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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/107573
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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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CONICET Digital (CONICET) |
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Consejo Nacional de Investigaciones Científicas y Técnicas |
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CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.mail.fl_str_mv |
dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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13.070432 |