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.
%0 Journal Article
%1 banerjee2023geometric
%A Banerjee, Souvik
%A Dorband, Moritz
%A Erdmenger, Johanna
%A Weigel, Anna-Lena
%D 2023
%J J. High Energy Phys.
%K a
%N 10
%P 26--
%R 10.1007/JHEP10(2023)026
%T Geometric phases characterise operator algebras and missing information
%U https://doi.org/10.1007/JHEP10(2023)026
%V 2023
%X 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.
@article{banerjee2023geometric,
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.},
added-at = {2023-11-28T11:19:19.000+0100},
author = {Banerjee, Souvik and Dorband, Moritz and Erdmenger, Johanna and Weigel, Anna-Lena},
biburl = {https://www.bibsonomy.org/bibtex/249bf9c9676ae155098375ac9bd6f27d5/ctqmat},
day = 05,
doi = {10.1007/JHEP10(2023)026},
interhash = {f98c203685085c7087d5b150736eb61b},
intrahash = {49bf9c9676ae155098375ac9bd6f27d5},
issn = {10298479},
journal = {J. High Energy Phys.},
keywords = {a},
month = {10},
number = 10,
pages = {26--},
refid = {Banerjee2023},
timestamp = {2023-11-28T11:19:19.000+0100},
title = {Geometric phases characterise operator algebras and missing information},
url = {https://doi.org/10.1007/JHEP10(2023)026},
volume = 2023,
year = 2023
}