PT-Symmetric Quantum Theory Defined in a Krein Space

TL;DR

This paper develops a rigorous mathematical framework for PT-symmetric quantum theory using Krein spaces, clarifying the properties of PT-symmetric operators and Hamiltonians in this context.

Contribution

It introduces a Krein space formulation for PT-symmetric quantum theory, establishing conditions for P-Hermiticity and transposition symmetry of operators.

Findings

PT-symmetric operators are P-Hermitian iff they have transposition symmetry.

The framework applies to systems on real lines and complex contours.

Properties of PT-symmetric Hamiltonians are derived from the Krein space structure.

Abstract

We provide a mathematical framework for PT-symmetric quantum theory, which is applicable irrespective of whether a system is defined on R or a complex contour, whether PT symmetry is unbroken, and so on. The linear space in which PT-symmetric quantum theory is naturally defined is a Krein space constructed by introducing an indefinite metric into a Hilbert space composed of square integrable complex functions in a complex contour. We show that in this Krein space every PT-symmetric operator is P-Hermitian if and only if it has transposition symmetry as well, from which the characteristic properties of the PT-symmetric Hamiltonians found in the literature follow. Some possible ways to construct physical theories are discussed within the restriction to the class K(H).