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arXiv · 2508.06923

Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space

Junichi Takahashi·Shousuke Ohmori
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Abstract

The rigged Hilbert space, a triplet extension of the Hilbert space, provides a mathematically rigorous foundation for quantum mechanics by extending the Hilbert space to accommodate generalized eigenstates. In this paper, we construct a triplet structure for the thermal space arising in Thermo Field Dynamics with the aid of the tensor product formulation of rigged Hilbert spaces, a formalism that reformulates thermal averages as pure-state expectation values in a doubled Hilbert space. We then induce the rigged Liouville space for Liouville space of density operators from the triplet structure for Thermo Field Dynamics; the rigged Liouville space corresponds isomorphically one-to-one with that of Thermo Field Dynamics. This correspondence offers a unified topological foundation for quantum statistical mechanics at finite temperature and establishes a framework for future generalizations to open and non-equilibrium quantum systems.

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Junichi Takahashi, Shousuke Ohmori. 2025-08-09. Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space. https://arxiv.org/abs/2508.06923

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