The Logic of Quantum Mechanics

  • Garrett Birkhoff
  • John Von Neumann
Part of the The University of Western Ontario Series in Philosophy of Science book series (WONS, volume 5a)

Abstract

One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).

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Notes

  1. 2a.
    L. Pauling and E. B. Wilson, An Introduction to Quantum Mechanics, McGraw-Hill, 1935, p. 422.Google Scholar
  2. 2b.
    Dirac, Quantum Mechanics, Oxford, 1930, §4.Google Scholar
  3. 3.
    For the existence of mathematical causation, cf. also p. 65 of Heisenberg’s Dirac The Physical Principles of the Quantum Theory, Chicago, 1929.Google Scholar
  4. 4.
    Cf. J. von Neumann, Mathematische Grundlagen der Quanten-mechanik, Berlin, 1931, p. 18.Google Scholar
  5. 7.
    Cf. J. von Neumann, ‘Operatorenmethoden in der klassischen Mechanik,’ Annals of Math. 33 (1932), 595–8. The difference of two sets S 1 , S 2 is the set (S 1+S 2) — S 1·S 2 of those points, which belong to one of them, but not to both.CrossRefGoogle Scholar
  6. 8.
    F. Hausdorff, Mengenlehre, Berlin, 1927, p. 78.Google Scholar
  7. 9.
    M. H. Stone, ‘Boolean Algebras and Their Application to Topology’, Proc. Nat. Acad. 20 (1934), 197.CrossRefGoogle Scholar
  8. 10a.
    Cf. von Neumann, op. cit., pp. 121CrossRefGoogle Scholar
  9. 10b.
    90 or Dirac, op. cit., 17. We disregard complications due to the possibility of a continuous spectrum. They are inessential in the present case.CrossRefGoogle Scholar
  10. 13a.
    F. J. Murray and J. v. Neumann, ‘On Rings of Operators’, Annals of Math., 37 (1936), 120.CrossRefGoogle Scholar
  11. 13b.
    It is shown on p. 141, loc. cit. (Definition 4.2.1 and Lemma 4.2.1), that the closed linear sets of a ring M — that is those, the “projection operators” of which belong to M — coincide with the closed linear sets which are invariant under a certain group of rotations of Hilbert space. And the latter property is obviously conserved when a set-theoretical intersection is formed.CrossRefGoogle Scholar
  12. 14.
    Thus in Section 6, closed linear subspaces of Hilbert space correspond one-many to experimental propositions, but one-one to physical qualities in this sense.Google Scholar
  13. 15.
    F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, 1914, Chap. VI, §1.Google Scholar
  14. 16.
    The final result was found independently by O. Öre, ‘The Foundations of Abstract Algebra. I, Annals of Math. 36 (1935), 406–37, and by H. MacNeille in his Harvard Doctoral Thesis, 1935.CrossRefGoogle Scholar
  15. 19.
    R. Dedekind, Werke, Braunschweig, 1931, vol. 2, p. 110.Google Scholar
  16. 20.
    G. Birkhoff, ‘On the Combination of Subalgebras’, Proc. Camb. Phil. Soc. 29 (1933), 441–64, §§23–4. Also, in any lattice satisfying L6, isomorphism with respect to inclusion implies isomorphism with respect to complementation; this need not be true if L6 is not assumed, as the lattice of linear subspaces through the origin of Cartesian n-space shows.CrossRefGoogle Scholar
  17. 21.
    M. H. Stone, ‘Boolean Algebras and Their Application to Topology’, Proc. Nat. Acad. 20 (1934), 197–202.CrossRefGoogle Scholar
  18. 22.
    A detailed explanation will be omitted, for brevity; one could refer to work of G. D. Birkhoff, J. von Neumann, and A. Tarski.Google Scholar
  19. 23.
    G. Birkhoff, op. cit., §28. The proof is easy. One first notes that since a ⊂ (ab) ∩ c if ac, and b ∩ c ⊂ (ab) ∩ c in any case, a ∪ (b ∩ c) ⊂(ab) ∩ c. Then one notes that any vector in (ab) ∩ c can be written ξ = α + β[α∈a, βb, ξc]. But β = ξα is in c(since ξc and α∈ac); hence ξ = α + βa∪(b∩c), and a∪(bc)⊃(ab)∩c, completing the proof.CrossRefGoogle Scholar
  20. 24.
    R. Dedekind, Werke, vol. 2, p. 255.Google Scholar
  21. 25.
    The statements of this paragraph are corollaries of Theorem 10.2 of G. Birkhoff, ‘On the Combination of Subalgebras’, Proc. Camb. Phil. Soc. 29 (1933), 441–64 op. cit.CrossRefGoogle Scholar
  22. 26.
    G. Birkhoff ‘Combinatorial Relations in Projective Geometries’, Annals of Math. 30 (1935), 743–8.CrossRefGoogle Scholar
  23. 27.
    O. Öre, op. cit., p. 419.CrossRefGoogle Scholar
  24. 28.
    Using the terminology of footnote,13 and of loc. cit. there: The ring MM′ should contain no other projection-operators than 0, 1, or: the ring M must be a “factor.” Cf. loc. cit. 13 p. 120.CrossRefGoogle Scholar
  25. 29.
    Cf. §§103–105 of B. L. Van der Waerden’s Moderne Algebra, Berlin, 1931, Vol. 2.Google Scholar
  26. 30a.
    n = 4, 5,… means of course n-1≧3, that is, that Q n-1 is necessarily a “Desarguesian” geometry. (Cf. O. Veblen and J. W. Young, Projective Geometry, New York, 1910, Vol. 1, page 41). Then F=F(Q n-1) can be constructed in the classical way. (Cf., Veblen and Young, Vol. 1, pages 141–150). The proof of the isomorphism between Q n-1 and the P n-1 (F) as constructed above, amounts to this: Introducing (not necessarily commutative) homogeneous coordinates x 1,…,x n from F in Q n-1, and expressing the equations of hyperplanes with their help. This can be done in the manner which is familiar in projective geometry, although most books consider the commutative (“Pascalian”) case only.Google Scholar
  27. 30b.
    D. Hilbert, Grundlagen der Geometrie, 7th edition, 1930, pages 96–103, considers the noncommutative case, but for affine geometry, and n-1=2, 3 only. Considering the lengthy although elementary character of the complete proof, we propose to publish it elsewhere.Google Scholar
  28. 30a.
    R. Brauer, ‘A Characterization of Null Systems in Projective Space’, Bull. Am. Math. Soc. 42 (1936), 247–54, treats the analogous question in the opposite case that \( X \cap X' \ne \circledcirc \) is postulated.CrossRefGoogle Scholar
  29. 31.
    In the real case, w(x)=x; in the complex case, w(x + iy) = xiy; in the quaternionic case, w(u + ix + jy + kz) = u — ix — jy — kz; in all cases, the 2, are 1. Conversely, A. Kolmogoroff, ‘Zur Begründung der projektiven Geometrie’, Annals of Math. 33 (1932), 175–6 has shown that any projective geometry whose k-dimensional elements have a locally compact topology relative to which the lattice operations are continuous, must be over the real, the complex, or the quaternion field.CrossRefGoogle Scholar
  30. 33a.
    J. von Neumann, ‘Continuous Geometries’, Proc. Nat. Acad. 22 (1936), 92–100CrossRefGoogle Scholar
  31. 33b.
    J. von Neumann, ‘Continuous Geometries’, Proc. Nat. Acad. 22 (1936), and 101–109. These may be a more suitable frame for quantum theory, than Hilbert space.CrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1975

Authors and Affiliations

  • Garrett Birkhoff
    • 1
  • John Von Neumann
    • 1
  1. 1.The Institute for Advanced StudyHarvard UniversityUSA

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