Hermitian–Non-Hermitian Interfaces in Quantum Theory

Abstract

In the global framework of quantum theory, the individual quantum systems seem clearly separated into two families with the respective manifestly Hermitian and hiddenly Hermitian operators of their Hamiltonian. In the light of certain preliminary studies, these two families seem to have an empty overlap. In this paper, we will show that whenever the interaction potentials are chosen to be weakly nonlocal, the separation of the two families may disappear. The overlaps alias interfaces between the Hermitian and non-Hermitian descriptions of a unitarily evolving quantum system in question may become nonempty. This assertion will be illustrated via a few analytically solvable elementary models.

Gauge invariance and the Englert-Brout-Higgs mechanism in non-Hermitian field theories

“…There has been much work in recent years on quantummechanical models with non-Hermitian, PT -symmetric Hamiltonians [1][2][3], which have become an important area of research in integrated photonics and other fields [4][5][6]see Ref. [7] for a review of relations to conventional models with Hermitian Hamiltonians. Quantum field theories (QFTs) with non-Hermitian Hamiltonians have also attracted interest, including a model with an iϕ 3 scalar interaction [8][9][10][11][12], which was shown in the framework of PT -symmetric QFT to have a physically meaningful effective potential despite its being unbounded from below [13], and a PTsymmetric −ϕ 4 model featuring asymptotic freedom [14].…”