Appendix A: Extension to Euclidean Models

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


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.


Euclidean model Circle product Change of variable Regulators 


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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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