Renormalization without regularization

A short review of recently developed renormalization schemes without regularization is presented. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp 330–336, November, 1998.     

Renormalization over lines

A class of renormalization schemes that are not based on the subtraction of counter-terms is described. In these schemes, the renormalized Feynman amplitude is constructed recurrently from the amplitudes for diagrams without one internal line. These renormalizations are shown to be equivalent to the Bogoliubov-Parasiuk R-operation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 399–415, March, 1997.     

Renormalization group methods

An especially intractable breed of problems in physics involves those with very many or an infinite number of degrees of freedom and in addition involve “renormalization.” Renormalization is explained as the existence of very many length or energy scales of importance in the physics of the problem. The renormalization group approach is a way of reducing the complexity of these problems to the point where numerical methods can be used to solve them. The Kondo problem (dilute magnetic alloys) is used as an illustration.     

Renormalization and central extensions

The stochastic limit of quantum theory [1] motivated a new approach to the renormalization program. Subsequent investigations brought to light unexpected connections with conformal field theory and some subtle relationships between renormalization and central extensions. In the present paper we review the path that has lead to these connections at the light of some recent results.     

Continuous Constructive Fermionic Renormalization

. We build the two dimensional Gross-Neveu model by a new method which requires neither cluster expansion nor discretization of phase-space. It simply reorganizes the perturbative series in terms of trees. With this method we can define non perturbatively the renormalization group differential equations of the model and at the same time construct explicitly their solution.     

Holographic Renormalization Group Flows

Theoretical and Mathematical Physics - We briefly review recent applications of holographic renormalization group flow equations in a hot and dense quark-gluon plasma (QGP). We especially focus on...     

Renormalization of multiple zeta values

Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually undefined. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara–Kaneko–Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.     

Renormalization Operators and Kneading Sequences

ＲｅｎｏｒｍａｌｉｚａｔｉｏｎＯｐｅｒａｔｏｒｓａｎｄＫｎｅａｄｉｎｇＳｅｑｕｅｎｃｅｓ￥ＬｉａｏＧｏｎｇｆｕ（廖公夫）ａｎｄＺｈａｎｇＡｉｈｕａ（张爱华）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ．ＪｉｉｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈａｎｇｃｈ...     

Renormalization Group and Stochastic PDEs

We develop a renormalization group (RG) approach to the study of existence and uniqueness of solutions to stochastic partial differential equations driven by space-time white noise. As an example, we prove well-posedness and independence of regularization for the \({\phi^4}\) model in three dimensions recently studied by Hairer and Catellier and Chouk. Our method is “Wilsonian”: the RG allows to construct effective equations on successive space-time scales. Renormalization is needed to control the parameters in these equations. In particular, no theory of multiplication of distributions enters our approach.     

Universal Invariant Renormalization for Supersymmetric Theories

We propose a universal gauge-invariant renormalization scheme for supersymmetric theories. We apply this scheme to the supersymmetric quantum electrodynamics.     

The Attractor of Fibonacci-like Renormalization Operator

In this paper we extend the Fibonacci-like maps to a wider class with the so-called "bounded combinatorics". The Fibonacci-like renormalization operator R is defined and we show that the orbit of each map from this class converges to a universal limit under iterates of R.     

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)⁺), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)⁺), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S¹(B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)⁺) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B)⁺) is shown to give the same results as the standard renormalization procedure for the scalar field.     

Renormalization Group and the Ricci Flow

We discuss from a geometric point of view the connection between the renormalization group flow for non–linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests a natural way for extending, beyond the weak coupling regime, the embedding of the Ricci flow into the renormalization group flow.     

Trees, Renormalization and Differential Equations

The Butcher group and its underlying Hopf algebra of rooted trees were originally formulated to describe Runge–Kutta methods in numerical analysis. In the past few years, these concepts turned out to have far-reaching applications in several areas of mathematics and physics: they were rediscovered in noncommutative geometry, they describe the combinatorics of renormalization in quantum field theory. The concept of Hopf algebra is introduced using a familiar example and the Hopf algebra of rooted trees is defined. Its role in Runge–Kutta methods, renormalization theory and noncommutative geometry is described.     

Renormalization of potentials and generalized centers

We generalize the Riesz potential of a compact domain in Rm${\mathbb{R}}^m$ by introducing a renormalization of the rα−m$r^{\alpha -m}$-potential for α?0$\alpha ?0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.     

Exponential Function

We give the meaning to expressions of the form a^r
Where the base is a a positive real number and the exponent is r a positive real number.