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Riemannian structure on manifolds of quantum states

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A metric tensor is defined from the underlying Hilbert space structure for any submanifold of quantum states. The case where the manifold is generated by the action of a Lie group on a fixed state vector (generalized coherent states manifold hereafter noted G.C.S.M.) is studied in details; the geometrical properties of some wellknown G.C.S.M. are reviewed and an explicit expression for the scalar Riemannian curvature is given in the general case. The physical meaning of such Riemannian structures (which have been recently introduced to describe collective manifolds in nuclear physics) is discussed. It is shown on examples that the distance between nearby states is related to quantum fluctuations; in the particular case of the harmonic oscillator group the condition of zero curvature appears to be identical to that of non dispersion of wave packets.

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Communicated by R. Haag

Equipe de Recherche Associée au C.N.R.S.

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Provost, J.P., Vallee, G. Riemannian structure on manifolds of quantum states. Commun.Math. Phys. 76, 289–301 (1980).

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