The metric for non-Hermitian Hamiltonians : a case study

[摘要] ENGLISH ABSTRACT: We are studying a possible implementation of an appropriate framework for a proper non-Hermitian quantum theory. We present the case where for a non-Hermitian Hamiltonian withreal eigenvalues, we define a new inner product on the Hilbert space with respect to whichthe non-Hermitian Hamiltonian is Quasi-Hermitian. The Quasi-hermiticity of the Hamiltonianintroduces the bi-orthogonality between the left-hand eigenstates and the right-hand eigenstates,in which case the metric becomes a basis transformation. We use the non-Hermitian quadraticHamiltonian to show that such a metric is not unique but can be uniquely defined by requiringto hermitize all elements of one of the irreducible sets defined on the set of all observables. Wecompare the constructed metric with specific known examples in the literature in which cases aunique choice is made.

[发布机构] Stellenbosch University