Foundations of Physics

, Volume 21, Issue 10, pp 1183–1236 | Cite as

The foundation of quantum theory and noncommutative spectral theory: Part II

  • Hans Kummer


The present paper comprises Sects. 5–8 of a work which proposes an axiomatic approach to quantum mechanics in which the concept of a filter is the central primitive concept. Having layed down the foundations in the first part of this work (which appeared in the last issue of this journal and comprises Sects. 0–4), we arrived at a dual pair 〈Y, M〉 consisting of abase norm space Y and anorder unit space M, being in order and norm duality with respect to each other. This is precisely the setting of noncommutative spectral theory, a theory which has been developed during the late nineteen seventies by Alfsen and Shultz. (2,3) In this part we add to the four axioms (Axioms S, DP, R, SP) of Sect. 3 three further axioms (Axioms E, O, L). These axioms are suggested by the work of Alfsen and Shultz and enable us to derive the JB-algebra structure of quantum mechanics (cf. Theorem 8.9).


Quantum Mechanic Quantum Theory Norm Space Spectral Theory Dual Pair 
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  1. 1.
    E. M. Alfsen,Compact Convex Sets and Boundary Integrals (Springer, New York, 1971).Google Scholar
  2. 2.
    E. M. Alfsen and W. F. Shultz, “Noncommutative spectral theory for affine function spaces on convex sets,”Mem. Am. Math. Soc. 6(172), (1976).Google Scholar
  3. 3.
    E. M. Alfsen and W. F. Shultz, “On noncommutative spectral theory and Jordan algebras,”Proc. London Math. Soc. 38(3), 497–516 (1979).Google Scholar
  4. 4.
    E. M. Alfsen, W. F. Shultz, and E. Störmer, “A Gelfand-Neumark theorem for Jordan algebras,”Adv. Math. 28, 11–56 (1978).Google Scholar
  5. 5.
    J. Bellissard and B. Iochum, “Self-dual cones versus Jordan algebras. The theory revisited,”Ann. Inst. Fourier (Grenoble) 28(1), 27–67 (1978).Google Scholar
  6. 6.
    H. Hanche-Olsen and E. Störmer,Jordan Operator Algebras (Pitman, London, 1984).Google Scholar
  7. 7.
    R. D. Schafer,An Introduction to Nonassociative Algebras (Academic Press, New York, 1966).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Hans Kummer
    • 1
  1. 1.Department of Mathematics and StatisticsQueen's UniversityKingstonCanada

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