Polar duality between pairs of transversal Lagrangian planes; applications to uncertainty principles

Abstract

We extend the notion of polar duality to pairs (ℓ,ℓ′) of transversal Lagrangian planes in the standard symplectic space. (R2n,ω). This allows us to show that polar duality has a natural symplectic interpretation. Our main results are the following: we first show that the oblique projections Ωℓ and Ωℓ′ on ℓ and ℓ′ of a centrally symmetric convex body Ω containing a symplectic ball with radius one satisfy the duality relation Ωℓo⊂Ωℓ′. We thereafter show that if, conversely, (Ωℓ,Ωℓ′)⊂ℓ×ℓ′ is such that Ωℓo=Ωℓ′ then Ωℓ and Ωℓ′ are the projections of a symplectic ball with radius one which is the John ellipsoid of Ωℓ×Ωℓ′o. We apply these results to the quantum principle of indeterminacy, which contains as a particular case the usual uncertainty principle.