Editorial: Non-Hermitian quantum mechanics
Abstract
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…
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