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The nonrelativistic limit of central-extended Poincaré group and some consequences for quantum actualization

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2009
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American Institute of Physics
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The nonrelativistic limit of the centrally extended Poincaré group is considered and their consequences in the modal Hamiltonian interpretation of quantum mechanics are discussed [O. Lombardi and M. Castagnino, Stud. Hist. Philos. Mod. Phys 39, No.2, 380–443 (2008); J. Phys, Conf. Ser. 128, 012014 (2008)]. Through the assumption that in quantum field theory the Casimir operators of the Poincar´e group actualize, the nonrelativistic limit of the latter group yields to the actualization of the Casimir operators of the Galilei group, which is in agreement with the actualization rule of previous versions of modal Hamiltonian interpretation
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O. Lombardi, M. Castagnino, Stud. Hist. Phil. Mod. Phys,39, 380-443, 2008. M. Castagnino, O. Lombardi, Jour. Phys. Conf. Series, 128, 012014, 2008. J. S. Ardenghi, M. Castagnino, O. Lombardi, submitted to Foundations of Physics, 2008. B. C. Van Fraassen, A formal approach to philosophy of science, Paradigms and paradoxes: The philosophical Challenge of the Quantum Domain. Pittsburgh: University of Pittsburgh Press, 303-366, (1972). S. Kochen, A new interpretation of quantum mechanics, Symposium on the Foundations of Modern Physics. Singapore: World Scientific, 151-169, (1985). J. Bub, Interpreting the Quantum World, Cambridge Univ. Press, Cambridge 1997. N. D. Mermin, Pramana 51, 549-565, 1998. J. F. Cariñena, M. A. del Olmo and M. Santander, J. Phys. A: Math. Gen. 14, 1 (1981) H. Weyl The Theory of Groups and Quantum Mechanics, (Dover, New York, 1931). L. Ballentine, Quantum Mechanics, a Modern Development, (World Scientific, Singapore, 1998). J.-M. Levy Léblond, Annals I.H.P., section A, 3 (1), 1-12, 1965. M. Bacry, J.-M. Levy Léblond, Journ. Math. Phys., 9 (10), 1605-1614, 1968. R. Haag, Local Quantum Physics, (Springer-Verlag, Berlin, 1993). Nevil F. Mott, ”The wave mechanics of alpha-ray tracks”, Proceedings of the Royal Society, London, A126, 79-84, 1929. M. Castagnino, R. Laura, Int. Journ. Theo. Phys.39, 1767, 2000; Phys Rev. A. 62, ♯ 022107, 2000. M. Castagnino. O. Lombardi, Phys. Rev. A, 72, # 012102, 2005, Stud. Hist. Phil. Mod. Phys. 35, 73, 2004. E. G. Beltrametti and A. Blasi A, Phys. Lett. 20 62-64, (1966) Generators Ji define the SO(3) group, generators Pi, Ji, define the ISO(3) group: the inhomogeneous rotation group in three dimensions. Actually (5) is the largest subgroup that remains invariant by the Inönü-Wigner contraction of the Poincaré on the Galilei group. By a trivial extension of a Lie algebra g we mean the direct sum g M, where M is an additional commuting generator. See also in [11] for a c = 1 deduction. For the same reasons explained in paper [3], where the transformation H ! W is introduced, and the non-relativistic case analyzed.
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