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Nonperturbative Quantum Field Theory pp 309–351Cite as

Multigrid Methods in Quantum Field Theory

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Part of the book series: Nato Science Series B: ((NSSB,volume 185))

Abstract

The multigrid formulation of Euclidean quantum field theory (= classical statistical mechanics) on the continuum or on a lattice of small lattice spacing is helpful both for analytical and numerical investigations. Basically it amounts to simultaneous performance of a sequence of renormalization group transformations. It permits Monte Carlo simulations for critical and nearly critical systems without critical slowing down. Combined with analytic tools - the theory of polymer systems on the multigrid - it offers a chance to perform computer simulations for continuum systems without UV-cutoff, rather than their lattice approximations.

  • I. Generalities: Renormalization group and multigrid

  • II. Multigrid Monte Carlo simulations for λϕ4-theory. Possible generalization to gauge theories

  • III. Towards computer simulation of continuum theories without UV-cutoff via simulation of polymer systems on the multigrid

  • IV. Analytic tools: Polymer systems on the multigrid. Renormalization and UV-convergence

I made an effort to make section II as selfcontained as possible. The multigrid reformulation of Euclidean quantum field theory as used here was introduced by A. Pordt and the author some years ago [1]. The numerical multigrid investigations of ϕ4-theory were done in collaboration with S. Meyer [2]. The polymer approach was developed jointly with A. Pordt, K. Pinn, and H.-J. Timme [1,3,4,5]. Financial support from Deutsche Forschungsgemeinschaft is gratefully acknowledged.

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Author information

Authors and Affiliations

  1. II Institut für Theoretische Physik der Universität, Hamburg, Germany

    Gerhard Mack

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  1. Gerhard Mack
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Editor information

Editors and Affiliations

  1. University of Utrecht, Utrecht, The Netherlands

    G. ’t Hooft

  2. Harvard University, Cambridge, Massachusetts, USA

    A. Jaffe

  3. University of Hamburg, Hamburg, Federal Republic of Germany

    G. Mack

  4. University of Paris VI, Paris, France

    P. K. Mitter

  5. LAPP, Annecy-le-Vieux, France

    R. Stora

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© 1988 Plenum Press, New York

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Mack, G. (1988). Multigrid Methods in Quantum Field Theory. In: ’t Hooft, G., Jaffe, A., Mack, G., Mitter, P.K., Stora, R. (eds) Nonperturbative Quantum Field Theory. Nato Science Series B:, vol 185. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0729-7_11

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  • DOI: https://doi.org/10.1007/978-1-4613-0729-7_11

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