Read e-book online Communications in Mathematical Physics - Volume 243 PDF

By M. Aizenman (Chief Editor)

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It is natural to interpret as the quantization of σ . Denote by Aut(h1 (M)) the linear Poisson isomorphisms of h1 (M). The set of all Poisson diffeomorphisms σ for which a as above exists, forms a subgroup Diff K (P , {·, ·}) of the Poisson diffeomorphism group Diff(P , {·, ·}) of P . The map E : Diff K (P , {·, ·}) → Aut(h1 (M)) so defined, is a group homomorphism which will be called, according to Odzijewicz [O2], Ehrenfest quantization. Consider now the flow σt of the Hamiltonian vector field Xh on M and assume that σt ∈ Diff K (P , {·, ·}) for all t.

9. 3), where N k=1 Pk = 1, with N ∈ N or N = ∞. 12) and if N < ∞ let N ρ= λk Pk , λk ∈ C, λk = λ for k = , λ1 = 0, rank Pk < ∞ for k ≥ 2. 13) Thus ρ ∈ L1 (M). It is easy to check that ∞ ∗ N L (M)ρ = im R = Pk XPk | X ∈ L∞ (M) . 13), g ∈ GL∞ (M), X, Y ∈ L∞ (M). g. 6) and hence the coadjoint orbit is also connected; thus it is a symplectic leaf of the Banach Lie-Poisson space L1 (M). The characteristic subspace Sρ = {[X, ρ] | X ∈ L∞ (M)} is contained in     Pk XP | X ∈ L∞ (M) = ker R ∗ , ker R   k= and if N ∈ N one has Pk XP = [ρ, Y ] k= for some Y ∈ L∞ (M) which is related to X through the system of equations Pk XP = (λk − λ )Pk Y P for all k = .

Flaschka, J. V. -P. Ortega, and J. Marsden for several useful discussions that improved the exposition. Special thanks to P. Bona for his interest in our work and his inspired remarks. The first author was partially supported by KBN under grant 2 PO3 A 012 19. The second author was partially supported by the European Commission and the Swiss Federal Government through funding for the Research Training Network Mechanics and Symmetry in Europe (MASIE) as well as the Swiss National Science Foundation.

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Communications in Mathematical Physics - Volume 243 by M. Aizenman (Chief Editor)

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