Abstract
A riemannian metric is introduced in the manifold representing the states of a generic physical system, under suitable assumptions of regularity on the “generalized transition probability” defined in [1]. From the mean values of the observables it is then possible to construct gradients and brackets, and in the special case of a system admitting a quantum-mechanical description the latter are shown to be related to the familiar commutators via a skew-symmetric tensor field which is part of the intrinsic geometry of the projective Hilbert space of physical states.
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Cantoni, V.: Commun. math. Phys.44, 125 (1975)
Mackey, G. W.: Mathematical foundations of quantum mechanics. New York: Benjamin 1963
Lang, S.: Differential manifolds. Reading: Addison Wesley 1972
Cantoni, V.: Intrinsic geometry of the quantum-mechanical “phase space”, Hamiltonian systems and correspondence principle. Rend. Accad. Nazl. Lincei, to appear
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Communicated by R. Haag
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Cantoni, V. The riemannian structure on the states of quantum-like systems. Commun.Math. Phys. 56, 189–193 (1977). https://doi.org/10.1007/BF01611503
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DOI: https://doi.org/10.1007/BF01611503