Abstract
The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or “Q-type” operators. “P-type” operators analogous to Schroedinger momenta, in that they transform theQ-type quantities, are bracing (Br), Clifford multiplication by a setX, and the creator ofX, represented by Grassmann multiplicationc(X) by the setX. Br and its adjoint Br* form a Bose-Einstein canonical pair, andc(X) and its adjointc(X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied to a quantum time space and a quantum automaton.
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References
Aristotle,De Interpretationes, Chaps. 11 and 12.
Bell, J. L. (1977).Boolean-valued models and independence proofs in set theory. Oxford University Press, New York.
Chevalley, C. C. (1954).The algebraic theory of spinors. Columbia University Press, New York.
Feynman, R. P. (1965).The character of physical laws. Massachusetts Institute of Technology, Cambridge, Massachusetts.
Finkelstein, D. (1969). “Space-Time Code,”Physical Review,184, 1261; (1982). “Cosmological Choices,”Synthese, to appear.
Ford, J. (1982). “How Random is a Coin Toss?,” inProceedings of Workshop on Longtime Prediction in Nonlinear Conservative Dynamical Systems, L. E. Reichl, ed. in press.
Hestenes, D. (1966).Space-Time Algebra. Gordon & Breach, New York.
Kaufman, A. (1975).Introduction to the theory of fuzzy subsets. Academic Press, New York.
Kripke, S. A. (1963). “Semantical Considerations on Modal Logics,”Acta Philosophica Fennica,16, 83.
Misner, C. A., Thorne, K., and Wheeler, J. A. (1973).Gravitation. W. H. Freeman, San Francisco.
Riesz, M. (1958?)Clifford numbers and spinors, University of Maryland Institute for Fluid Dynamics Lecture Series No. 38 (mineographed).
Sallngaros, M., and Dresden, M. (1979). “Properties of an associative algebra of tensor fields,”Physical Review,43, 1.
Takeuti, G. (1981). “Quantum set theory,” inCurrent Issues in Quantum Logic, E. Beltrametti and B. C. van Fraasen, eds. Plenum Press, New York.
Takeuti, G. (1978).Two Applications of Logic to Mathematics. Princeton University Press, Princeton, New Jersey.
von Weizsaecker, C. F. (1975, 1977, 1979, 1981). “Binary Alternatives and Space-Time Structure,” and other papers inQuantum Theory and the Structures of Time and Space I–IV., L. Castell et al., eds. Hanser, Munich, and work cited there.
Zadeh, L. A., et al., eds., (1975).US-Japan Seminar on Fuzzy Sets and Their Applications, Academic Press, New York.
Zuse, K. (1969).Rechnender Raum. Vieweg, Braunschweig.
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This material is based upon work supported in part by NSF Grant No. PHY8007921.
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Finkelstein, D. Quantum sets and clifford algebras. Int J Theor Phys 21, 489–503 (1982). https://doi.org/10.1007/BF02650180
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DOI: https://doi.org/10.1007/BF02650180