Skip to main content

An Axiomatic Basis for Quantum Mechanics

Abstract

In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in this derivation are the co-ordinatization of generalized geometries and a theorem of Solér which characterizes Hilbert spaces among the orthomodular spaces. A generalized Wigner theorem is applied to reduce some of the assumptions of Solér’s theorem to the theory of symmetry in quantum mechanics. Since this reduction is only partial we also point out the remaining open questions.

This is a preview of subscription content, access via your institution.

Notes

  1. Clearly, this assumption could also be posed under Axiom 1 but we refrain of doing it.

  2. The paper of Mielnik [45] contains an extensive analysis of possible state changes, including some nonlinear processes.

  3. This property has independently been introduced in [65] and [33] and it is known to be equivalent to the fact that each \(\alpha \in \mathbf S\) has a (unique) support in \(\mathbf L\) [3, Theorem11.4.3].

  4. Our proof is an adaption of the corresponding results in [3]. Another source leading to this conclusion is given by the results of Sects. 2.5.2 of [55].

  5. For a detailed discussion of this theorem, see, e.g. [63]

  6. If \(K=\mathbb R\) then \({}^*\) is the identity. For \(K=\mathbb C\) the map \({}^*\) cannot be the identity and if it is continuous then it is the complex conjugation. For \(K=\mathbb H\) the map is the quaternionic conjugation.

  7. Various definitions of the notion of symmetry in quantum mechanics are studied e.g. in [9, 48].

References

  1. Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  2. Baer, R.: Linear Algebra and Projective Geometry. Academic Press, New York (1952)

    MATH  Google Scholar 

  3. Beltrametti, E., Cassinelli, G.: The Logic of Quantum Mechanics, Addison-Wesley, Reading, 1981. Cambridge University Press 1985 (2010)

  4. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–702 (1935)

    Article  ADS  MATH  Google Scholar 

  6. Bugajska, K., Bugajski, S.: The projection postulate in quantum logic. Bull. Acad. Pol. Sci. Ser. des Sci. Math., Astron. Et Phys. 21, 873–877 (1973)

    MathSciNet  Google Scholar 

  7. Bugajski, S., Lahti, P.: Fundamental principles of quantum theory. Int. J. Theor. Phys. 19, 499–514 (1980)

    Article  MathSciNet  Google Scholar 

  8. Cassinelli, G., Beltrametti, E.: Ideal, first-kind measurements in a proposition-state structure. Commun. Math. Phys. 40, 7–13 (1975)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Cassinelli, G., De Vito, E., Lahti, P.J., Levrero, A.: The Theory of Symmetry Actions in Quantum Mechanics. Springer, LNP 654 (2004)

  10. Cassinelli, G., Lahti, P.: A theorem of Solér, the theory of symmetry, and quantum mechanics. Int. J. Geom. Methods Modern Phys., (9) 1260005(7) (2012)

  11. Chiribella, G., D’Ariano, G.M., Perinotti, P. : Probabilistic theories with purification. Phys. Rev. A. 81, 062348(40) (2010)

  12. Chiribella, G., D’Ariano, G.M., Perinotti, P.: Informational derivation of quantum theory. Phys. Rev. A, 84, 012311(39) (2011)

  13. Clifton, R., Bub, J., Halvorson, H.: Characterizing quantum theory in terms of information-theoretic constrains. arXiv:quant-ph/0211089v2 19 Feb 2003

  14. Cornette, W.M., Gudder, S.P.: The mixture of quantum states. J. Math. Phys. 15, 842–850 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  15. Davies, E.B., Lewis, J.L.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London (1976)

    MATH  Google Scholar 

  17. Dirac, P.A.M.: The Principles of Quantum Mechanics. Oxford University Press, London (1930)

    MATH  Google Scholar 

  18. Edwards, C.M.: The operational approach to algebraic quantum theory I. Commun. Math. Phys. 16, 207–230 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Edwards, C.M.: Classes of operations in quantum theory. Commun. Math. Phys. 20, 26–36 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Edwards, C.M.: The theory of pure operations. Commun. Math. Phys. 24, 260–288 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Edwards, C.M., Gerzon, M.A.: Monotone convergence in partially ordered vector spaces. Ann. Inst. Henri Poinceré 12, 323–328 (1970)

    MathSciNet  MATH  Google Scholar 

  22. Ellis, A.J.: The duality of partially ordered normed linear spaces. J. Lond. Math. Soc. 39, 730–744 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ellis, A.J.: Linear operators in partially ordered normed vector spaces. J. Lond. Math. Soc. 41, 323–332 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fillmore, F.A., Longstaff, W.E.: On isomorphisms of lattices of closed subspaces. Can. J. Math. XXXVI, 820–829 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Foulis, D.J., Randall, C.H.: The Empirical Logic Approach to the Physical Sciences, pp. 230–249. Academic Press, New York (1978)

    MATH  Google Scholar 

  26. Foulis, D.J., Randall, C.H.: Empirical logic and quantum mechanics. Synthese 29, 81–111 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957)

    MathSciNet  MATH  Google Scholar 

  28. Gross, H., Künzi, U.-M.: On a class of orthomodular quadratic spaces. L’Enseign. Math. 31, 187–212 (1985)

    MathSciNet  MATH  Google Scholar 

  29. Gudder, S.P.: Convex structures and operational quantum mechanics. Commun. Math. Phys. 29, 249–264 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  30. Gudder, S.P.: Stochastic Methods in Quantum Mechanics. Elsevier, Amsterdam (1979)

    MATH  Google Scholar 

  31. Holland, S.S.: Orthomodularity in infinite dimensions; a theorem of M. Solér. Bull. Am. Math. Soc. 32, 205–234 (1995)

    Article  MATH  Google Scholar 

  32. Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley, Reading (1968)

    MATH  Google Scholar 

  33. Jauch, J.M., Piron, C.: Can hidden variables be excluded in quantum mechanics? Helv. Phys. Acta 36, 827–837 (1963)

    MathSciNet  MATH  Google Scholar 

  34. Keller, H.: Ein nicht-klassischer Hilbertscher Raum. Math. Z. 172, 41–49 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  35. Keller, H.: Measures on Non-Classical Hilbertian Spaces, Notas Mathematicas No 16. Universidad Catoliga Santiago, Chile (1984)

    Google Scholar 

  36. Lahti, P., Bugajski, S.: Fundamental principles of quantum theory. II. From a convexity scheme to the DHB theory. Int. J. Theor. Phys. 24, 1051–1080 (1985)

    Article  MathSciNet  Google Scholar 

  37. Lahti, P.J., Ma̧czynski, M.J.: Heisenberg inequality and the complex field in quantum mechanics. J. Math. Phys. 28, 1764–1769 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  38. Lahti, P.J., Ma̧czynski, M.J.: Orthomodularity and quadratic transformations in probabilistic theories of physics. J. Math. Phys. 33, 4133–4138 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Loomis, L.H.: On the representation of \(\sigma \)-complete Boolean algebras. Bull. Am. Math. Soc. 35, 757–760 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ludwig, G.: Attempt of an axiomatic foundation of quantum mechanics and more general theories (II). Commun. Math. Phys 4, 331–348 (1967)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Ludwig, G.: Foundations of Quantum Mechanics I. Springer, New York (1983)

    Book  MATH  Google Scholar 

  42. Mackey, G.: Mathematical Foundations of Quantum Mecianics. W.A. Benjamin Inc., New York (1963)

    Google Scholar 

  43. Ma̧czynski, M.J.: The orthogonality postulate in axiomatic quantum mechanics. Int. J. Theor. Phys. 8, 353–360 (1973)

    Article  MathSciNet  Google Scholar 

  44. Maeda, F., Maeda, S.: Theory of Symmetric Lattices. Springer, Berlin (1970)

    Book  MATH  Google Scholar 

  45. Mielnik, B.: Theory of filters. Commun. Math. Phys. 15, 1–46 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Mielnik, B.: Generalized quantum mechanics. Commun. Math. Phys. 37, 115–152 (1974)

    Article  MathSciNet  Google Scholar 

  47. Mittelstaedt, P.: Quantum Logic. D. Reidel Publ. Co., Dordrecht (1978)

    Book  MATH  Google Scholar 

  48. Molnár, L.: Selected Preserver Problems in Algebraic Structures of Linear Operators and on Function Spaces. Springer, LNM 1895 (2007)

  49. Morash, R.P.: Angle bisection and orthoautomorphisms in Hilbert lattices. Can. J. Math. 25, 261–272 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  50. Piron, C.: Foundations of Quantum Physics. Benjamin, Reading (1976)

    Book  MATH  Google Scholar 

  51. Piziak, R.: Orthomodular lattices and quandratic spaces: a survey. Rocky Mt J. Math. 21, 951–992 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  52. Pulmannová, S.: Axiomatization of quantum logics. Int. J. Theor. Phys. 35, 2309–2319 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  53. Pool, J.C.T.: Baer\({}^*\)-semigroups and the logic of quantum mechanics. Commun. Math. Phys. 9, 118–141 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Pool, J.C.T.: Semimodularity and the logic of quantum mechanics. Commun. Math. Phys. 9, 212–228 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum logics. Kluwer, Dordrecht (1991)

    MATH  Google Scholar 

  56. Rédei, M.: Quantum Logic in Algebraic Approach, Fundamental Theories of Physics 91. Kluwer Academic, Dordrecht (1998)

    Book  Google Scholar 

  57. Schaefer, H.H.: Topological Vector Spaces. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  58. Schaefer, H.H.: Orderings of vector spaces. In: Hartkämper, A., Neumann, H. (eds.) Foundations of Quantum Mechanics and Ordered Linear Spaces, pp. 4–10. Springer, Berlin (1974)

  59. Solér, M.P.: Characterization of Hilbert spaces by orthomodular spaces. Commun. Algebr. 23, 219–243 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  60. Sikorski, R.: Boolean Algebras. Springer, Berlin (1964)

    MATH  Google Scholar 

  61. Stone, M.H.: The theory of representations of Boolean algebras. Trans. Am. Math. Soc. 40, 37–111 (1936)

    MathSciNet  MATH  Google Scholar 

  62. Stone, M.H.: Postulates for the barycentric calculus. Ann. Mat. Pura Appl., 29 (1949), 25 Ű30

  63. Varadarajan, V.S.: Geometry of Quantum Theory, n edn. Springer, New York (1985)

    MATH  Google Scholar 

  64. Wilbur, W.J.: On characterizing the standard quantum logic. Trans. Am. Math. Soc. 233, 265–281 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  65. Zierler, N.: Axioms for non-relativistic quantum mechanics. Pac. J. Math., 11, 1151–1169 (1961)

Download references

Acknowledgments

We are grateful to Drs. Paul Busch and Maciej Ma̧czynski for their valuable comments in earlier versions of this manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pekka Lahti.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cassinelli, G., Lahti, P. An Axiomatic Basis for Quantum Mechanics. Found Phys 46, 1341–1373 (2016). https://doi.org/10.1007/s10701-016-0022-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-016-0022-y

Keywords

  • Quantum mechanics
  • Axiomatic basis
  • Generalized probabilistic theories
  • Orthomodular spaces
  • Theorem of Solér
  • Generalized Wigner theorem