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Quantum mechanics as an infinite‐dimensional Hamiltonian system with uncertainty structure: Part I
1.R. Cirelli, P. Lanzavecchia, and A. Manià, “Normal pure states of the Von Neumann algebra of bounded operators as Kähler manifold,” J. Phys. A 16, 3829 (1983).
2.R. Cirelli and P. Lanzavecchia, “Hamiltonian vector fields in quantum mechanics,” Nuovo Cimento B 79, 271 (1984).
3.A. Heslot, “Une caractérization des espaces projectifs complexes,” C. R. Acad. Sci. Paris 298, 95 (1984).
4.A. Heslot, “Quantum mechanics as a classical theory,” Phys. Rev. D 31, 1341 (1985).
5.R. Cirelli and L. Pizzocchero, “On the integrability of quantum mechanics as an infinite‐dimensional Hamiltonian system,” to appear in Nonlinearity.
6.S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Wiley, New York, 1969), Vols. I and II.
7.Throughout the paper, by Riemann metric, we shall mean a strong Riemann metric. Analogously, by symplectic form, we shall always mean a strong symplectic form.
8.The quickest proof is obtained as follows: If X is a smooth vector field such that for every integral curve λ of X having as domain a bounded open interval the norm is constant along λ, and thus for From here and the completeness of the distance d induced by g, it follows that admits well defined limits as t tends to a, b, respectively. Now, using the existence theorem for differential equations, we can extend λ in a left neighborhood of a and in a right neighborhood of b. In conclusion, every integral curve of X with domain a bounded open interval can be extended. On the completeness of infinitesimal isometries, see also Ref. 6.
9.M. C. Abbati, R. Cirelli, P. Lanzavecchia, and A. Manià, “Pure states of general quantum‐mechanical systems as Kähler bundles,” Nuovo Cimento B 83, 43 (1984).
10.G. Fubini, “Sulle metriche definite da una forma Hermitiana,” Atti Ist. Veneto 6, 501 (1903);
10.E. Study, “Kürzeste Wege im komplexen Gebiete,” Math. Ann. 60, 321 (1905).
11.R. Hermann, “Topics in the mathematics of quantum mechanics,” in Interdisciplinary Mathematics (Mathematical Science, Brookline, MA, 1973), Vol. VI. Here and in Ref. 12, attention is mainly fixed on with the poorer structure of symplectic manifold.
12.R. Hermann, “Quantum mechanics and geometric analysis on manifolds,” Int. J. Theor. Phys. 21, 803 (1982).
13.R. Cirelli, A. Manià, and L. Pizzocchero, “A smooth functional representation for ‐algebras,” in preparation.
14.This is the mathematically precise formulation of the variational principle appearing in any standard textbook on quantum mechanics: The eigenstates of a self‐adjoint operator make stationary its mean value.
15.B. Mielnik, “Generalized quantum mechanics,” Commun. Math. Phys. 37, 221 (1974).
16.The fullness condition is linked with the homogeneity of the state manifold. At least in the finite‐dimensional case, it is not difficult to show that, if M is connected, fullness implies local homogeneity: For every pair x, y of points of M, there is a diffeomorphism between a neighborhood of x and a neighborhood of y that preserves ω, g and sends x into y. Homogeneity in the usual, global sense is granted under the additional assumption of completeness. As a partial converse, if M is connected, simply connected and (globally) homogeneous, then is full.
17.E. C. G. Stueckelberg, “Quantum theory in real Hilbert space,” Helv. Phys. Acta 33, 727 (1960);
17.E. C. G. Stueckelberg and M. Guenin, “Quantum theory in real Hilbert space II (Addenda and Errata),” Helv. Phys. Acta 34, 621 (1961).
18.R. Cirelli, A. Manià, and L. Pizzocchero, J. Math. Phys. 31, 2898 (1990).
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