Quasi-morphisms and the Poisson bracket

Entov, Michael; Polterovich, Leonid; Zapolsky, Frol

Quasi-morphisms and the Poisson bracket

Authors

Entov, Michael

Polterovich, Leonid

Zapolsky, Frol

Description

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.
minor changes, to appear in Pure and Applied Mathematics Quarterly (special issue dedicated to Gregory Margulis' 60th birthday)