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Rigged Hilbert space treatment for the bra-ket formalism of positive-definite metric non-Hermitian systems (Mathematical Aspects of Quantum Fields and Related Topics)

Shousuke Ohmori

Published May 1, 2025• 0 citations

Abstract

The rigged Hilbert space (RHS) treatment for the Dirac's bra-ket formalism in a non-Hermitian system with a positive-definite metric is reviewed. We propose an RHS related to the positive-definite metric induced from a positive operator η, called η-RHS. The nuclear spectral theorem for the η-RHS exhibits spectral expansions of the bra and kets by the generalized eigenvectors of an η-quasi Hermitian operator. We establish a complete bi-orthogonal system that endows the transformation theory between the Hermitian and non-Hermitian systems using the spectral expansions. The η-quasi Hermitian operator can be extended to the general operator on the bra-ket space while preserving its symmetric structure.

DOI: 10.17983/301914

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