Abstract
Homeomorphisms of the unit-sphere of states are studied. Generalizations of the Piron statement and Wigners' theorem are obtained. It is shown that if the semigroup of the unitary operations of quantum theory were extended by introducing any non-linear operation, amobility phenomenon would occur consisting of a possibility of moving any two states to any two surroundings on the unit sphere. For the resulting “non-linear wave packets” the complementarity doctrine would become impossible because of “fluidity” of the space of states under the dynamical transformations.
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Ruelle, D.: Applications conservant une mesure absolument continue par rapport àdx sur [0, 1]. Commun. Math. Phys.55, 47 (1977)
Eckmann, J.-P.: Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys.53, 643 (1981)
Piña, E.: Talk at the Phys. Dept., CINVESTAV, México (1983)
Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848 (1964)
Lamb, W.E.: Phys. Today,22, No. 4, 23 (1969)
Mielnik, B.: Generalized quantum mechanics. Commun. Math. Phys.37, 221 (1974)
Lubkin, E.: A physical system which can be forced to execute an arbitrary unitary transformation, and its use to perform arbitrary tests. J. Math. Phys.15, 673 (1974)
Mielnik, B.: Mobility of nonlinear systems. J. Math. Phys.21, 44 (1980)
Waniewski, J.: “Globalna dynamika i Struktura wypukła stanóww teoriach kwantowych i klasycznych”, Warszawa 1983 (manuscript of Ph. D. thesis)
Waniewski, J.: “Time inversion and mobility of many particle systems. Commun. Math. Phys.76, 27 (1980)
Mielnik, B.: Global mobility of Schrödinger's particle. Rep. Math. Phys.12, 331 (1977)
Haag, R., Bannier, U.: Comments on Mielnik's generalized (non-linear) quantum mechanics. Commun. Math. Phys.60, 1 (1978)
Piron, C.: Axiomatique Quantique. Helv. Phys. Acta37, 439 (1964)
Piron, C.: Foundations of quantum physics. New York: Benjamin 1976
Wigner, E.P.: Ann. Math.40, 149 (1939)
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Communicated by R. Haag
Supported by COSNET/SEP
On leave of absence from Institute of Physics, Warsaw University, Warsaw, Poland
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Mielnik, B. Phenomenon of mobility in non-linear theories. Commun.Math. Phys. 101, 323–339 (1985). https://doi.org/10.1007/BF01216093
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DOI: https://doi.org/10.1007/BF01216093