Hybrid form of quantum theory with non-Hermitian Hamiltonians

Abstract

In Schrödinger picture the unitarity of evolution is usually guaranteed by the Hermiticity of the Hamiltonian operator h=h† in a conventional Hilbert space Htextbook. After a Dyson-inspired operator-transformation (OT) non-unitary preconditioning Ω:h→H the simplified Hamiltonian H is, in its manifestly unphysical Hilbert space Hauxiliary, non-Hermitian. Besides its natural OT-based physical interpretation it can also be “Hermitized” (i.e., made compatible with the unitarity) via a metric-amendment (MA) change of the Hilbert space, Hauxiliary→Hphysical. In our present letter we propose another, third, hybrid form (HF) of the Hermitization of H in which the change involves, simultaneously, both the Hamiltonian and the metric. Formally this means that the original Dyson map is assumed factorizable, Ω=ΩMΩH. A key practical advantage of the new HF approach lies in the model-dependent adaptability of such a factorization. The flexibility and possible optimality of the balance between the MA-related (i.e., metric-amending) factor ΩM and the OT-related (i.e., Hamiltonian-changing) factor ΩH are explicitly illustrated via an elementary two-state quantum model.