# Non-Hermitian quantum mechanics

In physics, non-Hermitian quantum mechanics describes quantum mechanical systems where Hamiltonians are not Hermitian.

## History

The first paper that has "non-Hermitian quantum mechanics" in the title was published in 1996 by Naomichi Hatano and David R. Nelson. The authors mapped a classical statistical model of flux-line pinning by columnar defects in high-T$_c$ superconductors to a quantum model by means of an inverse path-integral mapping and ended up with a non-Hermitian Hamiltonian with an imaginary vector potential in a random scalar potential. They further mapped this into a lattice model and came up with a tight-binding model with asymmetric hopping, which is now widely called the Hatano-Nelson model. The authors showed that there is a region where all eigenvalues are real despite the non-Hermiticity.
Parity–time (PT) symmetry was initially studied as a specific system in non-Hermitian quantum mechanics. In 1998, physicist Carl Bender and former graduate student Stefan Boettcher published a paper where they 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, a PT-symmetric Hamiltonian's eigenfunctions have positive norms and exhibit unitary time evolution, requirements for quantum theories. Bender won the 2017 Dannie Heineman Prize for Mathematical Physics for his work.
A closely related concept is that of pseudo-Hermitian operators, which were considered by physicists Paul Dirac, Wolfgang Pauli, and Tsung-Dao Lee and Gian Carlo Wick. Pseudo-Hermitian operators were discovered (or rediscovered) almost simultaneously by mathematicians Mark Krein and collaborators as G-Hamiltonian in the study of linear dynamical systems. The equivalence between pseudo-Hermiticity and G-Hamiltonian is easy to establish.
In the early 1960s, Olga Taussky, Michael Drazin, and Emilie Haynsworth demonstrated that the necessary and sufficient criteria for a finite-dimensional matrix to have real eigenvalues is that said matrix is pseudo-Hermitian with a positive-definite metric.
In 2002, Ali Mostafazadeh showed that diagonalizable PT-symmetric Hamiltonians belong to the class of pseudo-Hermitian Hamiltonians. In 2003, it was proven that in finite dimensions, PT-symmetry is equivalent to pseudo-Hermiticity regardless of diagonalizability, thereby applying to the physically interesting case of non-diagonalizable Hamiltonians at exceptional points. This 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, metamaterials, electric circuits, and nuclear magnetic resonance. 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."