Editorial: Non-Hermitian quantum mechanics (original) (raw)

2020, Progress of theoretical and experimental physics

We all learn in elementary quantum mechanics class that all measurable physical quantities, and specifically the Hamiltonian operator, must be Hermitian, because the eigenvalues must be real. This is perhaps von Neumann's mathematical dogma, to which two main physical counterarguements exist. Presumably the first attempt to extend the Hamiltonian to a non-Hermitian operator was made to explain nuclear decay. Alpha or beta particles emitted from a radioactive nuclide in an experimental apparatus travel through free space outside the nuclide and, theoretically, are lost to infinite space, or experimentally, are absorbed by detectors that surround the free space. In either case, the emitted particles never come back to the nuclide, and therefore the energy of the nuclide is not conserved. In the present terminology, the radioactive nuclide is an open quantum system; it is not closed but open to the environment of infinite free space for alpha and beta particles or macroscopic detectors. The same situation can emerge at a mesoscopic scale; in a typical experimental setup, a quantum dot is open to quantum wires, which are terminated outside by electrodes. The Hamiltonian of the whole universe may be Hermitian according to von Neumann, but a part of it, for example, a radioactive nuclide, a quantum dot, or whatever is connected to the rest of the macroscopic universe, does not conserve energy, and hence can be described by an effective non-Hermitian Hamiltonian after eliminating the environmental degrees of freedom [1]. The non-Hermitian Hamiltonian obtained in this way produces complex eigenvalues of resonant states. It is somewhat infamous that the eigenfunction of a resonant state diverges spatially, and hence is unnormalizable, but it in fact means that the environment is macroscopic. The other counterargument to von Neumann's dogma is more ambitious. The celebrated theory of parity-time (PT) symmetry [2] assumed, at least originally, that the Hamiltonian of the whole universe is non-Hermitian but in a parameter regime of real eigenvalues. The symmetry under the combination of time-reversal and parity operations, or more generally an antilinear operation combined with linear operations, produces either real or conjugate pairs of complex eigenvalues, which may be made exclusively real if the parameters are so tuned. The simplest example would be the Hamiltonian for a two-site tight-binding model, H = iγ −t hop −t hop −iγ , (1) where both γ and t hop are real parameters. This non-Hermitian Hamiltonian is PT symmetric in the sense that the combination of the antilinear complex conjugation operation T = * and the linear