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

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.

arXiv comment32 pages, no figures; explanation, discussion and references added

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Cited by in corpus (11)

• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• $J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach
• PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
• 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)
• p-Adic Schrödinger-Type Operator with Point Interactions
• 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