Uncertainty principle and geometry of the infinite Grassmann manifold

Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.

Detalles Bibliográficos
Autores principales: Andruchow, Esteban, Corach, Gustavo
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: Polish Academy of Sciences. Institute of Mathematics 2024
Materias:
Projections
Pair of projections
Grassmann maniffold
Uncertainty principle
Acceso en línea:http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1803
Aporte de:
Repositorio Institucional Digital de Acceso Abierto (UNGS) de Universidad Nacional de General Sarmiento
id I71-R177-UNGS-1803
record_format dspace
spelling I71-R177-UNGS-18032024-12-23T13:21:46Z Uncertainty principle and geometry of the infinite Grassmann manifold Andruchow, Esteban Corach, Gustavo Projections Pair of projections Grassmann maniffold Uncertainty principle Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina. Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina. We study the pairs of projections PIf=?If,QJf=(?Jf) ?, f?L^2(R^n), where I,J?R^n are sets of finite positive Lebesgue measure, ?I,?J denote the corresponding characteristic functions and ?, ? denote the Fourier-Plancherel transformation L^2(R^n)?L^2(R^n) and its inverse. These pairs of projections have been widely studied by several authors in connection with the mathematical formulation of Heisenberg´s uncertainty principle. Our study is done from a differential geometric point of view. We apply known results on the Finsler geometry of the Grassmann manifold P(H) of a Hilbert space H to establish that there exists a unique minimal geodesic of P(L^2(R^n)), which is a curve of the ?(t)=e^{itXI,J}P^{Ie?itXI,J} which joins PI and QJ and has length ?/2. Here X_I,J is a selfadjoint operator determined by the sets I,J. As a consequence we deduce that if H is the logarithm of the Fourier-Plancherel map, then ?[H,PI]???/2. The spectrum of X_I,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue ?(X_I,J) which satisfies cos(?(X_I,J))=?PIQJ?. 2024-12-23T13:21:46Z 2024-12-23T13:21:46Z 2019 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion Andruchow, E. y Corach, G. (3-2019). Uncertainty principle and geometry of the infinite Grassmann manifold. Studia Mathematica, 248(1), 31-44. 0039-3223 http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1803 eng 10.4064/sm170915-27-12 info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf application/pdf Polish Academy of Sciences. Institute of Mathematics Studia Mathematica. Mar. 2019; 248(1): 31-44 https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/studia-mathematica/all/248
institution Universidad Nacional de General Sarmiento
institution_str I-71
repository_str R-177
collection Repositorio Institucional Digital de Acceso Abierto (UNGS)
language Inglés
orig_language_str_mv eng
topic Projections
Pair of projections
Grassmann maniffold
Uncertainty principle
spellingShingle Projections
Pair of projections
Grassmann maniffold
Uncertainty principle
Andruchow, Esteban
Corach, Gustavo
Uncertainty principle and geometry of the infinite Grassmann manifold
topic_facet Projections
Pair of projections
Grassmann maniffold
Uncertainty principle
description Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
format Artículo
Artículo
publishedVersion
author Andruchow, Esteban
Corach, Gustavo
author_facet Andruchow, Esteban
Corach, Gustavo
author_sort Andruchow, Esteban
title Uncertainty principle and geometry of the infinite Grassmann manifold
title_short Uncertainty principle and geometry of the infinite Grassmann manifold
title_full Uncertainty principle and geometry of the infinite Grassmann manifold
title_fullStr Uncertainty principle and geometry of the infinite Grassmann manifold
title_full_unstemmed Uncertainty principle and geometry of the infinite Grassmann manifold
title_sort uncertainty principle and geometry of the infinite grassmann manifold
publisher Polish Academy of Sciences. Institute of Mathematics
publishDate 2024
url http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1803
work_keys_str_mv AT andruchowesteban uncertaintyprincipleandgeometryoftheinfinitegrassmannmanifold
AT corachgustavo uncertaintyprincipleandgeometryoftheinfinitegrassmannmanifold
_version_ 1824528696708431872

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