Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics

Chia-Yi Ju1,2, Adam Miranowicz3,4, Yueh-Nan Chen5,6,7, Guang-Yin Chen8, and Franco Nori4,9,10

1Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
2Center for Theoretical and Computational Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
3Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
4Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama, 351-0198, Japan
5Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
6Center for Quantum Frontiers of Research & Technology, NCKU, Tainan 70101, Taiwan
7Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
8Department of Physics, National Chung Hsing University, Taichung 40227, Taiwan
9Quantum Computing Center, RIKEN, Wakoshi, Saitama, 351-0198, Japan
10Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA

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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.

In this study, we show that if a system depends on a continuous parameter, the quantum states vary with the parameter described by a Schroedinger-like equation, which formally resembles a parallel transport or evolution equation along the dimension described by the parameter. Moreover, we derive the governing equation for the geometry/metric of the underlying Hilbert space along the parameter-formed dimension. Rather than solely engaging in a formal study of the properties of these emergent dimensions, we also explore their applications across various fields in quantum physics.

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[2] Miloslav Znojil, "Hybrid form of quantum theory with non-Hermitian Hamiltonians", Physics Letters A 457, 128556 (2023).

[3] Miloslav Znojil, "Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation", Physics Letters A 462, 128655 (2023).

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