International Journal of Theoretical Physics

, Volume 57, Issue 7, pp 2093–2102 | Cite as

Remarks on a New Possible Discretization Scheme for Gauge Theories

  • Jean-Pierre Magnot


We propose here a new discretization method for a class of continuum gauge theories which action functionals are polynomials of the curvature. Based on the notion of holonomy, this discretization procedure appears gauge-invariant for discretized analogs of Yang-Mills theories, and hence gauge-fixing is fully rigorous for these discretized action functionals. Heuristic parts are forwarded to the quantization procedure via Feynman integrals and the meaning of the heuristic infinite dimensional Lebesgue integral is questioned.


Gauge invariance Yang-Mills theory Discretized model 


  1. 1.
    Albeverio, S., Hoegh-Krohn, R., Mazzuchi, S.: Mathematical theory of Feynman Path Integrals; an introduction 2nd edition; Lecture Notes in Mathematics 523, Springer (2005)Google Scholar
  2. 2.
    Albeverio, S., Zegarlinski, B.: Construction of convergent simplicial approximations of quantum fileds on Riemannian manifolds. Comm. Math. Phys. 132, 39–71 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hahn, A: The Wilson loop observables of Chern-Simons theory on \(\mathbb {R}^{3}\) in axial gauge. Comm. Math. Phys. 248(3), 467–499 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Huber, M., Campagnari, D., Reinhardt, H.: Vertex functions of Coulomb gauge Yang–Mills theory. Phys. Rev. D 91, 025014 (2015)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Lim: Non-abelian gauge theory for Chern-Simons path integral on \(\mathbb {R}^{3}\), Journal of Knot Theory and its Ramifications 21 no. 4. articleID1250039 (24p) (2012)Google Scholar
  6. 6.
    Magnot, J.-P.: The mean value for infinite volume measures, infinite products and heuristic infinite dimensional Lebesgue measures. J. Math. 2017, 14 (2017). Article ID 9853672MathSciNetCrossRefGoogle Scholar
  7. 7.
    Reinhardt, H.: Yang-mills in axial gauge. Phys. Rev. D 55, 2331–2346 (1997)ADSCrossRefGoogle Scholar
  8. 8.
    Rovelli, C., Vittodo, F.: Covariant loop quantum gravity. Cambridge university press, Cambridge (2014)CrossRefGoogle Scholar
  9. 9.
    Sen, S., Sen, S., Sexton, J.C., Adams, D.H.: A geometric discretisation scheme applied to the Abelian Chern-Simons theory. Phys. Rev. E 61, 3174–3185 (2000)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Sengupta, A.N. In: Arous, G.B., Cruzeiro, A.B., Jan, Y.L., Zambrini, J.-C. (eds.) : Gauge Theory in Two Dimensions: Topological, Geometric and Probabilistic Aspects. Pages in 109–129 inStochastic Analysis in Mathematical Physics. World Scientific, Singapore (2008)Google Scholar
  11. 11.
    Sengupta, A.N.: Yang-mills in Two Dimensions and Chern-Simons in Three, in Chern- Simons Theory: 20 years after. In: Anderson, J.E., Boden, H.U., Hahn, A., Himpel, B. (eds.) AMS/IP Studies in Advanced Mathematics, pp. 311–320 (2011)Google Scholar
  12. 12.
    Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LAREMAUniversité d’AngersClermont-FerrandFrance

Personalised recommendations