Abstract
We attempt to give apedagogical introduction to perturbative renormalization. Our approach is to first describe, following Linstedt and Poincaré, the renormalization of formal perturbation expansions for quasi-periodic orbits in Hamiltonian mechanics. We then discuss, following [FT1, FT2], the renormalization of the formal ground state energy density of a many Fermion system. The construction of formal quasi-periodic orbits is carried out in detail to provide a relatively simple model for the considerably more involved, and perhaps less familiar, perturbative analysis of a field theory.
As we shall see, quasi-periodic orbits and many Fermion systems have a number of important features in common. In particular, as Poincaré observed in the classical case and [FT1, FT2] pointed out in the latter, the formal expansions considered here both contain divergent subseries.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A1] V. I. Arnold,Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics60, Springer-Verlag, Berlin, 1980.
[A2] V. I. Arnold,Small denominators II, Proof of a theorem of A.N. Kolomogorov on the preservation of conditionally-periodic motions under a small perturbation of the Hamiltonian, Russian Math. Surveys18 (1963).
[E1] H. Eliaisson,Absolutely convergent series expansions for quasi periodic motions, Stockholm, preprint, 1988.
[E2] H. Eliaisson,Generalization of an estimate of small divisors by Siegel inAnalysis, Et Cetera, ed. Z. Rabinowitz Academic Press, New York, 1990.
[FT1] J. Feldman and E. Trubowitz,Perturbation theory for many fermion systems, Helvetica Physica Acta63 (1990).
[FT2] J. Feldman, and E. Trubowitz,The flow of an electron-phonon system to the superconducting state, Helvetica Physica Acta64 (1991).
[FW] A. L. Fetter and J. D. Walecka,Quantum Theory for of Many-Particle Systems, McGraw-Hill, New York, 1971.
[GJ] I. P. Goulden and D. M. Jackson,Combinatorial Enumeration, Wiley, New York, 1983.
[K] A. N. Kolmogorov,On the preservation of conditionally-periodic motions for small change in Hamilton's function, Dokl. Akad. Nauk SSSR98 (1954).
[P] H. Poincaré,Les Méthodes Nouvelles de la Méchanique Cleleste, Vol. 1, Dover, New York, 1957.
[Sc] W. M. Schmidt,Diophantine Approximation, Lecture Notes in Mathematics785, Springer-Verlag, Berlin, 1980.
[Si] C. L. Siegel,Iteration of analytic functions, Ann. of Math.43 (1942), 607–612.
Author information
Authors and Affiliations
Additional information
Dedicated to Professor Shmuel Agmon
Research supported in part by the Natural Sciences and Engineering Research Council of Canada.
Rights and permissions
About this article
Cite this article
Feldman, J., Trubowitz, E. Renormalization in classical mechanics and many body quantum field theory. J. Anal. Math. 58, 213–247 (1992). https://doi.org/10.1007/BF02790365
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02790365