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Renormalization Group in Different Fields of Theoretical Physics KEK—91-13 JP9207322 KEK Report 91-13 February 1992 H Renormalization Group in Different Fields of Theoretical Physics D.V. Shirkov NATIONAL LABORATORY FOR HIGH ENERGY PHYSICS to National Laboratory for High Energy Physics, 1992 KEK Reports are available from: Technical Information & Library National Laboratory for High Energy Physics 1-1 Oho, Tsukuba-shi [baraki-ken, 305 JAPAN Phone: 0298-64-1171 Telex: 3652-534 (Domestic) (0)3652-534 (International) Fax: 0298-64-4604 Cable: KEKOHO RENORMALIZATION GROUP IN DIFFERENT FIELDS OF THEORETICAL PHYSICS D.V. SHIRKOV Lab. Thcor. Phys., Joint Inst, for Nucl. Research, Dubna, USSR Lecture presented at KEK theory division 2 - 3 April, 1991 ABSTRACT A very simple and general approach to the symmetry that is widely known as a Rcnormalizalion Group symmetry is presented. It essentially uses a functional formulation of group transformations that can be considered as a generalization of self-similarity transformations well known in mathematical physics since last century. This generalized Functional Self-Similarity symmetry and corresponding group transformations are discussed first for a number of simple physical problems taken from diverse fields of classical physics as well as for QED. Then we formulate the Rcnorm-Gnup Method as a regular procedure that essentially improves the approximate solutions near the singularity. After that we discuss relations between different formulations of Renormal- ization Group as they appear in various parts of a modern theoretical physics. Finally we present several topics of RG.M application in modern QFT. - 1 - CONTESTS I. Renormalization Group as a Functional Self-Similarity Group (4-33) $1-1. Introduction 4 §1-2. Mathematical preliminaries 7 §1-3. Dcfiniton of the Renorm-Group 10 §1-1. Simple illustration 14 §1-5. Functional self-similarity 16 §1-6. Other illustrations from classical physics 18 6a. Weak shock wave (IS) 6b. Transfer theory (20) §1-7. RG in QED 23 7a. Effective electron charge (23) 7b. RG transformation (28) 7c. Dyson transformation (30) §1-8. The Nature of RG=FSS Symmetry 32 II. Renornialization Group Method (34-47) §0-1. Basic idea 34 §11-2. Differential formulation 35 §11-3. General solutions 38 §11-4. Technology of RGM 40 §11-5. Illustrations of RGM from classical physics 42 ... 5«. Transfer problem (\2) 5b. Shock wave ($3) §11-6. RGM usage in QFT II §11-7. Essence of RGM 46 III. Different Faces of Renormalization Group (48-63) pi-1. Critical phenomena 48 §111-2. Polymer theory 51 §111-3. Noncoherent radiation transfer 54 §HI-4. Turbulence 56 §M-5. Paths of RG expansion 57 §111-6. Two faces of RG in QFT 59 §HI-7. Summary 63 IV Some RG Applications in QFT (64-82) §FV-1. General UV analysis 64 ... la. One-cc~up!:ng case (64) lb. Multi-coupling case (68) §IV-2. Perturbative appioach to the UV asymptote 70 ... 2a. Structure of RG results (70) 2b. The ghost-pole trouble (72) §IV-3. Reliability of results in QFT 73 ... 3a. Ghost pole in QED (73) 3b. Scalar quariic model (74) §IV-4. Asymptotic series Borel summation 77 §IV-5. Gauge fantoms in QCD 80 References (83-85) -3 - I. RENORMALIZATION GROUP as FUNCTIONAL SELF-SIMILARITY GROUP § 1-1. Introduction Specific symmetry underlying so called Renormalization Group (RG) was discovered in 1953 by Stiickelberg and Peterman [1] in the course of analysis of a finite arbitrariness arising in Quantum Field Theory (QFT) after subtraction of UltraViolet(UV) divergencies. They have realized the group structure of fi­ nite renormalizations and stressed the importance of differential group equations which they wrote down in a general form. Quite independently Gell-Mann and Low by manipulating with Dyson renor- malization transformations have succeeded [2] in obtaining functional equations for QED propagators and performing on their basis a general analysis of short- distance behavior of electromagnetic interaction. - 4 - 'The next important step was made by Bogoliubov and Shirkov [;i] in 19-"i5. These authors established close relation between approaches of [1] and [2]. de­ rived functional equations in a massive case and first introduced the differential formulation of RG symmetry. Their most important result is the formulation of a Renormalization Group Method (R.GM), a regular method of improving per­ turbation approximations by combinig them with RG differential Lie equations. This technique enabled the summation of leading log terms and have been used to deal with UV and IR behaviours in QED and other models. The most pop­ ular results of RG application in QFT obtained in the mid-fifties were related to short-distance behaviour. During about 15 years afterwards, RG and RGM were considered as very specific for only renormalized QFT with its divergencies and renormalizations. It is widely known that in the beginning of 70-ies just the RG method was used for discovery of a famous asymptotic freedom property in nonabelian quantum fields models. At the very same time K. Wilson (1971) [4] has managed to transfer RG ideology from QFT into statistical mechanics for the analysis of phase transition phenomena in spin lattices. This new version of RG based on a Kaclanoff's trick of spin blocking appeared to be much more transparent physically and apprehensible for a broad physical audience. Due to this, quite rapidly in the course of 70-ies, RG algorithms were successively transferred into other theoretical fields : the theory of polymers, turbulence, transfer and some others. Technically researchers in these diverse fields usually used a bit different formulation and terminology. Such expressions as "real-space R.G", "dynamic RG", "Monte-Carlo RG", 'RG chaos" are in use. - 5 - In such a situation it was quite natural to suppose that there .should exist some simple and general basement underlying RG different, faces in various fields of theoretical physics. An attempt to formulate such an universal picture was undertaken about ten years ago [5j. It turned out to be possible to discuss the RG type of symmetry in general terms of a classical mathematical physics. This symmetry can be defined as a symmetry of solution (not of equation!) for a physical system with respect to the transformation involving boundary condition parameters. In this short set of lectures we present a review of the RG ideas used in modern physics based on a rather simple foundation (we call it junctional self- similarity) which we illustrate by examples from classical physics and discuss the interrelation between RG different formulations. The idea to create a written version of these lectures belongs to Professor Yoshimitsu Shimizu. I use this possibility to thank him and Professor Hirotaka Sugawara, the KEK director, for the hospitality during my stay in Japan. The text of lectures was written with the help of Drs. Chong-Sa Lim, Masato Jirnbo and Professor Nobuya Nakazawa. -6- 'i J 2. Mathematical preliminaries Let us start with MMIO simple statements which can be supposed to be widely known. In QFT the H.(! usually is associated with possibility to present any phys­ ical quantity. F(Q-.g) calculated under definite renonnalization prescription 2 ( in a massless case ) in a form /• '(Q //r, <\;J) with renormalized coupling con­ stant g,, definition attached to some rcriormalization point (or renormalization momentum scale) Q = //. The differential RG equation is usually said to be driven from the condition that F does not depend on the choice of/( , £ = 0. (1-1, d/i. The coupling constant g^ dependence on /:, normalization momenta value, is described by a specific function </(Q"') known as effective coupling (sometimes - effective coupling constant) : g^ = <y(/<"). Equation (1) can be written down in the form of partial linear Differential Equation(DE) [*4--0(9)jr]F(*-9) = O • (1-: ox Og ! where x = Q //r, g stands for gti and beta function is defined as 0to,) = --^r ^ .- = ,<- (1-3) The effective coupling(EC) g also can be considered as a function of two arguments: g~{Q~j;r,g,L) with the boundary condition ~g{\,g) = g. Besides (2) ii satisfies the nonlinear DE x r; = .J .'/('•.'/)) • (1-U ox that is nothing else but characteristic equation for (2). To employ this formalism one has to give the /9(//). Usually for this one uses renormalized perturbation theory. The foregoing' can he considered as a "RG folklore". For brevity we gave it in the simplest massless version that corresponds to the IV case. A less popular result is the Functional Equation (FE) for the 7j which in I'V case has the form a(x,9)=ji{x/t,a(t,g)) . (i-5) This equation which follows from Dyson renormalization transformations repre­ sents a basement of differential RG formulation. Popular DE (!) can be directly obtained from it by differtiating over x and putting / = x. On the other hand by differentiating (5) with respect to / at / = 1 we get partial DE [x^-/3(g)^}g(xl9) = Q . (1-6) analogous to (2). Hence the FE (5) as well as similar FEs for propagators and vertex func­ tions must be considered as the most adequate and general formulation of RG symmetry in QFT. We shall call it the basic RG equation. - S - However in reality, group 1'1's do not contain any physics at all. being just the refleciinn of nothing else but. the group composition law! Namely, ,ve can regard the change of a reference coupling as an operation of group element 1\ defined by I'tfi = Ji(t.'i). If we set .r = -/ , then the l.h.s. of (5) can achieved from !l by 7Vi. while the r.h.s. may be identified as TTTtc/. The content of Y.<\. (•!) now is just the group composition law. 7"r< = 7"r7i . Thus the properly of the basic RG equation (5) is the formal condition for trasformations I!t to form a group.
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