2006/05/31 by Toshiaki Tanaka
Mathematics · Physics and Astronomy · #Generalization #Hermitian matrix #Hilbert space #Observable #Operator (biology) #POVM #Quantization (signal processing) #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum and Classical Electrodynamics #Quantum state #Space (punctuation) #Spectral Theory in Mathematical Physics #hep-th #math-ph #math.FA #math.MP #quant-ph
paper · pdf · doi:10.1088/0305-4470/39/45/025
published as J.Phys.A39:14175-14203,2006 · 32 pages, no figures; explanation, discussion and references added
arxiv created 2006/08/25 · openalex publication_date 2006/10/24 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In our previous work, we proposed a mathematical framework for -symmetric quantum theory, and in particular constructed a Krein space in which -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 properties and -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 -symmetric quantum system, which indicates that -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 this framework.