Uncertainty principle and geometry of the infinite Grassmann manifold
Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Artículo publishedVersion |
| Lenguaje: | Inglés |
| Publicado: |
Polish Academy of Sciences. Institute of Mathematics
2024
|
| Materias: | |
| Acceso en línea: | http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1803 |
| Aporte de: |
| 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 |