A hint of renormalization

H.

Bethe

, “,” , – ().

S. Weinberg, The Quantum Theory of Fields (Cambridge U.P., Cambridge, 1995).

J. Collins, Renormalization (Cambridge U.P., Cambridge, 1984).

L. Ryder, Quantum Field Theory (Cambridge U.P., Cambridge, 1985).

M. Le Bellac, Quantum and Statistical Field Theory (Oxford U.P., Oxford, 1992).

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).

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison–Wesley, Reading, MA, 1992).

Conceptual Foundations of Quantum Field Theory, edited by T. Cao (Cambridge U.P., Cambridge, 1999).

K. G.

Wilson

and

J.

Kogut

, “,” , – ().

Some phase transitions are triggered by quantum fluctuations. This subtlety plays no role in what follows.

The short distance physics in statistical systems is given by Hamiltonians describing, for instance, interactions among magnetic ions or molecules of a fluid.

G. Lepage, “What is renormalization?,” TASI’89 Summer School, Boulder, ASI, 1989, p. 483.

P.

Kraus

and

D.

Griffiths

, “,” , – ().

I.

Mitra

,

A.

DasGupta

, and

B.

Dutta-Roy

, “,” , – ().

P.

Gosdzinsky

and

R.

Tarrach

, “,” , – ().

L.

Mead

and

J.

Godines

, “,” , – ().

K.

Adhikari

and

A.

Ghosh

, “,” , – ().

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.

Actually for QED it is a four-dimensional integral over four-momenta and the integrand is a product of propagators.

R.

Delbourgo

, “,” , – ().

S.

Nyeo

, “,” , – ().

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).

M.

Hans

, “,” , – ().

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.

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 $Λ→∞.$ This procedure is known as field renormalization.

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).

D.

Shirkov

and

V.

Kovalev

, “,” , – ().

The elements of the group are the functions: $gt=f(⋅,t)$ for $t∈R.$ 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 $gt′.gt=f(f(⋅,t),t′).$ It obeys trivially the identity: $gt′.gt=gt+t′$ which is nothing but Eq. (44). The identity is $gt=0$ and the inverse is $g−t.$ The law is associative as it should be for a group.

If we had not omitted in Eq. (52) the finite parts, we would have found $F(x=Λ)=g0+ag02+bg03+⋯ .$ Thus $g0$ is in general not associated exactly with the scale Λ, but with Λ up to a factor of order unity.

V.

Kovalev

and

D.

Shirkov

, “,” , – ().

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.

E.

Raposo

,

S.

de Oliveira

,

A.

Nemirovsky

, and

M.

Coutinho-Filho

, “,” , – ().

H.

Stanley

, “,” , – ().

J.

Tobochnik

, “,” , – ().

C.

Bagnuls

and

C.

Bervillier

, “,” , – ().

More precisely, working with nonrenormalizable couplings would require us to fine-tune infinitely many of them at scale $Λ−1$ to unnatural values. Most of the time, such a finely-tuned model is no longer predictive.

G.

Jona-Lasinio

, “,” cond-mat/0009219.

N.

Tetradis

and

C.

Wetterich

, “,” , – ().

Nonperturbatively, the existence of the limit $Λ→∞$ 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 $g0→0$ when $Λ→∞,$ corresponds to asymptotically free theories, that is, in four space–time dimensions, to non-Abelian gauge theories.

C.

Schmidhuber

, “,” , – ().