Quasi-morphisms and the Poisson Bracket

Abstract

For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold.This functional has a number of interesting properties.On the one hand, it is Lipschitz with respect to the uniform norm.On the other hand, it serves as a measure of non-commutativity of functions in the sense of the Poisson bracket, the operation which involves first derivatives of the functions.Furthermore, the same functional gives rise to a non-trivial lower bound for the error of the simultaneous measurement of a pair of non-commuting Hamiltonians.These results manifest a link between the algebraic structure of the group of Hamiltonian diffeomorphisms and the function theory on a symplectic manifold.The above-mentioned functional comes from a special homogeneous quasi-morphism on the universal cover of the group, which is rooted in the Floer theory.

Locations

• Pure and Applied Mathematics Quarterly - View - PDF
• arXiv (Cornell University) - View - PDF