Foundations of Physics

, Volume 45, Issue 5, pp 557–590 | Cite as

A Reconstruction of Quantum Mechanics

Article
  • 350 Downloads

Abstract

We show that exactly the same intuitively plausible definitions of state, observable, symmetry, dynamics, and compound systems of the classical Boolean structure of intrinsic properties of systems lead, when applied to the structure of extrinsic, relational quantum properties, to the standard quantum formalism, including the Schrödinger equation and the von Neumann–Lüders Projection Rule. This approach is then applied to resolving the paradoxes and difficulties of the orthodox interpretation.

Keywords

Reconstruction Quantum paradoxes Quantum mechanics  Classical mechanics 

References

  1. 1.
    Bargmann, V.: Note on Wigner’s theorem on symmetry operations. J. Math. Phys. 5, 862 (1964)CrossRefADSMATHMathSciNetGoogle Scholar
  2. 2.
    Beltrametti, E.G., Cassinelli, G.: The Logic of Quantum Mechanics. Addison-Wesley, Reading, MA (1981)MATHGoogle Scholar
  3. 3.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823 (1936)CrossRefGoogle Scholar
  4. 4.
    Bohm, A.: Quantum Mechanics: Foundations and Applications. Springer, New York (2001)Google Scholar
  5. 5.
    Bohr, N.: Causality and complementarity. Philos. Sci. 4, 289 (1937)CrossRefGoogle Scholar
  6. 6.
    Conway, J., Kochen, S.: The geometry of the quantum paradoxes. In: Bertlemann, R.A., Zeilinger, A. (eds.) Quantum [Un]speakables, p. 257. Springer, Berlin (2002)CrossRefGoogle Scholar
  7. 7.
    Conway, J., Kochen, S.: The strong free will theorem. Am. Math. Soc. Not. 56, 226 (2009)MATHMathSciNetGoogle Scholar
  8. 8.
    Faddeev, L.D., Yakubovskii, O.A.: Lectures on Quantum Mechanics for Mathematics Students. Am. Math. Soc, Providence, RI (2009)Google Scholar
  9. 9.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 3. Addison-Wesley, Reading, MA (1966)Google Scholar
  10. 10.
    Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 36 (1948)CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Finkelstein, D.: The logic of quantum physics. Trans. N. Y. Acad. Sci. 25, 621 (1963)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885 (1957)MATHMathSciNetGoogle Scholar
  13. 13.
    Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley, Reading, MA (1968)CrossRefMATHGoogle Scholar
  14. 14.
    Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59 (1967)MATHMathSciNetGoogle Scholar
  15. 15.
    Kochen, S., Specker, E.P.: Logical structures arising in quantum mechanics. In: Symposium at Berkeley, The Theory of Models, p. 177 (1965)Google Scholar
  16. 16.
    Kochen, S., Specker, E.P.: The calculus of partial propositional functions. In: Congress at Jerusalem, Methodology and Philosophy of Science, p. 45 (1964)Google Scholar
  17. 17.
    Koppelberg, S.: Handbook of Boolean Algebras, vol. 1. North-Holland, Amsterdam (1989)MATHGoogle Scholar
  18. 18.
    Mackey, G.W.: Mathematical Foundations of Quantum Mechanics. Benjamin, Amsterdam (1963)MATHGoogle Scholar
  19. 19.
    Piron, C.: Foundations of Quantum Physics. Benjamin, Reading, MA (1976)CrossRefMATHGoogle Scholar
  20. 20.
    Reck, M., Zeilinger, A., Bernstein, H.J., Bertani, P.: Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58 (1994)CrossRefADSGoogle Scholar
  21. 21.
    Uhlhorn, U.: Representation of symmetric transformations in quantum mechanics. Arkiv Fysik 23, 307 (1963)MATHGoogle Scholar
  22. 22.
    Varadarajan, V.S.: The Geometry of Quantum Theory. Van Nostrand, Princeton, NJ (1968)CrossRefGoogle Scholar
  23. 23.
    Wrede, E.: Über die Ablenkung von Molekularstrahlen elektrischer Dipolmolekule iminhomogenen elektrischen Feld. Z. Phys. A 44, 261 (1927)CrossRefGoogle Scholar
  24. 24.
    Zukowski, M., Zeilinger, A., Horne, M.A.: Realizable higher-dimensional two-particle entanglements via multiport beam splitters. Phys. Rev. A 55, 2564 (1997)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA

Personalised recommendations