959 jointly by the American Physical Society and American Institute of Physics. It is established by the Heineman Foundation in honour of Dannie Heineman. As of 2010 ...

for his work. A closely related concept is that of pseudo-Hermitian operators, which were considered by physicists Dirac, Pauli, and Lee and Wick. Pseudo-Hermitian operators were discovered (or rediscovered) almost simultaneously by mathematicians Krein et al. as G-Hamiltonian in the study of linear dynamical systems. The equivalence between pseudo-Hermiticity and G-Hamiltonian is easy to establish. In 2002, Ali Mostafazadeh showed that every non-Hermitian Hamiltonian with a real spectrum is pseudo-Hermitian. He found that PT-symmetric non-Hermitian Hamiltonians that are diagonalizable belong to the class of pseudo-Hermitian Hamiltonians. However, this result is not useful because essentially all interesting physics happens at the exception points where the systems are not diagonalizable. It was proven recently that in finite dimensions PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability, which indicates that the mechanism of PT-symmetry breaking at exception points, where the Hamiltionian is usually not diagonalizable, is the Krein collision between two eigenmodes with opposite signs of actions. In 2005, PT symmetry was introduced to the field of optics by the research group of Gonzalo Muga by noting that PT symmetry corresponds to the presence of balanced gain and loss. In 2007, the physicist Demetrios Christodoulides and his collaborators further studied the implications of PT symmetry in optics. The coming years saw the first experimental demonstrations of PT symmetry in passive and active systems. PT symmetry has also been applied to

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...

metamaterial A metamaterial (from the Greek word μετά ''meta'', meaning "beyond" or "after", and the Latin word ''materia'', meaning "matter" or "material") is any material engineered to have a property that is not found in naturally occurring materials. ...

electric circuit An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...

s, and

nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...

. In 2017, a non-Hermitian PT-symmetric Hamiltonian was proposed by Dorje Brody, and Markus Müller that "formally satisfies the conditions of the

Hilbert–Pólya conjecture In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theor ...

References

{{reflist Quantum optics