Uncertainty principle and geometry of the infinite Grassmann manifold

ESTEBAN ANDRUCHOW, GUSTAVO CORACH

STUDIA MATHEMATICA

POLISH ACAD SCIENCES INST MATHEMATICS

Lugar: VARSOVIA; Año: 2019 vol. 248 p. 31 - 44

0039-3223

We study the pairs of projectionsPIf=χIf,QJf=(χJf?) ˇ,f∈L2(Rn),where I,J⊂Rn are sets of finite positive Lebesgue measure, χI,χJ denote the corresponding characteristic functions and ?, ˇ denote the Fourier?Plancherel transformation L2(Rn)→L2(Rn) 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(L2(Rn)), which is a curve of the formδ(t)=eitXI,JPIe−itXI,Jwhich joins PI and QJ and has length π/2. Here XI,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]∥≥π/2.The spectrum of XI,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue γ(XI,J) which satisfiescos(γ(XI,J))=∥PIQJ∥.We study the pairs of projectionsPIf=χIf,QJf=(χJf?) ˇ,f∈L2(Rn),where I,J⊂Rn are sets of finite positive Lebesgue measure, χI,χJ denote the corresponding characteristic functions and ?, ˇ denote the Fourier?Plancherel transformation L2(Rn)→L2(Rn) 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(L2(Rn)), which is a curve of the formδ(t)=eitXI,JPIe−itXI,Jwhich joins PI and QJ and has length π/2. Here XI,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]∥≥π/2.The spectrum of XI,J is denumerable and symmetric with respect to the origin, and it has a smallest positive eigenvalue γ(XI,J) which satisfiescos(γ(XI,J))=∥PIQJ∥.