Abstract
We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.
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Acknowledgments
We thank Vijay Balasubramanian, Pablo Basteiro, Arpan Bhattacharyya, Saurya Das, Giuseppe Di Giulio, Ro Jefferson, René Meyer, Djordje Minic, Flavio Nogueira, Joris Raeymaekers, Shubho Roy, Eric Sharpe and Gideon Vos for useful discussions.
We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 390858490), via the SFB 1170 ToCoTronics (project-id 258499086) and via the German-Israeli Project Cooperation (DIP) grant ‘Holography and the Swampland’. This research was also supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science.
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Banerjee, S., Dorband, M., Erdmenger, J. et al. Geometric phases characterise operator algebras and missing information. J. High Energ. Phys. 2023, 26 (2023). https://doi.org/10.1007/JHEP10(2023)026
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DOI: https://doi.org/10.1007/JHEP10(2023)026