Fortschritte der Physik / Progress of Physics: Band 25, Heft 8 1977 [Reprint 2021 ed.] 9783112519509, 9783112519493

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Zeitschrift „Fortschritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche« Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrog der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3—4; Fernruf: 2 23 62 21 und 2 23 62 29; Telex-Nr. 114420; Postscheckkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R »104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gcsamthcrsteliung: V E B Druckbaus „Maxim Gorki", D D R - 7 4 Altenburg, Carl-von-Ossietzky-StraDe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 15«— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/25/8. (r) 1977 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618

Fortschritte der Physik 27, 4 5 9 - 5 0 0 (1977)

Renormalized Composite Fields in Quantum Field Theory S . A . ANIKIN

Lebedev Physical Institute, Moscow,

USSR

and M . C. POLIVANOV, 0 . 1 .

Steclov Mathematical

ZAVIALOV

Institute, Moscow,

USSR

Dynamics of Zimmermann composite fields is considered from general point of view based on "global" resolution of the combinatorial structure of the Bogolubov-Parasiuk renormalization that is on explicit formulae for i?-operation as applied to the formal sum of perturbation series.

Contents 1. Introduction 2. Heisenberg operators 3. ii-operation 4. Basic notions. Subtraction operator M 5. Structure of counterterms 6. Zimmermann identities 7. Wilson expansions 8. Equations of motion for composite fields 9. Renormalization group equations. Callan-Symanzik equations 10. Equations for regularized Green functions Literature

459 460 464 466 472 479 482 485 491 494 500

1. Introduction

Local Heisenberg operators, having non-linear (in elementary in-field) objects as their asymptotic images play an important role in all dynamical schemes of quantum field theory. Currents in the Lagrangian picture or in the theory of currents, particle fields in the composite models (say, quark models) give the natural examples of such operators. The normal product algorithm developed by Z I M M E R M A N N , L Ö W E N S T E I N and others proves to be perfect tool in numerous applications of composite fields. This algorithm is based on the standard notions of Lagrangian perturbation theory. But the main objects of the formalism have the form of "total" Heisenberg operators, i.e. of formally summed series of perturbation expansions. This suggests an idea of maximal possible absteining from perturbation theory. 34

Zeitschrift „Fortschritte der Physik", Heft 8

460

S . A . ANIKIN, M . C . POLIVANOV, 0 . 1 . ZAVIALOV

In the following we give presentation of methods and results of the normal product algorithm from this general point of view. This presentation, based on papers [5, 10, 11] has as its foundation direct computation of the counterterms necessary for renormalization of the Heisenberg operators, or of their vacuum expectations. Thus, though a possibility of Feynman graph expansion is an initial presumption of the method — it is the only way to apply the ii-operation — m fact, we never use Feynman graphs explicitly and even in the intermediate steps we use only " t o t a l " Heisenberg operators in the whole. In this way of reasoning we avoid some combinatorial difficulties which makes things clearer and simplifies the handling of renormalized Heisenberg operators. I t permits to obtain some additional new results and to make old results more transparent. When we need a concrete model we always consider a scalar quartic model but all the methods are applicable for the general case. As to the content of the paper it may be read from the names of Sections given above. 2. Heisenberg operators 1. We start with an operator of a free asymptotic m-field