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arXiv:2405.02848 · Journal of Mathematical Physics

Rigged Hilbert space formulation for quasi-Hermitian composite systems

Shousuke Ohmori

Published Dec 1, 2024• 1 citation
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Abstract

The discussion in this study delves into Dirac’s bra-ket formalism for a quasi-Hermitian quantum composite system based on the rigged Hilbert space (RHS). We establish an RHS with a positive-definite metric suitable for a quasi-Hermitian composite system. The obtained RHS is utilized to construct the bra and ket vectors for the non-Hermitian composite system and produce the spectral decomposition of the quasi-Hermitian operator. We show that the symmetric relations regarding quasi-Hermitian operators can be extended to dual spaces, and all descriptions obtained using the bra-ket formalism are completely developed in the dual spaces. Our methodology is applied to a non-Hermitian harmonic oscillator composed of conformal multi-dimensional many-body systems.

DOI
10.1063/5.0218063

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The discussion in this study delves into Dirac’s bra-ket formalism for a quasi-Hermitian quantum composite system based on the rigged Hilbert space (RHS). We establish an RHS with a positive-definite metric suitable for a quasi-Hermitian composite system. The obtained RHS is utilized to construct the bra and ket vectors for the non-Hermitian composite system and produce the spectral decomposition of the quasi-Hermitian operator. We show that the symmetric relations regarding quasi-Hermitian operators can be extended to dual spaces, and all descriptions obtained using the bra-ket formalism are completely developed in the dual spaces. Our methodology is applied to a non-Hermitian harmonic oscillator composed of conformal multi-dimensional many-body systems.
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.
We discuss a super bra-ket formalism for quasi-Hermitian Liouvillian operators within the framework of rigged Hilbert spaces (RHS). An RHS in terms of the Liouville space, referred to as a rigged Liouville space (RLS), is reconstructed by exploiting the mathematical fact that the space of Hilbert-Schmidt operators is unitarily equivalent to the tensor product of Hilbert spaces. The obtained RLS endows a rigorous foundation of the construction for the super bra-ket and for the spectral decompositions of both Hermitian and quasi-Hermitian Liouville operators, which are characterized by the generalized eigenvectors in the dual spaces. Furthermore, within this framework, the non-Hermitian Liouvillian operator and its adjoint can be constructed symmetrically, with their symmetric structure preserved. As an application of our RLS methodology, we examine the Liouville operators corresponding to Hermitian and non-Hermitian harmonic oscillators and elucidate the differences between their spectral decomposition forms.
We investigate Dirac’s bra–ket formalism based on a rigged Hilbert space for a non-Hermitian quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by a positive-definite metric, is established. With the aid of the nuclear spectral theorem for the obtained rigged Hilbert space, spectral expansions are shown for the bra–kets by the generalized eigenvectors of a quasi-Hermitian operator. The spectral expansions are utilized to endow the complete bi-orthogonal system and the transformation theory between the Hermitian and non-Hermitian systems. As an example of application, we show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.
This study discussed Dirac's bra-ket formalism for the identical particles system based on the rigged Hilbert space reformulated by R. Madrid [J. Phys A:Math. Gen. 37, 8129 (2004)]. The bra and ket vectors for a composite system that form the basis of an identical particle system are described in dual and anti-dual spaces for the tensor product of rigged Hilbert spaces. The permutation operator that characterizes the symmetry of identical particles is constructed as the operator on such dual spaces. We also show that the nuclear spectral theorem in the tensor product of rigged Hilbert spaces endows the spectral expansion of the self-adjoint operator in the dual and anti-dual spaces and the expansion is consistent with the identicle particle system when the permutation operator commutes the self-adjoint operator.
In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasi-hermitian operators. The necessary mathematics to understand rigged Hilbert spaces and their representations is presented. Using the mathematics, we give rigorous meaning to Dirac's kets and bras, using the momentum operator as an example. The mathematical theory culminates with the nuclear spectral theorem(Gelfand-Maurin) which guarantees an eigenket decomposition for the wave functions. Next, we illustrate the theory by applying it to the quantum system defined by a potential barrier. We then introduce non-hermitian operators of interest in Quantum Mechanics, specifically PT-symmetric Hamiltonians. It is known that PT-symmetric Hamiltonians are necessarily pseudo-hermitian. We restrict our attention to a subclass of pseudo-hermitian operators, the quasi-hermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasi-hermitian operator and the metric operator, guarantees an eigenket decomposition for a dense subset of the Hilbert space on which the operator is defined. We discuss how the extension may help put quasi-hermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasi-hermitian operators with respect to an operator-valued measure exists. This is an extension of the spectral decomposition of self-adjoint operators with respect to a projection-valued measure.
A class of pseudo-Hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators that are Hermitian with respect to a pre-determined positive-definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-Hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark (Zeeman) effect with non-Hermitian interaction, non-Hermitian general quadratic form of N boson (fermion) operators, symmetric and asymmetric XXZ spin chain in the complex magnetic field, non-Hermitian Haldane–Shastry spin chain and Lipkin–Meshkov–Glick model.