Uncertainty principle and geometry of the infinite Grassmann manifold

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?.