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Model and Set-Theoretic Aspects of Exotic Smoothness Structures on \(\mathbb {R}^4\)

  • Jerzy KrólEmail author
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 183)

Abstract

Model-theoretic aspects of exotic smoothness were studied long ago uncovering unexpected relations to noncommutative spaces and quantum theory. Some of these relations were worked out in detail in later work. An important point in the argumentation was the forcing construction of Cohen but without a direct application to exotic smoothness. In this article we assign the set-theoretic forcing on trees to Casson handles and characterize small exotic smooth \(R^4\) from this point of view. Moreover, we show how models in some Grothendieck toposes can help describing such differential structures in dimension 4. These results can be used to obtain the deformation of the algebra of usual complex functions to the noncommutative algebra of operators on a Hilbert space. We also discuss the results in the context of the Epstein-Glaser renormalization in QFT.

Keywords

Full Binary Tree Trivial Automorphism Wild Automorphism Atomless Boolean Algebra External Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The author appreciates much the important and fruitful discussions with Torsten Asselmeyer-Maluga and Krzysztof Bielas within the years about the wide range of topics appearing in the Chapter.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Physics, University of SilesiaKatowicePoland

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