Uncertainty principle and geometry of the infinite Grassmann manifold

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dc.creator Andruchow, Esteban
dc.creator Corach, Gustavo
dc.date.accessioned 2024-12-23T13:21:46Z
dc.date.available 2024-12-23T13:21:46Z
dc.date.issued 2019
dc.identifier.citation Andruchow, E. y Corach, G. (3-2019). Uncertainty principle and geometry of the infinite Grassmann manifold. Studia Mathematica, 248(1), 31-44.
dc.identifier.issn 0039-3223
dc.identifier.uri http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1803
dc.description Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
dc.description Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina.
dc.description Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina.
dc.description.abstract 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?.
dc.format application/pdf
dc.language eng
dc.publisher Polish Academy of Sciences. Institute of Mathematics
dc.relation 10.4064/sm170915-27-12
dc.rights info:eu-repo/semantics/openAccess
dc.rights https://creativecommons.org/licenses/by-nc-nd/4.0/
dc.source Studia Mathematica. Mar. 2019; 248(1): 31-44
dc.source.uri https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/studia-mathematica/all/248
dc.subject Projections
dc.subject Pair of projections
dc.subject Grassmann maniffold
dc.subject Uncertainty principle
dc.title Uncertainty principle and geometry of the infinite Grassmann manifold
dc.type info:eu-repo/semantics/article
dc.type info:ar-repo/semantics/artículo
dc.type info:eu-repo/semantics/publishedVersion


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