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A Non-Standard Quantum Logic with a Strong Set of States

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Current Issues in Quantum Logic

Abstract

Since the discovery of non-Euclidian geometries the academic community has widely recognized the importance of non-standard models of axiomatic systems. This paper presents a non-standard quantum logic, call it (L44, M22). Previous examples5,6 of non-standard quantum logics were non-standard by reason of the fact that the states were not strongly order determining. M22 is strongly order determining on L44. The property violated in L44 but satisfied in Hilbert space is a variant of Desargues’ Theorem. It is called the ortho- Arguesian law and was first formulated by Alan Day.

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References

  1. G. Birkhoff, “Lattice Theory”, (Third (new) Edition), AMS Colloquium Publications, Vo. X XV (1967).

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  2. Alan Day, personal communication.

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  3. E.R. Gerelle et al., Weights on spaces, in “Physical Reality and Mathematical Description”, C.P. Enz and J. Mehra eds., Reidel, Dordrecht (1974), pp. 169–192.

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  4. R.J. Greechie, Orthomodular lattices admitting no states, J. Combinatorial Theory 10 (1971), pp. 119–132.

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  5. R.J. Greechie, An orthomodular poset with a full set of states not embeddable in Hilbert Space, Caribbean J. Math. 1, no. 2 (1969), pp. 15–26.

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  6. R.J. Greechie, Another non standard quantum logic (and how I found it), in “Mathematical Foundation of Quantum Theory”, A.R. Marlow ed.. Academic Press (1978), pp. 71–85.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Kansas State University, Manhattan, Kansas, 66506, USA

    Richard J. Greechie

Authors
  1. Richard J. Greechie
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Editor information

Editors and Affiliations

  1. University of Genoa, Genoa, Italy

    Enrico G. Beltrametti

  2. Department of Philosophy, University of Toronto, Toronto, Ontario, Canada

    Bas C. van Fraassen

  3. University of Southern California, Los Angeles, California, USA

    Bas C. van Fraassen

  4. Istituto di Scienze Fisiche, Via Benedetto XV, 5, 16132, Genova, Italy

    Enrico G. Beltrametti

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© 1981 Plenum Press, New York

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Greechie, R.J. (1981). A Non-Standard Quantum Logic with a Strong Set of States. In: Beltrametti, E.G., van Fraassen, B.C. (eds) Current Issues in Quantum Logic. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3228-2_25

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  • DOI: https://doi.org/10.1007/978-1-4613-3228-2_25

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3230-5

  • Online ISBN: 978-1-4613-3228-2

  • eBook Packages: Springer Book Archive

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