TY - JOUR

T1 - Quasi-morphism and the poisson bracket

AU - Entov, Michael

AU - Polterovich, Leonid

AU - Zapolsky, Frol

PY - 2007/10

Y1 - 2007/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=42949145602&partnerID=8YFLogxK

U2 - 10.4310/PAMQ.2007.v3.n4.a9

DO - 10.4310/PAMQ.2007.v3.n4.a9

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AN - SCOPUS:42949145602

SN - 1558-8599

VL - 3

SP - 1037

EP - 1055

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

IS - 4

ER -