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
L. Pauling and E. B. Wilson, An Introduction to Quantum Mechanics, McGraw-Hill, 1935, p. 422.
Dirac, Quantum Mechanics, Oxford, 1930, §4.
For the existence of mathematical causation, cf. also p. 65 of Heisenberg’s Dirac The Physical Principles of the Quantum Theory, Chicago, 1929.
Cf. J. von Neumann, Mathematische Grundlagen der Quanten-mechanik, Berlin, 1931, p. 18.
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.
F. Hausdorff, Mengenlehre, Berlin, 1927, p. 78.
M. H. Stone, ‘Boolean Algebras and Their Application to Topology’, Proc. Nat. Acad. 20 (1934), 197.
Cf. von Neumann, op. cit., pp. 121
90 or Dirac, op. cit., 17. We disregard complications due to the possibility of a continuous spectrum. They are inessential in the present case.
F. J. Murray and J. v. Neumann, ‘On Rings of Operators’, Annals of Math., 37 (1936), 120.
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.
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.
F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, 1914, Chap. VI, §1.
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.
R. Dedekind, Werke, Braunschweig, 1931, vol. 2, p. 110.
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.
M. H. Stone, ‘Boolean Algebras and Their Application to Topology’, Proc. Nat. Acad. 20 (1934), 197–202.
A detailed explanation will be omitted, for brevity; one could refer to work of G. D. Birkhoff, J. von Neumann, and A. Tarski.
G. Birkhoff, op. cit., §28. The proof is easy. One first notes that since a ⊂ (a∪ b) ∩ c if a ⊂ c, and b ∩ c ⊂ (a ∪ b) ∩ c in any case, a ∪ (b ∩ c) ⊂(a∪ b) ∩ c. Then one notes that any vector in (a∪ b) ∩ c can be written ξ = α + β[α∈a, β∈b, ξ∈c]. But β = ξ – α is in c(since ξ∈c and α∈a⊂c); hence ξ = α + β∈a∪(b∩c), and a∪(b∩c)⊃(a∪b)∩c, completing the proof.
R. Dedekind, Werke, vol. 2, p. 255.
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.
G. Birkhoff ‘Combinatorial Relations in Projective Geometries’, Annals of Math. 30 (1935), 743–8.
O. Öre, op. cit., p. 419.
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.
Cf. §§103–105 of B. L. Van der Waerden’s Moderne Algebra, Berlin, 1931, Vol. 2.
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.
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.
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.
In the real case, w(x)=x; in the complex case, w(x + iy) = x — iy; 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.
J. von Neumann, ‘Continuous Geometries’, Proc. Nat. Acad. 22 (1936), 92–100
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.
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Birkhoff, G., Von Neumann, J. (1975). The Logic of Quantum Mechanics. In: Hooker, C.A. (eds) The Logico-Algebraic Approach to Quantum Mechanics. The University of Western Ontario Series in Philosophy of Science, vol 5a. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1795-4_1
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