Abstract
In this note, we present an elementary example of an ODE problem which clearly illustrates how various renormalization group approaches are used in theoretical physics. We attempt to clarify relations between the perturbative and constructive renormalizations through the example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N.N. Bogoliubov and D.V. Sirkov, Introduction to the Theory of Quantized Fields. Wiley, New York, 1980.
J. Bricmont, A. Kupiainen and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Comm. Pure Appl. Math.,XLVII (1994), 893–922.
J. Carr, Applications of Centre Manifold Theory. Applied Math. Sci.35, Springer-Verlag, New York, 1981.
L.Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. E,54, No.1 (1996), 376–393.
R. Courant and D. Hilbert, Methods of Mathematical Physics (Vol. 2), Interscience Publishers, New York, 1962.
S-I. Ei, K. Fujii and T. Kunihiro, Renormalization group method for reduction of evolution equations; invariant manifolds and envelops. Ann. Physics,280, No.2 (2000), 236–298.
M. Gell-Mann and F.E. Low, Quantum electrodynamics at small distances. Phys. Rev.,95 (1953), 1300–1312.
P. Hartman, Ordinary Differential Equations (2nd. Ed.). Birkhäuser, Boston Basel Stuttgart, 1982.
L.P. Kadanoff, The applications of renormalization group techniques to quarks and strings. Rev. Mod. Phys.,49 (1977), 267–296.
T. Kunihiro, A geometrical formulation of the renormalization group method for global analysis. Progr. Theoret. Phys.,94 (1995), 503–514.
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence. Synergetics19, Springer-Verlag, New York, 1984.
Y. Oono, Renormalization and asymptotics (Mathematical Physics Summer School at Gakushuin University, Tokyo, Japan, Sep. 23–25, 1999). Preprint, 1999.
S. Smale and W. Hirsh, Diffferential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York, 1974.
E. Stuckelberg and A. Petermann, La normalisation des constantes dans la theorie des quanta. Helv. Phys. Acta,26 (1953), 499–520.
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics. Applied Math. Sci.68, Springer-Verlag, New York, 1988.
K.G. Wilson, Renormalization group and critical phenomena, I Renormalization group and the Kadanoff scaling picture. Phys. Rev. B,4 (1971), 3174–3183; II Phase space cell analysis of critical behavior. Phys. Rev. B,4 (1971), 3184–3205.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Kuwamura, M. A perspective of renormalization group approaches. Japan J. Indust. Appl. Math. 18, 739–768 (2001). https://doi.org/10.1007/BF03167412
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03167412