Relativistic Non-Hermitian Quantum Mechanics

Abstract

The basic structure of quantum mechanics was delineated in the early days of the theory and has not been modified since. Still it is interesting to ask whether that basic structure can be altered or generalized. In the last decade Bender et al have shown that one of the fundamental assumptions of quantum mechanics, that operators are represented by Hermitian matrices, can to an extent be relaxed. In this theory, the parity (P) and time-reversal (T) operators play a role analogous to the Hermitian conjugate. Recently we have extended the realm of `PT quantum mechanics' to include systems that are odd under time-reversal (T^2 = - 1), in the interest of constructing the PT-analogue of the Dirac equation. We find that the fundamental representation of the Dirac equation, which describes relativistic fermions, remains unchanged in the generalization to the non-Hermitian theory. Higher dimensional representations, which ordinarily decouple into pairs of Dirac fermions in Hermitian quantum mechanics, here describe new types of particles with extremely compelling properties. Most notably we have constructed a toy model representing two generations of massless neutrinos that nonetheless undergo flavor oscillations; furthermore this model is Lorentz invariant and unitary in time. The Standard Model requires that the neutrino be massive in order to accomodate the observed flavor oscillations, thus this toy model represents a significant departure from Standard Model physics.