Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbo...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2024-03-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2024-03-13-1277/pdf/ |
| _version_ | 1797263347332153344 |
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| author | Chia-Yi Ju Adam Miranowicz Yueh-Nan Chen Guang-Yin Chen Franco Nori |
| author_facet | Chia-Yi Ju Adam Miranowicz Yueh-Nan Chen Guang-Yin Chen Franco Nori |
| author_sort | Chia-Yi Ju |
| collection | DOAJ |
| description | Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports. |
| first_indexed | 2024-04-25T00:11:34Z |
| format | Article |
| id | doaj.art-b1d73bcd1319457a936cefb07539123f |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-04-25T00:11:34Z |
| publishDate | 2024-03-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-b1d73bcd1319457a936cefb07539123f2024-03-13T11:15:28ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2024-03-018127710.22331/q-2024-03-13-127710.22331/q-2024-03-13-1277Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanicsChia-Yi JuAdam MiranowiczYueh-Nan ChenGuang-Yin ChenFranco NoriStudies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.https://quantum-journal.org/papers/q-2024-03-13-1277/pdf/ |
| spellingShingle | Chia-Yi Ju Adam Miranowicz Yueh-Nan Chen Guang-Yin Chen Franco Nori Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics Quantum |
| title | Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics |
| title_full | Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics |
| title_fullStr | Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics |
| title_full_unstemmed | Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics |
| title_short | Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics |
| title_sort | emergent parallel transport and curvature in hermitian and non hermitian quantum mechanics |
| url | https://quantum-journal.org/papers/q-2024-03-13-1277/pdf/ |
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