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Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization
We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Springer US
2021
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8549959/ https://www.ncbi.nlm.nih.gov/pubmed/34720131 http://dx.doi.org/10.1007/s10701-021-00465-6 |
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author | Gosson, Maurice A. de |
author_facet | Gosson, Maurice A. de |
author_sort | Gosson, Maurice A. de |
collection | PubMed |
description | We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity. |
format | Online Article Text |
id | pubmed-8549959 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Springer US |
record_format | MEDLINE/PubMed |
spelling | pubmed-85499592021-10-29 Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization Gosson, Maurice A. de Found Phys Article We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity. Springer US 2021-05-21 2021 /pmc/articles/PMC8549959/ /pubmed/34720131 http://dx.doi.org/10.1007/s10701-021-00465-6 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Gosson, Maurice A. de Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization |
title | Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization |
title_full | Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization |
title_fullStr | Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization |
title_full_unstemmed | Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization |
title_short | Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization |
title_sort | quantum polar duality and the symplectic camel: a new geometric approach to quantization |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8549959/ https://www.ncbi.nlm.nih.gov/pubmed/34720131 http://dx.doi.org/10.1007/s10701-021-00465-6 |
work_keys_str_mv | AT gossonmauriceade quantumpolardualityandthesymplecticcamelanewgeometricapproachtoquantization |