A hint of renormalization

American Journal of Physics

Volumn 72, Issue 2, 2004, Pages 170-184

  Delamotte, Bertrand ^a  

^a LPTHE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1042279725     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1624112     Document Type: Article

References (40)

1. H. Bethe, "The electromagnetic shift of energy levels," Phys. Rev. 72, 339-341 (1947).
2. S. Weinberg, The Quantum Theory of Fields (Cambridge U.P., Cambridge, 1995).
3. J. Collins, Renormalization (Cambridge U.P., Cambridge, 1984).
4. L. Ryder, Quantum Field Theory (Cambridge U.P., Cambridge, 1985).
5. M. Le Bellac, Quantum and Statistical Field Theory (Oxford U.P., Oxford, 1992).
6. J. Binney, N. Dowrick, A. Fisher, and M. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford U.P., Oxford, 1992).
7. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, Reading, MA, 1992).
8. Conceptual Foundations of Quantum Field Theory, edited by T. Cao (Cambridge U.P., Cambridge, 1999).
9. K. G. Wilson and J. Kogut, "The renormalization group and the e-expansion," Phys. Rep. C 12, 75-199 (1974).
10. Some phase transitions are triggered by quantum fluctuations. This subtlety plays no role in what follows.
11. The short distance physics in statistical systems is given by Hamiltonians describing, for instance, interactions among magnetic ions or molecules of a fluid.
12. G. Lepage, "What is renormalization?," TASI'89 Summer School, Boulder, ASI, 1989, p. 483.
13. P. Kraus and D. Griffiths, "Renormalization of a model quantum field theory," Am. J. Phys. 60, 1013-1023 (1992).
14. I. Mitra, A. DasGupta, and B. Dutta-Roy, "Regularization and renormalization in scattering from Dirac delta-potentials," Am. J. Phys. 66, 1101-1109 (1998).
15. P. Gosdzinsky and R. Tarrach, "Learning quantum field theory from elementary quantum mechanics," Am. J. Phys. 59, 70-74 (1991).
16. L. Mead and J. Godines, "An analytical example of renormalization in two-dimensional quantum mechanics," Am. J. Phys. 59, 935-937 (1991).
17. K. Adhikari and A. Ghosh, "Renormalization in non-relativistic quantum mechanics," J. Phys. A 30, 6553-6564 (1997).
18. 0 does not have in general an intuitive meaning in this case. Because this subtlety plays no role in our discussion, we ignore this difficulty in the following.
19. Actually for QED it is a four-dimensional integral over four-momenta and the integrand is a product of propagators.
20. R. Delbourgo, "How to deal with infinite integrals in quantum field theory," Rep. Prog. Phys. 39, 345-399 (1976).
21. S. Nyeo, "Regularization methods for delta-function potential in two-dimensional quantum mechanics," Am. J. Phys. 68, 571-575 (2000).
22. R, the renormalized quantities like F(x) are finally expressed only in terms of physical quantities that are independent of the regularization scheme (Ref. 15).
23. M. Hans, "An electrostatic example to illustrate dimensional regularization and renormalization group technique," Am. J. Phys. 51, 694-698 (1983).
24. R can themselves be nonconvergent. Most of the time they are at best asymptotic. In some cases they can be resummed using Borel transform and Padé approximants.
25. QFT, it is in general also necessary to change the normalization of the analog of the function F - the Green functions - by a factor that diverges in the limit Λ → ∞. This procedure is known as field renormalization.
26. Let us emphasize that there is a subtlety if dimensional regularization is used. Actually, this regularization also introduces a dimensional parameter λ, which is not directly a regulator as is the cut-off Λ in the integral of Eq. (5). The analog of Λ in this regularization is given by Λ = λ exp(1/ε), where ε=4 - D and D is the spatial dimension. It often is convenient to take λ ∼ μ. We mention that dimensional regularization kills all nonlogarithmic divergences (Ref. 14).
27. D. Shirkov and V. Kovalev, "Bogoliubov renormalization group and symmetry of solutions in mathematical physics," Phys. Rep. 352, 219-249 (2001).
28. -t. The law is associative as it should be for a group.
29. 0 is in general not associated exactly with the scale Λ, but with Λ up to a factor of order unity.
30. V. Kovalev and D. Shirkov, "Functional self-similarity and renormalization group symmetry in mathematical physics," Theor. Math. Phys. 121, 1315-1332 (1999).
31. It is quite similar to the Compton wavelength of the pion which is the typical range of the nuclear force between hadrons like protons and nucleons.
32. E. Raposo, S. de Oliveira, A. Nemirovsky, and M. Coutinho-Filho, "Random walks: A pedestrian approach to polymers, critical phenomena and field theory," Am. J. Phys. 59, 633-645 (1991).
33. H. Stanley, "Scaling, universality, and renormalization: Three pillars of modern critical phenomena," Rev. Mod. Phys. 71, S358-S366 (1999).
34. J. Tobochnik, "Resource Letter CPPPT-1: Critical point phenomena and phase transitions," Am. J. Phys. 69, 255-263 (2001).
35. C. Bagnuls and C. Bervillier, "Exact renormalization group equations: An introductory review," Phys. Rep. 348, 91-157 (2001).
36. -1 to unnatural values. Most of the time, such a finely-tuned model is no longer predictive.
37. G. Jona-Lasinio, "Renormalization group and probability theory," cond-mat/0009219.
38. N. Tetradis and C. Wetterich, "Critical exponents from the effective average action," Nucl. Phys. B [FS] 422, 541-592 (1994).
39. 0 → 0 when Λ → ∞, corresponds to asymptotically free theories, that is, in four space-time dimensions, to non-Abelian gauge theories.
40. C. Schmidhuber, "On water, steam, and string theory," Am. J. Phys. 65, 1042-1050 (1997).