An elementary introduction to perturbative renormalization and renormalization group is presented. No prior knowledge of field theory is necessary because we do not refer to a particular physical theory. We are thus able to disentangle what is specific to field theory and what is intrinsic to renormalization. We link the general arguments and results to real phenomena encountered in particle physics and statistical mechanics.

## REFERENCES

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 ε-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.

Actually, the analog of the function in Eq. (1) would be a correlation function of four density or spin fields taken at four different points. These functions are not easily measurable and thus $g0$ 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.

It is nontrivial to prove in full generality that the results obtained after renormalization are independent of the regularization scheme. However, it is easy to grasp the idea behind it. Because renormalization consists in eliminating parameters like $g0$ and in replacing them by measurable couplings like $gR,$ 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.

Note that the (renormalized) series in $gR$ 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.

In 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 $\Lambda \u2192\u221e.$ 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 $\Lambda =\lambda \u200aexp(1/\epsilon ),$ where $\epsilon =4\u2212D$ and $D$ is the spatial dimension. It often is convenient to take $\lambda \u223c\mu .$ 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.

The elements of the group are the functions: $gt=f(\u22c5,t)$ for $t\u2208R.$ They transform an initial condition $r0$ into the solution at a later time interval $t$ of the differential equation we consider [see Eq. (43) in our example]: $gt(r0)=f(r0,t).$ The composition law is thus $gt\u2032.gt=f(f(\u22c5,t),t\u2032).$ It obeys trivially the identity: $gt\u2032.gt=gt+t\u2032$ which is nothing but Eq. (44). The identity is $gt=0$ and the inverse is $g\u2212t.$ The law is associative as it should be for a group.

29.

If we had not omitted in Eq. (52) the finite parts, we would have found $F(x=\Lambda )=g0+ag02+bg03+\cdots \u200a.$ Thus $g0$ 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.

More precisely, working with nonrenormalizable couplings would require us to fine-tune infinitely many of them at scale $\Lambda \u22121$ to unnatural values. Most of the time, such a finely-tuned model is no longer predictive.

37.

38.

N.

Tetradis

and C.

Wetterich

, “Critical exponents from the effective average action

,” Nucl. Phys. B [FS]

422

, 541

–592

(1994

).39.

Nonperturbatively, the existence of the limit $\Lambda \u2192\u221e$ is more subtle than perturbatively. The renormalization group flow must be controlled in this limit and this is achieved if nonperturbatively $g0$ has a finite limit, that is, if there exists an ultraviolet fixed point of the RG flow. The case $g0\u21920$ when $\Lambda \u2192\u221e,$ 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

).
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