PT-Symmetric Quantum Theory Defined in a Krein Space

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).

8 pages, no figures; Refs. added, minor revision

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Cited by in corpus (12)

• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems
• General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space
• PT Symmetric, Hermitian and P-Self-Adjoint Operators Related to Potentials in PT Quantum Mechanics
• On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)
• Krein-unitary Schrieffer-Wolff transformation and band touchings in bosonic Bogoliubov-de Gennes and other Krein-Hermitian Hamiltonians
• Parasupersymmetry and N-fold Supersymmetry in Quantum Many-Body Systems I. General Formalism and Second Order
• Parasupersymmetry and N-fold Supersymmetry in Quantum Many-Body Systems II. Third Order
• N-fold Parasupersymmetry
• The operational foundations of PT-symmetric and quasi-Hermitian quantum theory
• Operator based approach to PT-symmetric problems on a wedge-shaped contour