Uncertainty principle and geometry of the infinite Grassmann manifold

Autores
Andruchow, Esteban; Corach, Gustavo
Año de publicación
2019
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We study the pairs of projections PIf=χIf,QJf=(χJf) ˇ, f∈L^2(R^n), where I,J⊂R^n are sets of finite positive Lebesgue measure, χI,χJ denote the corresponding characteristic functions and ˇ, ˇ 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 δ(t)=e^{itXI,J}P^{Ie−itXI,J} which joins PI and QJ and has length π/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]∥≥π/2. The spectrum of X_I,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue γ(X_I,J) which satisfies cos(γ(X_I,J))=∥PIQJ∥.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina
Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina
Materia
PROJECTIONS
PAIR OF PROJECTIONS
GRASSMANN MANIFOLD
UNCERTAINTY PRINCIPLE
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/107573

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spelling Uncertainty principle and geometry of the infinite Grassmann manifoldAndruchow, EstebanCorach, GustavoPROJECTIONSPAIR OF PROJECTIONSGRASSMANN MANIFOLDUNCERTAINTY PRINCIPLEhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We study the pairs of projections PIf=χIf,QJf=(χJf) ˇ, f∈L^2(R^n), where I,J⊂R^n are sets of finite positive Lebesgue measure, χI,χJ denote the corresponding characteristic functions and ˇ, ˇ 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 δ(t)=e^{itXI,J}P^{Ie−itXI,J} which joins PI and QJ and has length π/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]∥≥π/2. The spectrum of X_I,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue γ(X_I,J) which satisfies cos(γ(X_I,J))=∥PIQJ∥.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; ArgentinaPolish Academy of Sciences. Institute of Mathematics2019-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/107573Andruchow, Esteban; Corach, Gustavo; Uncertainty principle and geometry of the infinite Grassmann manifold; Polish Academy of Sciences. Institute of Mathematics; Studia Mathematica; 248; 1; 3-2019; 31-440039-32231730-6337CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.impan.pl/get/doi/10.4064/sm170915-27-12info:eu-repo/semantics/altIdentifier/doi/10.4064/sm170915-27-12info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1701.03733info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:18:03Zoai:ri.conicet.gov.ar:11336/107573instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:18:03.275CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Uncertainty principle and geometry of the infinite Grassmann manifold
title Uncertainty principle and geometry of the infinite Grassmann manifold
spellingShingle Uncertainty principle and geometry of the infinite Grassmann manifold
Andruchow, Esteban
PROJECTIONS
PAIR OF PROJECTIONS
GRASSMANN MANIFOLD
UNCERTAINTY PRINCIPLE
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
dc.creator.none.fl_str_mv Andruchow, Esteban
Corach, Gustavo
author Andruchow, Esteban
author_facet Andruchow, Esteban
Corach, Gustavo
author_role author
author2 Corach, Gustavo
author2_role author
dc.subject.none.fl_str_mv PROJECTIONS
PAIR OF PROJECTIONS
GRASSMANN MANIFOLD
UNCERTAINTY PRINCIPLE
topic PROJECTIONS
PAIR OF PROJECTIONS
GRASSMANN MANIFOLD
UNCERTAINTY PRINCIPLE
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We study the pairs of projections PIf=χIf,QJf=(χJf) ˇ, f∈L^2(R^n), where I,J⊂R^n are sets of finite positive Lebesgue measure, χI,χJ denote the corresponding characteristic functions and ˇ, ˇ 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 δ(t)=e^{itXI,J}P^{Ie−itXI,J} which joins PI and QJ and has length π/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]∥≥π/2. The spectrum of X_I,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue γ(X_I,J) which satisfies cos(γ(X_I,J))=∥PIQJ∥.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina
Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; Argentina
description We study the pairs of projections PIf=χIf,QJf=(χJf) ˇ, f∈L^2(R^n), where I,J⊂R^n are sets of finite positive Lebesgue measure, χI,χJ denote the corresponding characteristic functions and ˇ, ˇ 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 δ(t)=e^{itXI,J}P^{Ie−itXI,J} which joins PI and QJ and has length π/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]∥≥π/2. The spectrum of X_I,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue γ(X_I,J) which satisfies cos(γ(X_I,J))=∥PIQJ∥.
publishDate 2019
dc.date.none.fl_str_mv 2019-03
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/107573
Andruchow, Esteban; Corach, Gustavo; Uncertainty principle and geometry of the infinite Grassmann manifold; Polish Academy of Sciences. Institute of Mathematics; Studia Mathematica; 248; 1; 3-2019; 31-44
0039-3223
1730-6337
CONICET Digital
CONICET
url http://hdl.handle.net/11336/107573
identifier_str_mv Andruchow, Esteban; Corach, Gustavo; Uncertainty principle and geometry of the infinite Grassmann manifold; Polish Academy of Sciences. Institute of Mathematics; Studia Mathematica; 248; 1; 3-2019; 31-44
0039-3223
1730-6337
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.impan.pl/get/doi/10.4064/sm170915-27-12
info:eu-repo/semantics/altIdentifier/doi/10.4064/sm170915-27-12
info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1701.03733
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Polish Academy of Sciences. Institute of Mathematics
publisher.none.fl_str_mv Polish Academy of Sciences. Institute of Mathematics
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
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