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Uncertainty principle and geometry of the infinite Grassmann manifold

Volume 248 / 2019

Esteban Andruchow, Gustavo Corach Studia Mathematica 248 (2019), 31-44 MSC: Primary 58B20, 47B15; Secondary 42A38, 47A63. DOI: 10.4064/sm170915-27-12 Published online: 1 March 2019

Abstract

We study the pairs of projections $$ P_If=\chi_If ,\quad Q_Jf= (\chi_J \hat{f})\check{\ }, \quad f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite positive Lebesgue measure, $\chi_I, \chi_J$ denote the corresponding characteristic functions and $\hat{\ } , \check{\ }$ denote the Fourier–Plancherel transformation $L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ 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 $\mathcal P(\mathcal H)$ of a Hilbert space $\mathcal H$ to establish that there exists a unique minimal geodesic of $\mathcal P(L^2(\mathbb{R}^n))$, which is a curve of the form $$ \delta(t)=e^{itX_{I,J}}P_Ie^{-itX_{I,J}} $$ which joins $P_I$ and $Q_J$ and has length $\pi/2$. 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 $$ \|[H,P_I]\|\ge \pi/2. $$ The spectrum of $X_{I,J}$ is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue $\gamma(X_{I,J})$ which satisfies $$ \cos(\gamma(X_{I,J}))=\|P_IQ_J\|. $$

Authors

  • Esteban AndruchowInstituto de Ciencias
    Universidad Nacional de General Sarmiento
    J. M. Gutierrez 1150
    1613 Los Polvorines, Argentina
    and
    Instituto Argentino de Matemática ‘Alberto P. Calderón’, CONICET
    Saavedra 15, 3er. piso
    1083 Buenos Aires, Argentina
    e-mail
  • Gustavo CorachFacultad de Ingeniería
    Universidad de Buenos Aires
    Paseo Colón 850
    1063 Buenos Aires, Argentina
    and
    Instituto Argentino de Matemática ‘Alberto P. Calderón’, CONICET
    Saavedra 15, 3er. piso
    1083 Buenos Aires, Argentina
    e-mail

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