A selection rule for transitions in PT-symmetric quantum theory

TL;DR

This paper establishes a selection rule in PT-symmetric quantum theory, showing that transitions between states of different PT-norm are forbidden under time-dependent Hamiltonians, with illustrative matrix and continuum examples.

Contribution

It introduces a novel selection rule for transitions in PT-symmetric quantum systems, extending understanding of state dynamics under time-dependent Hamiltonians.

Findings

Transitions between states of different PT-norm are forbidden.

The selection rule is demonstrated with matrix models and continuum examples.

The rule applies to time-dependent PT-symmetric Hamiltonians.

Abstract

Carl Bender and collaborators have developed a quantum theory governed by Hamiltonians that are PT-symmetric rather than Hermitian. To implement this theory, the inner product was redefined to guarantee positive norms of eigenstates of the Hamiltonian. In the general case, which includes arbitrary time-dependence in the Hamiltonian, a modification of the Schrodinger equation is necessary as shown by Gong and Wang to conserve probability. In this paper, we derive the following selection rule: transitions induced by time dependence in a PT-symmetric Hamiltonian cannot occur between normalized states of differing PT-norm. We show three examples of this selection rule in action: two matrix models and one in the continuum.