Fortschritte der Physik / Progress of Physics: Volume 34, Number 9 [Reprint 2022 ed.] 9783112613849, 9783112613832

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FORTSCHRITTE DER Volume 34-1986 PHYSIK Number 9 PROGRESS OF PHYSICS Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J. Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J . Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay

CONTENTS: I. L. Buohbinder Renormalization of Quantum Field Theory in Curved Space-Time and Renormalization Group Equations

605-628

P. Moylan An Integral Transform in de Sitter Space

629-647

AKADEMIE-VERLAG • BERLIN ISSN 0015 - 8208

Fortsohr. Phys., Berlin 84 (1986) 9, 605-647

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the authors name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p,...), elementary mathematical functions like Re, Im, sin, cos, exp,...): black underlined Greek letters: red underlined Boldface Greek letters: red interlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c 0, h K, o 0, p P, s S, u U, v V, w W, x X, y Y, z Z). It will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, b\, Mil, WVf Please differentiate between following symbols: a, a; a, a , oo; a, d; c, C, a; e, e, k, K, x; x, x, X, x, X ; 1,1; o, 0, a, 0; p, q; u, U, IJ ; v, v, V; cp, ,&,0; &, 0, Q. 8. Each paper should be followed by a list of references with consecutive numeration. The numbers should be placed on the line between squared brackets or typewritten inclined lines. Articles in journals should be cited with author's name and abbreviated first name, title of journal volume number (underlined), year of publication (in brackets) and page number. In case of books author's name and abbreviated first name, title, place, and year of publication should be given 9. Manuscripts should be submitted fit for printing; badly arranged manuscripts are returned. 10. Proof reading should be limited to the correction of typographical errors. Deletions and insertions are not permitted in proof reading. 11. Of every paper published the author(s) will receive 30 reprints free of charge. 12. Within the framework of legal protection the publishers have the right of publication, distribution and translation reserved. Without express permission it is not allowed to produce photocopies, microfilms etc. of this journal or parts of it.

FORTSCHRITTE DER PHYSIK VOLUME 34

1986

NUMBER 9

Fortschr. Phys. 84 (1986) 9, 6 0 5 - 6 2 8

Renormalization oi Quantum Field Theory in Curved Space-Time and Renormalization Group Equations I . L . BUCHBINDER

Tomsk Pedagogical Institute, Tomsk, USSR Contents 1. 2. 3. 4. 5. 6. 7.

Introduction 605 Structure of quantum field theory renormalization in curved space-time 607 Renormalization group equations 613 Behaviour of effective charges in concrete theories. Asymptotic conformai invariance . . 615 Renormalization group equations and effective potential in external gravitational field . 619 Quantum field theory renormalization in external gravitational field with torsion . . . . 623 Conclusion. . . 626 References 627

Abstract The structure of quantum field theory renormalization in curved space-time is investigated. The equations allowing us to investigate the behaviour of vacuum energy and vertex functions in the limit of small distances in the external gravitational field are established. The behaviour of effective charges corresponding tó the parameters of nonminimal coupling of the matter with the gravitational field is studied and the conditions under which asymptotically free theories become asymptotically conformally invariant are found. The examples of asymptotically conformally invariant theories are given. On the basis of a direct solution of renormalization group equations the effective potential in the external gravitational field and the effective action in the gravity with the high derivatives are obtained. The expression for the cosmological constant in terms of ii 2 -gravity Lagrangian parameters is given which does not contradict the observable data. Renormalization and renormalization group equations for the theory in curved space-time with torsion are investigated.

1.

Introduction

L a t e l y different aspects of quantum field theory in curved spacetime have intensively been studied. The interest in this theory is caused, on the one hand, by its application for cosmology and physics of black holes. On the other hand, the theory in which the quantum dynamics of the m a t t e r is studied on a classical gravitational field background can be considered as the first approximation to not yet constructed consistent quantum g r a v i t y theory. One of the major problems of quantum field theory is t h a t of renormalizations. A great number of papers, most of which referring to free field theories, are devoted to the discussion of this problem in the presence of the external gravitational field. Essential 1

Fortschr. Phys. 34 (1986) 9

606

I. L. Buchbinder, Renormalization of QFT

progress made in the investigation of free theory renormalizations is shown in reviews [1-4]. Quantum theory of interacting fields in curved space-time began to develop not long ago. The main characteristics of such a theory were pointed out in papers [5—9] (see also [4]). Renormalization of interacting field theories was studied in papers [10—39]. General analysis of renormalization structure [10—13, 19, 22, 25, 26, 38, 39] was fulfilled and for a number of concrete theories one-loop [14, 17, 18, 21, 23, 24, 27 — 30, 32—34, 36, 37, 79] and two-loop [14, 20] counterterms were found. On the whole the, analysis carried out in papers [10—39] shows that renormalizability of the theory in curved space does not follow from that of plane space. Nevertheless, if the theory is multiplicatively renormalized in plane space, one can achieve its multiplicative renormalizability in curved space too. Here field and parameters renormalization constants having analogues in plane space are the same as those in plane space. However, multiplicative renormalizability of the theory in curved space requires in addition: a) adding to the bare Lagrangian the Lagrangian of the external field with bare parameters and renormalization of these parameters with the help of new renormalization constants; b) adding the terms providing nonmmimal coupling of the matter with a gravitational field to the bare matter Lagrangian and renormalization of nonminimal coupling parameters with the aid of new renormalization constants. Multiplicative renormalizability allows us to use for studying theory properties the method of the renormalized group [40—42]. One of the main applications of the renormalized group in plane space is connected with the possibility of studying vertex functions behaviour at great momenta. In curved space in which it is impossible to fulfill the transition to momentum respresentation without perturbation theory over the external field such a possibility seems lost. However, in paper [31, 33 — 37, 39] it was noted that the momenta transformation p -s- kp {k = const.) used in plane space can be replaced by scale metric transformation of the external gravitational field g a p-> k~2gaf. When k -> oo all typical distances decrease, which corresponds to the increase of typical momenta. Renormalization group equations allowing us to study the vertex functions behaviour a t the transition to still smaller typical distances were dealt with in papers [31, 33 — 37] and their general analysis is given in paper [39]. In doing so it was shown that some asymptotically free theories are asymptotically conformally invariant. In such theories effective charges corresponding to nonminimal coupling of the matter with the gravitational field have ultravioletly fixed points corresponding to conformally invariant theories. Renormalization group equations can be written directly for the effective action and, consequently, for the effective potential. As a result there is a possibility of finding the effective potential in curved space-time by direct solving renormalization group equations [32, 43]. Renormalization group equations are suitable for investigating different asymptotics of the effective potential [44]. The presence of the nonminimal interaction of the matter with the gravitational field is essential for multiplicative renormalizability of the theory. If torsion is absent, the only admissible nonminimal interaction is provided by the term where R is scalar curvature, are nonminimal coupling parameters,