Abstract
Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states. This geometrical structure can accommodate generalizations of quantum mechanics, including the nonlinear relativistic models recently proposed. It is shown that any such generalization satisfying a few physically reasonable conditions would reduce to ordinary quantum mechanics for states that are “near” the vacuum. In particular the origin of complex structure is described.
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References
Abraham, R., Marsden, J.E.: Foundations of mechanics. New York, Amsterdam: Benjamin 1967
Maclane, S.: Geometrical mechanics (duplicated lecture notes). University of Chicago (1968)
Souriau, J.M.: Commun. math. Phys.1, 374 (1966)
Souriau, J.M.: Ann. Inst. Henri Poincaré A6, 311 (1967)
Kostant, B.: Quantization and unitary representations. Lecture notes in mathematics, Vol. 170. Berlin, Heidelberg, New York: Springer 1970
Chernoff, P.R., Marsden, J.E.: Properties of infinite-dimensional Hamiltonian systems. Lecture notes in mathematics, Vol. 425. Berlin, Heidelberg, New York: Springer 1974
Haag, R., Kastler, D.: J. Math. Phys.5, 848 (1964)
Edwards, C.M.: Commun. math. Phys.20, 26 (1971)
Davis, E.B., Lewis, J.L.: Commun. math. Phys.17, 239 (1970)
Birkhoff, G., Neumann, J. von: Ann. Math.37, 823 (1936)
Varadarajan, V.S.: Geometry of quantum theory, Vol. 1. Princeton, NJ: van Nostrand 1968
Mielnik, B.: Commun. math. Phys.37, 221 (1974)
Kibble, T.W.B.: Commun. math. Phys.64, 73–82 (1978)
Haag, R., Bannier, U.: Commun. math. Phys.60, 1 (1978)
Penrose, R.: Gen. Rel. Grav.7, 31 (1976);7, 171 (1976)
Jauch, J.M.: Foundations of quantum mechanics. Reading, Mass.: Addison-Wesley 1968
Piron, C.: Foundations of quantum physics. New York, Amsterdam: Benjamin 1976
Bogolubov, N.N., Logunov, A.A., Todorov, I.T.: Introduction to axiomatic quantum field theory. Reading, Mass.: Benjamin 1975
Lang, S.: Introduction to differentiable manifolds. New York, London: Interscience 1962
Yamamuro, S.: Differential calculus in topological linear spaces. Lecture notes in mathematics, Vol. 374. Berlin, Heidelberg, New York: Springer 1974
Kijowski, J., Szczyrba, W.: Studia Math.30, 247 (1968)
Penot, J.-P.: Studia Math.47, 1 (1973)
Lloyd-Smith, J.: Trans. Am. Math. Soc.187, 249 (1974)
Choquet-Bruhat, Y., de Witt-Morette, C., Willard-Bleick, M.: Analysis, manifolds and physics. Amsterdam: North-Holland 1977
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Communicated by R. Haag
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Kibble, T.W.B. Geometrization of quantum mechanics. Commun.Math. Phys. 65, 189–201 (1979). https://doi.org/10.1007/BF01225149
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DOI: https://doi.org/10.1007/BF01225149