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Advances in Mathematical Physics
Volume 2018 (2018), Article ID 7496762, 5 pages
https://doi.org/10.1155/2018/7496762
Review Article

A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons

1LAREMA, UMR CNRS 6093, Université d’Angers, 2 bd Lavoisier, 49045 Angers Cedex 1, France
2Lycée Jeanne d’Arc, 40 avenue de Grande Bretagne, 63000 Clermont-Ferrand, France

Correspondence should be addressed to Jean-Pierre Magnot; moc.liamg@tongam.pj

Received 2 October 2017; Revised 16 December 2017; Accepted 26 December 2017; Published 23 January 2018

Academic Editor: Andrei D. Mironov

Copyright © 2018 Jean-Pierre Magnot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. W. Koczkodaj and J-P. Magnot, https://pdfs.semanticscholar.org/4c22/f469004fd0e03172e661c1611945eb3348a6.pdf.
  2. J.-P. Magnot, “On pairwise comparisons with values in a group: algebraic structures,” 2016, https://arxiv.org/abs/1608.07468. View at Google Scholar
  3. J.-P. Magnot, Remarks on a new possible discretization scheme for gauge theories, https://arxiv.org/abs/1706.09518.
  4. W. W. Koczkodaj, J. Szybowski, and E. Wajch, “Inconsistency indicator maps on groups for pairwise comparisons,” International Journal of Approximate Reasoning, vol. 69, pp. 81–90, 2016. View at Publisher · View at Google Scholar · View at Scopus
  5. Z. Duszak and W. W. Koczkodaj, “Generalization of a new definition of consistency for pairwise comparisons,” Information Processing Letters, vol. 52, no. 5, pp. 273–276, 1994. View at Publisher · View at Google Scholar · View at Scopus
  6. W. W. Koczkodaj, “A new definition of consistency of pairwise comparisons,” Mathematical and Computer Modelling, vol. 18, no. 7, pp. 79–84, 1993. View at Publisher · View at Google Scholar · View at Scopus
  7. W. W. Koczkodaj and R. Szwarc, “On axiomatization of inconsistency indicators for pairwise comparisons,” Fundamenta Informaticae, vol. 132, no. 4, pp. 485–500, 2014. View at Publisher · View at Google Scholar · View at Scopus
  8. S. Albeverio and R. Høegh-Krohn, “Brownian Motion, Markov Cosurfaces, Higgs Fields,” in Fundamental Aspects of Quantum Theory, vol. 144 of NATO ASI Series, pp. 95–104, Springer US, Boston, MA, 1986. View at Publisher · View at Google Scholar
  9. S. Albeverio, R. Hoegh-Krohn, and H. Holden, “Markov cosurfaces and gauge fields,” Acta Physica Austriaca, no. Supplement 26, pp. 211–231, 1984. View at Google Scholar
  10. S. A. Albeverio, R. J. Høegh-Krohn, and S. Mazzuchi, Mathematical Theory of Feynman Path Integrals; An Introduction, vol. 523 of Lecture Notes in Mathematics 523, Springer, 2005.
  11. S. Albeverio and B. Zegarlinski, “Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds,” Communications in Mathematical Physics, vol. 132, no. 1, pp. 39–71, 1990. View at Publisher · View at Google Scholar · View at Scopus
  12. A. Hahn, “The Wilson loop observables of Chern-Simons theory on ℝ3 in axial gauge,” Communications in Mathematical Physics, vol. 248, no. 3, pp. 467–499, 2004. View at Publisher · View at Google Scholar · View at Scopus
  13. M. Q. Huber, D. R. Campagnari, and H. Reinhardt, “Vertex functions of Coulomb gauge Yang-Mills theory,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 91, article 025014, no. 2, 2015. View at Publisher · View at Google Scholar · View at Scopus
  14. A. P. Lim, “Non-abelian gauge theory for Chern-Simons path i ntegral on R3,” Journal of Knot Theory and Its Ramifications, vol. 21, article 1250039, no. 04, 2012. View at Google Scholar
  15. H. Reinhardt, “Yang-Mills in axial gauge,” Physical Review, vol. 55, pp. 2331–2346, 1997. View at Google Scholar
  16. C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity, Cambridge University Press, Cambridge, UK, 2014. View at Publisher · View at Google Scholar
  17. S. Sen, S. Sen, J. C. Sexton, and D. H. Adams, “Geometric discretization scheme applied to the Abelian Chern-Simons theory,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 61, no. 3, pp. 3174–3185, 2000. View at Publisher · View at Google Scholar
  18. A. N. Sengupta, “Yang-Mills in two dimensions and Chern-Simons in three,” in Chern-Simons Theory: 20 years after, J. Anderson, H. U. Boden, A. Hahn, and B. Himpel, Eds., pp. 311–320, AMS/IP Studies in Advanced Mathematics, 2011. View at Google Scholar
  19. A. N. Sengupta, “Connections over two-dimensional cell complexes,” Reviews in Mathematical Physics, vol. 16, no. 3, pp. 331–352, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, NJ, USA, 1957. View at MathSciNet