Abstract
We consider the logic needed for models of quantum gravity, taking as our starting point a simple pregeometric toy model based on graph theory. First a discussion of quantum logic seen in the light of canonical quantum gravity is given, then a simple toy model is proposed and the logical structure underlying it exposed. It is then shown that this logic is nonclassical and in fact contains quantum logics as special cases. We then go on to show how Yang-Mills theory and quantum mechanics fits in. A single mathematical structure is proposed capable of containing all these subjects in a natural and elegant way. Causality plays an important role. The mere presence of a causal relation almost inevitably yields this kind of logic.
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References
Antonsen, F. (1992). Pregeometry, Thesis, Niels Bohr Institute, University of Copenhagen, unpublished.
Antonsen, F. (1994).International Journal of Theoretical Physics,33, 1189–1205.
Antonsen, F. (n.d.). Models of pregeometry, InProceedings of the 2nd A. A. Friedmann International Seminar on Gravitation and Cosmology, St. Petersburg, 1992, to appear.
Bell, J. L. (1988).Topases and Local Set Theories: An Introduction, Clarendon Press, Oxford.
Binder, J. and Pták, P. (1990).Acta Universitatis Carolinae Mathematica et Physica,31, 21–26.
Bombelli, L., and Meyer, D. A. (1989).Physics Letters A,141, 226–228.
Bombelli, L., Lee, L., Meyer, D., and Sorkin, R. D. (1987).Physical Review Letters,59, 521–524.
Borchers, H.-J., and Sen, R. N. (1990).Communications in Mathematical Physics,132, 593–611.
Brightwell, G., and Gregory, R. (1991).Physical Review Letters,66, 260–263.
Chapman, J., and Rowbotton, F. (1992).Relative Category Theory and Geometric Mor-Phisms—A Logical Approach, Clarendon Press, Oxford.
Choquet-Bruhat, Y., DeWitt-Morette, C., and Dillard-Bleick, M. (1982).Analysis, Manifolds and Physics (2nd edition), North-Holland, Amsterdam.
Doebner, H. D., and Lücke, W. (1991).Journal of Mathematical Physics,32, 250–253.
Dummet, M. (1977).Elements of Intuitionism, Clarendon Press, Oxford.
Finkelstein, D. (1987).International Journal of Theoretical Physics,27, 473–519.
Finkelstein, D. (1989).International Journal of Theoretical Physics,28, 441–467.
Finkelstein, D., and Finkelstein, S. R. (1983).International Journal of Theoretical Physics,22, 753–779.
Finkelstein, D., and Hallidy, W. H. (1991).International Journal of Theoretical Physics,30, 463–486.
Froggat, C., and Nielsen, H. B. (1991).Origin of Symmetries, World Scientific, Singapore.
Garden, R. W. (1984).Modern Logic and Quantum Mechanics, Adam Hilger, Bristol.
Göckeler, M., and Schücker, T. (1989).Differential Geometry, Gauge Theory, and Gravity, Cambridge University Press, Cambridge.
Goldblatt, R. (1984).Topoi. The Categorial Analysis of Logic (revised edition), North-Holland, Amsterdam.
Gudder, S. (1979).Stochastic Methods in Quantum Mechanics, North-Holland, Amsterdam.
Haag, R., and Kastler, D. (1964).Journal of Mathematical Physics,5, 848–861.
Hawking, S., and Ellis, J. (1973).The Large-Scale Structure of Space-Time, Cambridge University Press, Cambridge.
Kelley, J. L. (1975).General Topology, Springer-Verlag, Berlin.
Kleene, S. (1980).Introduction to Metamathematics, North-Holland, Amsterdam.
Müller, A. (1992).Communications in Mathematical Physics,149, 495–512.
Nielsen, H. B., and Brene, N. (n.d.). Gauge glass, NBI-HE-84-47.
Pitowsky, I. (1989).Quantum Probability —Quantum Logic, Springer-Verlag, Berlin.
Preuss, G. (1988).Theory of Topological Structure. An Approach to Categorial Topology, Reidel, Dordrecht.
Pulmannová, S., and Majernik, V. (1992).Journal of Mathematical Physics,33(6), 2173–2178.
Rosenthal, K. I., and Niefield, S. B. (1989). Connections between ideal theory and the theory of locales,Annals of the New York Academy of Sciences,552, 138–151, and references therein.
Rudin, W. (1973).Functional Analysis, McGraw-Hill, New York.
Šostak, A. P. (1990).Radovi Matematički,6, 249–263.
t'Hooft, G. (1990).Nuclear Physics B,342, 471–485.
Troesltra, A. S., and van Dalen, D. (1988).Constructivism in Mathematics: An Introduction I–II, North-Holland, Amsterdam.
Varadarajan, V. S. (1985).Geometry of Quantum Theory, Springer-Verlag, New York.
Vickers, S. (1989).Topology via Logic, Cambridge University Press, Cambridge.
Warner, F. K. (1983).Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York.
Woodhouse, N. M. J. (1973).Journal of Mathematical Physics,14, 495–501.
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Antonsen, F. Logics and quantum gravity. Int J Theor Phys 33, 1985–2017 (1994). https://doi.org/10.1007/BF00675166
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DOI: https://doi.org/10.1007/BF00675166