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Appendix A: Extension to Euclidean Models

  • Roland Bauerschmidt
  • David C. Brydges
  • Gordon Slade
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2242)

Abstract

We give an introduction to some of the modifications needed to extend the renormalisation group method from the hierarchical to the Euclidean setting, and point out where in the literature these extensions can be found in full detail. Although the renormalisation group philosophy remains the same for the hierarchical and Euclidean models, the Euclidean setting requires significant adjustments. These include: an extended version of the perturbative coordinate, introduction of the circle product, analysis of field gradients within blocks, and more sophisticated norms involving regulators. We discuss all these items. The transfer of relevant contributions from the nonperturbative coordinate to the perturbative coordinate, in order to ensure the crucial contraction of the renormalisation group map, is now carried out via a sophisticated change of variables procedure. We also discuss this change of variables.

Keywords

Euclidean model Circle product Change of variable Regulators 

References

  1. 1.
    A. Abdesselam, A complete renormalization group trajectory between two fixed points. Commun. Math. Phys. 276, 727–772 (2007)MathSciNetCrossRefGoogle Scholar
  2. 19.
    R. Bauerschmidt, D.C. Brydges, G. Slade, Scaling limits and critical behaviour of the 4-dimensional n-component |φ|4 spin model. J. Stat. Phys. 157, 692–742 (2014)MathSciNetCrossRefGoogle Scholar
  3. 22.
    R. Bauerschmidt, D.C. Brydges, G. Slade, A renormalisation group method. III. Perturbative analysis. J. Stat. Phys. 159, 492–529 (2015)CrossRefGoogle Scholar
  4. 23.
    R. Bauerschmidt, D.C. Brydges, G. Slade, Structural stability of a dynamical system near a non-hyperbolic fixed point. Ann. Henri Poincaré 16, 1033–1065 (2015)MathSciNetCrossRefGoogle Scholar
  5. 42.
    D.C. Brydges, G. Slade, A renormalisation group method. I. Gaussian integration and normed algebras. J. Stat. Phys. 159, 421–460 (2015)zbMATHGoogle Scholar
  6. 43.
    D.C. Brydges, G. Slade, A renormalisation group method. II. Approximation by local polynomials. J. Stat. Phys. 159, 461–491 (2015)zbMATHGoogle Scholar
  7. 44.
    D.C. Brydges, G. Slade, A renormalisation group method. IV. Stability analysis. J. Stat. Phys. 159, 530–588 (2015)zbMATHGoogle Scholar
  8. 45.
    D.C. Brydges, G. Slade, A renormalisation group method. V. A single renormalisation group step. J. Stat. Phys. 159, 589–667 (2015)zbMATHGoogle Scholar
  9. 47.
    D.C. Brydges, H.-T. Yau, Grad ϕ perturbations of massless Gaussian fields. Commun. Math. Phys. 129, 351–392 (1990)MathSciNetCrossRefGoogle Scholar
  10. 53.
    D.C. Brydges, P.K. Mitter, B. Scoppola, Critical (Φ 4)3,𝜖. Commun. Math. Phys. 240, 281–327 (2003)CrossRefGoogle Scholar
  11. 69.
    J. Dimock, T.R. Hurd, A renormalization group analysis of correlation functions for the dipole gas. J. Stat. Phys. 66, 1277–1318 (1992)MathSciNetCrossRefGoogle Scholar
  12. 96.
    K. Gawȩdzki, A. Kupiainen, Block spin renormalization group for dipole gas and (∇φ)4. Ann. Phys. 147, 198–243 (1983)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Roland Bauerschmidt
    • 1
  • David C. Brydges
    • 2
  • Gordon Slade
    • 2
  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  2. 2.Department of MathematicsThe University of British ColumbiaVancouverCanada

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