Abstract
Given a state on an algebra of bounded quantummechanical observables, we investigate those subalgebrasthat are maximal with respect to the property that thegiven state's restriction to the subalgebra is a mixture of dispersion-free states —what we call maximal beable subalgebras (borrowingterminology due to J. S. Bell). We also extend ourresults to the theory of algebras of unboundedobservables (as developed by Kadison), and show how ourresults articulate a solid mathematical foundation forcertain tenets of the orthodox Copenhagen interpretationof quantum theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
J. Anderson, Extensions, restrictions, and representations of states on C*-algebras, Transactions of the American Mathematical Society 249 (1979), 303-329.
J. L. Bell, Logical reflections on the Kochen-Specker theorem, in Perspectives on Quantum Reality (R. Clifton, ed.), Kluwer, Dordrecht (1996), pp. 227-235.
J. S. Bell, On the problem of hidden variables in quantum mechanics, Reviews of Modern Physics 38 (1966), 447-452.
J. S. Bell, Subject and object, in The Physicist's Conception of Nature (J. Mehra, ed.), Reidel, Dordrecht (1973), pp. 687-690.
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge (1987).
D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables, Parts I and II, Physical Review 85 (1952), 166-193.
N. Bohr, Can quantum-mechani cal description of physical reality be considered complete? Physical Review 48 (1935), 696-702.
N. Bohr, On the notions of causality and complementarity, Dialectica 2 (1948), 312-319.
J. Bub, Complementarity and the orthodox (Dirac-von Neumann) interpretation of quantum mechanics, in Perspectives on Quantum Reality (R. Clifton, ed.), Kluwer, Dordrecht (1996), pp. 211-226.
J. Bub, Interpreting the Quantum World, Cambridge University Press, Cambridge (1997).
J. Bub and R. Clifton, A uniqueness theorem for no-collapse interpretations of quantum mechanics, Studies in History and Philosophy of Modern Physics 27 (1996), 181-219.
R. Clifton, Getting contextual and nonlocal elements-of-reality the easy way, American Journal of Physics 61 (1993), 443-447.
R. Clifton, Independently motivating the Kochen-Dieks modal interpretation of quantum mechanics, British Journal for Philosophy of Science 46 (1995), 33-57.
R. Clifton, Making sense of the Kochen-Dieks no-collapse interpretation of quantum mechanics independent of the measurement problem, Annals of the New York Academy of Sciences 775 (1995), 570-578.
R. Clifton, Beables in algebraic quantum theory, in From Physics to Philosophy (J. Butterfield and C. Pagonis, eds.), Cambridge University Press, Cambridge, in press.
P. A. M. Dirac, Quantum Mechanics, 4th ed., Clarendon Press, Oxford (1958).
G. G. Emch, Mathematical and Conceptual Foundations of 20th Century Physics, North-Holland, Amsterdam (1984).
A. M. Gleason, Measures on closed subspaces of Hilbert space, Journal of Mathematics and Mechanics 6 (1957), 885-893.
S. S. Horuzhy, Introduction to Algebraic Quantum Field Theory, Kluwer, Boston (1990).
D. Howard, What makes a classical concept classical? Toward a reconstruction of Niels Bohr's philosophy of physics, in Niels Bohr and Contemporary Philosophy (J. Faye and H. J. Folse, eds.), Kluwer, Dordrecht (1994), pp. 201-229.
R. Jost, Measures on the finite dimensional subspaces of a Hilbert space, in Studies in Mathematical Physics: Essays in Honour of Valentine Bergmann, Princeton University Press, Princeton, New Jersey (1976), pp. 209-228.
R. V. Kadison, Algebras of unbounded functions and operators, Expositiones Mathematicae 4 (1986), 3-33.
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, American Mathematical Society, Providence, Rhode Island (1997).
R. V. Kadison and I. M. Singer, Extensions of pure states, American Journal of Mathematics 81 (1959), 383-400.
S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics 17 (1967), 59-87.
N. P. Landsman, Poisson spaces with a transition probability, Reviews of Mathematical Physics 9 (1997), 29-57.
N. P. Landsman, Mathematical Topics between Classical and Quantum Mechanics, Springer-Verlag, New York (1998).
D. B. Lowdenslager, On postulates for general quantum mechanics, Proceedings of the American Mathematical Society 8 (1957), 88-91.
B. Misra, When can hidden variables be excluded in quantum mechanics? Nuovo Cimento XLVIIA (1967), 841-859.
A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht (1993).
A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Physical Review 47 (1935), 777-78 0.
W. Rudin, Real and Complex Analysis, McGraw-Hill, New York (1987).
E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften 23 (1935), 807-812, 823-828, 844-849.
I. E. Segal, Postulates for general quantum mechanics, Annals of Mathematics 48 (1947), 930-948.
P. Teller, On the problem of hidden variables for quantum mechanical observables with continuous spectra, Philosophy of Science 44 (1977), 475-477.
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, New Jersey (1955).
J. Zimba and R. Clifton, Valuations on functionally closed sets of quantum-mechanical observables and Von Neumann's no-hidden-variables theorem, in The Modal Interpretation of Quantum Mechanics (D. Dieks and P. Vermaas, eds.), Kluwer, Dordrecht (1998), pp. 69-101.
Rights and permissions
About this article
Cite this article
Halvorson, H., Clifton, R. Maximal Beable Subalgebras of Quantum Mechanical Observables. International Journal of Theoretical Physics 38, 2441–2484 (1999). https://doi.org/10.1023/A:1026628407645
Issue Date:
DOI: https://doi.org/10.1023/A:1026628407645