Abstract
In this work we expand the foundational perspective of category theory on quantum event structures by showing the existence of an object of truth values in the category of quantum event algebras, characterized as subobject classifier. This object plays the corresponking role that the two-valued Boolean truth values object plays in a classical event structure. We construct the object of quantum truth values explicitly and argue that it constitutes the appropriate choice for the valuation of propositions describing the behavior of quantum systems.
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REFERENCES
V. S. Varadarajan, Geome'ry of Quantum Mechanics, Vol. 1 (Van Nostrand, Princeton, New Jersey, 1968).
G. Birkhoff and J. von Neumann, “The logic of quantum mechanics.” Ann. Math. 37, 823 (1936).
E. Zafiris, “On quantum event structures. Part I: The categorical scheme,” Found. Phys. Lett. 14(2), 147 (2001).
E. Zafiris, “On quantum event structures. Part II: Interpretational aspects,” Found. Phys. Lett. 14(2), 167 (2001).
E. Zafiris, “Quantum event structures from the perspective of Grothendieck topoi,” to appear in Found. Phys.
6._ S. Kochen and E. Specker, “The problem of hidden variables in quantum mechanics,” J. Math. and Mech. 17, 59 (1967).
F. W. Lawvere and S. H. Schanuel, Conceptual Mathematics (University Press, Cambridge, 1997).
S. MacLane, Categories for the Working Mathematician (Springer, New York, 1971).
S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic (Springer, New York, 1992).
J. L. Bell, Toposes and Local Set Theories (University Press, Oxford, 1988).
M. Artin, A. Grothendieck, and J. L. Verdier, Theorie de topos et cohomologie etale des schemas (Springer LNM 269 and 270) (Springer, Berlin, 1972).
J. L. Bell, “From absolute to local mathematics,” Synthese 69 (1986).
J. L. Bell, “Categories, toposes and sets,” Synthese 51(3) (1982).
J. Butterfield and C. J. Isham, “A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations,” Intern. J. Theoret. Phys. 37, 2669 (1998).
J. Butterfield and C. J. Isham, “A topos perspective on the Kochen-Specker theorem: II. Conceptual aspects and classical analogues,” Intern. J. Theoret. Phys. 38, 827 (1999).
J. P. Rawling and S. A. Selesnick:, “Orthologic and quantum logic, models and computational elements,” J. Assoc. Computing Machinery 47, 721 (2000).
I. Raptis, “Presheaves, sheaves, and their topoi in quantum gravity and quantum logic,” gr-qc/0l10064.
J. Butterfield and C. J. Isham, “Some possible roles for topos theory in quantum theory and quantum gravity,” Found. Phys. 30, 1707 (2000).
G. Takeuti, “Two applications of logic to mathematics,” Mathematical Society of Japan 13, Kano Memorial Lectures 3 (1978).
M. Davis, “A relativity principle in quantum mechanics,” Internat. J. Theoret. Phys. 16, 867 (1977).
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Zafiris, E. On Quantum Event Structures. Part III: Object of Truth Values. Found Phys Lett 17, 403–432 (2004). https://doi.org/10.1023/B:FOPL.0000042696.35112.78
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DOI: https://doi.org/10.1023/B:FOPL.0000042696.35112.78