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A Renormalisation Group Method. I. Gaussian Integration and Normed Algebras

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Abstract

This paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. Our immediate motivation is a specific model, involving both boson and fermion fields, which arises as a representation of the continuous-time weakly self-avoiding walk. In this paper, we define normed algebras suitable for a renormalisation group analysis, and develop methods for performing analysis on these algebras. We also develop the theory of Gaussian integration on these normed algebras, and prove estimates for Gaussian integrals. The concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series.

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Acknowledgments

The work of both authors was supported in part by NSERC of Canada. DB gratefully acknowledges the support and hospitality of the Institute for Advanced Study at Princeton and of Eurandom during part of this work. GS gratefully acknowledges the support and hospitality of the Institut Henri Poincaré, and of the Kyoto University Global COE Program in Mathematics, during stays in Paris and Kyoto where part of this work was done. We thank Benoît Laslier for many helpful comments, and an anonymous referee for numerous pertinent suggestions.

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  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    David C. Brydges & Gordon Slade

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  1. David C. Brydges
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  2. Gordon Slade
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Correspondence to David C. Brydges.

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Brydges, D.C., Slade, G. A Renormalisation Group Method. I. Gaussian Integration and Normed Algebras. J Stat Phys 159, 421–460 (2015). https://doi.org/10.1007/s10955-014-1163-z

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  • DOI: https://doi.org/10.1007/s10955-014-1163-z

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