General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space

TL;DR

This paper investigates the mathematical and physical foundations of PT-symmetric quantum theory within Krein spaces, addressing spectral properties, time evolution, and observables, and proposing a new quantization scheme alternative to traditional Hermitian quantum mechanics.

Contribution

It introduces a comprehensive analysis of PT-symmetric quantum theory in Krein spaces, highlighting its role as an alternative quantization method and establishing criteria for physical observables.

Findings

PT-symmetric operators naturally act in Krein spaces.

The theory can be constructed from any real classical system.

A postulate for physical observables is proposed.

Abstract

In our previous work, we proposed a mathematical framework for PT-symmetric quantum theory, and in particular constructed a Krein space in which PT-symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral property and PT symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical-quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding PT-symmetric quantum system, which indicates that PT-symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in the framework.