Abstract
We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.
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REFERENCES
C.J. Isham and J. Butterfield, “A topos perspective on the Kochen-Specker theorem: I.Quantum states as generalised valuations,” Internat.J.Theoret.Phys. 37, 2669–2733 (1998), quant-ph/980355.
J. Butterfield and C.J. Isham, “A topos perspective on the Kochen-Specker theorem: II.Conceptual aspects, and classical analogues,” Internat.J.Theoret.Phys. 38, 827–859 (1999), quant-ph/9808067.
J. Hamilton, C.J. Isham, and J. Butterfield, “A topos perspective on the Kochen_Specker theorem: III.Von Neumann algebras as the base category,” forthcoming in Internat.J.Theoret.Phys. (2000), quant-ph/9911020.
S. Kochen and E. Specker, “The problem of hidden variables in quantum mechanics,” J.Math.Mech. 17, 59–87 (1967).
G. Birkhoff and J. von Neumann, “The logic of quantum mechanics,” Ann.Math. 37, 823–843 (1936).
M.L.Dalla Chiara and R.Giuntini, “Quantum logics,” in Handbook of Philosophical Logic, D.Gabbay and F.Guenthner, eds.(Kluwer Academic, Dordrecht, to appear).
M.L. Dalla Chiara and R. Giuntini, “Unsharp quantum logics,” Found.Phys. 24, 1161–1177 (1994).
G. Cattaneo and F. Laudisa, “Axiomatic unsharp quantum logis,” Found.Phys. 24, 631–684 (1994).
J. Butterfield and C.J. Isham, “Spacetime and the philosophical challenge of quantum gravity,” in Physics meets Philosophy at the Planck Scale, C. Callender and N. Huggett, eds.(Cambridge University Press, Cambridge, 2000), gr-qc 9903072.
M. Farrukh, “Application of nonstandard analysis to quantum mechanics,” J.Math.Phys. 16, 177–200 (1975).
R. Lavendhomme, Basic Concepts of Synthetic Differential Geometry (Kluwer, Dordrecht, 1996).
K.Savvidou, “The action operator for continuous-time histories,” forthcoming in J.Math.Phys., gr-qc 9811078.
F.Markopoulou, “The internal description of a causal set: What the universe looks like from the inside,” forthcoming, gr-qc 9811053.
C.J. Isham, “Quantum logic and the consistent histories approach to quantum theory,” J.Math.Phys. 23, 2157–2185 (1994).
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Isham, C.J., Butterfield, J. Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity. Foundations of Physics 30, 1707–1735 (2000). https://doi.org/10.1023/A:1026406502316
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DOI: https://doi.org/10.1023/A:1026406502316