Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915437389807616 |
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| author | Takahashi, Junichi Ohmori, Shousuke |
| author_facet | Takahashi, Junichi Ohmori, Shousuke |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space Takahashi, Junichi Ohmori, Shousuke Mathematical Physics 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. |
| title | Rigged Hilbert Space Formulation of Quantum Thermo Field Dynamics and Mapping to Rigged Liouville Space |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2508.06923 |