Uncertainty principle and geometry of the infinite Grassmann manifold

• Autores: Andruchow, Esteban; Corach, Gustavo
• Año de publicación: 2019
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Studia Mathematica. Mar. 2019; 248(1): 31-44; https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/studia-mathematica/all/248
• Materia: Projections; Pair of projections; Grassmann maniffold; Uncertainty principle
• Institución: Universidad Nacional de General Sarmiento

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?.