PT-Symmetric Quantum Theory Defined in a Krein Space

Details PT-Symmetric Quantum Theory Defined in a Krein Space PT-Symmetric Quantum Theory Defined in a Krein Space

2006-03-13

arxiv.org/abs/hep-th/0603096v2

J.Phys.A39:L369,2006

Physics

High Energy Physics

High Energy Physics - Theory

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

Scientific paper

10.1088/0305-4470/39/22/L04

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).

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