CPT-Frames for PT-symmetric Hamiltonians

TL;DR

This paper explores the mathematical foundations of PT-symmetric quantum mechanics by introducing and analyzing PT-frames and CPT-frames, aiming to establish a consistent inner product structure for PT-symmetric Hamiltonians.

Contribution

It proposes the concepts of PT-frames and CPT-frames on Hilbert spaces and discusses their existence and construction, advancing the mathematical framework of PT-symmetric quantum theory.

Findings

Defined PT-frames and CPT-frames on Hilbert spaces.

Discussed conditions for the existence of these frames.

Provided methods for constructing PT-frames and CPT-frames.

Abstract

PT-symmetric quantum mechanics is an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose and complex conjugate) is replaced by the physically transparent condition of space-time reflection symmetry (PT-symmetry). A Hamiltonian H is said to be PT-symmetric if it commutes with the operator PT. The key point of PT-symmetric quantum theory is to build a new positive definite inner product on the given Hilbert space so that the given Hamiltonian is Hermitian with respect to the new inner product. The aim of this note is to give further mathematical discussions on this theory. Especially, concepts of PT-frames, CPT-frames on a Hilbert space and for a Hamiltonian are proposed, their existence and constructions are discussed.