Abstract
This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice \({\mathfrak{D}}(e,R)\) , which is a sublattice of the lattice \({\mathfrak{L}}\) of all quantum propositions and is determined by a quantum state e and a preferred determinate observable R. Topos-theoretic extension is made in the functor category \(\mathbf{Sets}^{\mathcal{C}_{R}}\) of which base category \({\mathcal{C}_{R}}\) is determined by R. Each true atom, which determines truth values, true or false, of all propositions in \({\mathfrak{D}}(e,R)\) , generates also a multi-valued valuation function of which domain and range are \({\mathfrak{L}}\) and a Heyting algebra given by the subobject classifier in \(\mathbf{Sets}^{\mathcal{C}_{R}}\) , respectively. All true propositions in \({\mathfrak{D}}(e,R)\) are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category \(\mathbf{Sets}^{\mathcal{C}}\) . Here, the base category \(\mathcal{C}\) includes all \({\mathcal{C}_{R}}\) ’s as subcategories. Although \(\mathbf{Sets}^{\mathcal{C}}\) has a structure apparently different from \(\mathbf{Sets}^{\mathcal{C}_{R}}\) , a subobject semi-classifier of \(\mathbf{Sets}^{\mathcal{C}}\) gives valuations completely equivalent to those in \(\mathbf{Sets}^{\mathcal{C}_{R}}\) ’s.
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References
Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)
van Fraassen, B.: Quantum Mechanics: A Empiricist View. Clarendon Press, Oxford (1991)
Bub, J.: Interpreting the Quantum World. Cambridge University Press, Cambridge (1997)
Vermaas, P.: A Philosopher’s Understanding of Quantum Mechanics. Cambridge University Press, Cambridge (1999)
Isham, C.J.: Topos theory and coherent histories. Int. J. Theor. Phys. 36, 785–814 (1997)
Isham, C.J., Butterfield, J.: A topos perspective on the Kochen-Specker theorem: I. Internal valuations. Int. J. Theor. Phys. 37, 2669–2733 (1998)
Butterfield, J., Isham, C.J.: A topos perspective on the Kochen-Specker theorem: II. Conceptual aspects, and classic analogues. Int. J. Theor. Phys. 38, 837–859 (1999)
Hamilton, J., Butterfield, J., Isham, C.J.: A topos perspective on the Kochen-Specker theorem: III. Von Neumann algebras as the base category. Int. J. Theor. Phys. 39, 1413–1436 (2000)
Butterfield, J., Isham, C.J.: A topos perspective on the Kochen-Specker theorem: IV. Internal valuations. Int. J. Theor. Phys. 41, 613–639 (2002)
Isham, C.J.: A topos perspective on state-vector reduction. Int. J. Theor. Phys. 45, 1524–1551 (2006)
Goldblatt, R.: Topoi: The Categorical Analysis of Logic. North-Holland, Amsterdam (1979)
MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, New York (1992)
Döring, A., Isham, C.J.: A topos foundation for theoretical physics: I. Formal language for physics. quant-ph/0703060v1 (2007)
Döring, A., Isham, C.J.: A topos foundation for theoretical physics: II. Daseinization and the liberation of quantum theory. quant-ph/0703062v1 (2007)
Döring, A., Isham, C.J.: A topos foundation for theoretical physics: III. The representation of physical quantities with arrows \(\breve{\delta}^{o}(A):\underline{\Sigma}\to \underline{\mathbb{R}^{\succeq}}\) . quant-ph/0703060v1 (2007)
Döring, A., Isham, C.J.: A topos foundation for theoretical physics: IV. Categories of systems. quant-ph/0703066v1 (2007)
Lambek, J., Scott, P.J.: Introduction to Higher Order Categorical Logic. Cambridge University Press, Cambridge (1986)
Bell, J.L.: Toposes and Local Set Theories. Clarendon Press, Oxford (1988)
Bub, J.: Quantum Mechanics without the Projection Postulate. Found. Phys. 22, 737–754 (1992)
Davey, B.A., Priestly, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)
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Nakayama, K. Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics. Int J Theor Phys 47, 2065–2094 (2008). https://doi.org/10.1007/s10773-008-9649-6
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DOI: https://doi.org/10.1007/s10773-008-9649-6