Euclidean Quantum Gravity 9810205155, 9810205163

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EUCLIDEAN QUANTUM GRAVITY

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EUCLIDEAN QUANTUM GRAVITY EDITORS

G W GIBBONS S W HAWKING Department of Applied Mathematics and Theoretical Physics University of Cambridge England

V f c World Scientific ™^

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

The editors and publisher are grateful to the authors and the following publishers for their assistance and their permission to reproduce the articles found in this volume: The American Physical Society (Phys. Rev. D and Phys. Rev. Lett.) Elsevier Science Publishers (Nucl. Phys. B and Phys. Lett.) Springer-Verlag (Commun. Math. Phys.) The American Mathematical Society Cambridge University Press Gauther-Villars Lehigh University Plenum Press The Royal Society of London

EUCLIDEAN QUANTUM GRAVITY Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic ormechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 981-02-0515-5 981-02-0516-3 (pbk)

Printed in Singapore by Utopia Press.

CONTENTS

Introduction

IX

I. GENERAL FORMALISM 1. One-Loop Diverg5pncies in the Theory of Gravitation by G. 't Hooft K M. Veltman, Ann. Inst. Henri Poincare 20, 69-94 (1974) 2. The Path-Integral Approach to Quantum Gravity by S. W. Hawking, in General Relativity: An Einstein Centenary Survey, eds. S. W. Hawking & W. Israel, (Cambridge University Press, 1979).

29

3. Euclidean Quantum Gravity by S. W. Hawking, in Recent Developments in Gravitation Cargese Lectures, eds. M. Levy k S. Deser, (Plenum, 1978).

73

4. Path Integrals and the Indefiniteness of the Gravitational Action by G. W. Gibbkns, S. W. Hawking k M. J. Perry, Nuci Phys. B138, 141-150 (1978).

102

5. Proof of the Positive Action Conjecture in Quantum Relativity by R. Schoen M S. T. Yau, Phys. Rev. Lett. 42, 547-548 (1979).

112

6. Zeta Function Regularization of Path Integrals in Curved Spacetime by S. W. Hawking, Commun. Math. Phys. 56, 133-148 (1977).

114

7. The Conformal Rotation in Perturbative Gravity by K. Schleich, Phys. Rev. D36, 2342-2363 (1987).

130

8. Quantum Tunneling and Negative Eigenvalues by S. Coleman, Nucl. Phys. B298, 178-186 (1988).

152

9. A Relation Between Volume, Mean Curvature and Diameter by R. L. Bishop| , Notices Amer. Math. Soc. 10, 364 (1963).

161

10. The Cosmological Constant is Probably Zero by S. W. Hawkibg, Phys. Lett. 134B, 403-404 (1984).

162

II. BLACK HOLES 1. Particle Creation by Black Holes by S. W. Hawkii.kg, Commun. Math. Phys. 43, 199-220 (1975).

I

167

2. Path Integral Derivation of Black-Hole Radiance by J. B. Hartle k S. W. Hawking, Phys. Rev. D13, 2188-2203 (1976).

189

3. Black Holes and Thermal Green Functions by G. W. Gibbons & M. J. Perry, Proc. Roy. Soc. London A358, 467-494 (1978).

205

4. Action Integrals and Partition Functions in Quantum Gravity by G. W. Gibbons k S. W. Hawking, Phys. Rev. D15, 2752-2756 (1977).

233

5. Instability of Flat Space at Finite Temperature by D. J. Gross, M. J. Perry k L. G. Yaffe, Phys. Rev. D25, 330-355 (1982).

238

6. Thermal Stress Tensor in Static Einstein Spaces by D. N. Page, Phys. Rev. D25, 1499-1509 (1982).

264

7. Quantum Stess Tensor in Schwarzschild Space-Time by K. W. Howard k P. Candelas, Phys. Rev. Lett 53, 403-406 (1984).

275

III. Q U A N T U M COSMOLOGY 1. Cosmological Event Horizons, Thermodynamics and Particle Creation by G. W. Gibbons k S. W. Hawking, Phys. Rev. D15, 2738-3751 (1977).

281

2. Gravitational Effects on and of Vacuum Decay by S. Coleman k F. de Luccia, Phys. Rev. D21, 3305-3315 (1980).

295

3. Supercooled Phase Transitions in the Very Early Universe by S. W. Hawking k I. G. Moss, Phys. Lett HOB, 35-38 (1983).

306

4. Wave Function of the Universe by J. B. Hartle k S. W. Hawking, Phys. Rev. D28, 2960-2975 (1983).

310

5. The Quantum State of the Universe by S. W. Hawking, Nucl. Phys. B239, 257-276 (1984).

326

6. Origin of Structure in the Universe by J. J. Halliwell k S. W. Hawking, Phys. Rev. D31, 1777-1791 (1985).

346

VI

IV. WORMHOLES 1. Wormholes in Space time by S. W. Hawking, Phys. Rev. D37, 904-910 (1988).

363

2. Axion-Induced Topology Change in Quantum Gravity and String Theory by S. B. Giddings k A. Strominger, Nucl. Phys. B306, 890-907 (1988).

370

3. Why There is Nothing Rather than Something: A Theory of the Cosmological Constant by S. Coleman, Nucl. Phys. B310, 643-668 (1988).

388

4. Wormholes and the Cosmological Constant by I. Klebanov, L. Susskind k T. Banks, Nucl. Phys. B317, 665-692(1989).

414

5. Wormholes in Spacetime and the Constants of Nature by J. Preskill, Nucl. Phys. B232, 141-186 (1989).

442

6. Wormholes in Spacetime and 0QCD by J. Preskill, S. P. Trivedi & M. B. Wise, Phys. Lett. 223B, 26-31 (1989).

488

V. GRAVITATIONAL I N S T A N T O N S 1. Asymptotically Flat Self-Dual Solutions to Euclidean Gravity by T. Eguchi k A. J. Hanson, Phys. Lett. 74B, 249-251 (1978).

497

2. Gravitational Multi-Instantons by G. W. Gibbons k S. W. Hawking, Phys. Lett. 75B, 430-432 (1978).

500

3. The Positive Action Conjecture and Asymptotically Euclidean Metrics in Quantum Gravity by G. W. Gibbons k C. N. Pope, Commun. Math. Phys. 66, 267-290 (1979).

503

4. Polygons and Gravitons by N. J. Hitchin, Math. Proc. Camb. Phil. Soc. 85, 465-476 (1979).

527

5. The Construction of ALE Spaces as Hyper-Kahler Quotients by P. B. Kronheimer, /. Diff. Geom. 29, 665-683 (1989).

539

6. A Compact Rotating Gravitational Instanton by D. N. Page, Phys. Lett. 79B, 235-238 (1979).

558

VII

7. Classification of Gravitational Instanton Symmetries by G. W. Gibbons k S. W. Hawking, Commun. Math. Phys. 66, 291-310 (1979).

562

8. Low Energy Scattering of Non-Abelian Monopoles by M. F. Atiyah k N. J. Hitchin, Phys. Lett 107A, 21-25 (1985).

582

vjji

INTRODUCTION

Historical Origins development for the classical The 1960's and early 1970's was a period of rapid deve theory of general relativity. Stimulated by the experimental experimen discoveries of quasars, pulsars and the cosmic microwave background, attention turned to the problem of and to the problem the gravitational collapse of stars and more massive bodies be of the big bang. This period culminated in the proofs ol of the singularity theorems (see e.g. [1]) showing that under rather general conditions conditio spacetime singularities are, according to the classical theory, inevitable both in th the collapse of stars to form what became known as black holes and at some time in the past during the very of important results earliest phases of the universe. Shortly afterwards a number nui were obtained which provided a broad and satisfactory picture p of the formation of black holes and their basic properties. These early results and the resulting picture were described in a coherent form for the first time in the ]1972 Les Houches Summer School Proceedings [2]. Among the most intriguing results, resu and as it turned out, most significant for future research, were the clear analogies between the properties analog] and thermodynamics [3]. of black holes according to classical general relativity anc It was natural, therefore, to attempt to extend these purely classical results to the realm of the quantum theory. Since at that time, and anc indeed to this day, there was no satisfactory fully quantized theory of gravity, the 1first steps in this direction were taken using quantum field theory on a purely classical clas background. It had been realized for some time previously [4] that one of the most important physical effects that the quantum theory introduces is the production produ of particle pairs by strong gravitational fields. When applied to black holes for the first time in [5, 6] it was discovered that the previously noted analogy between the properties of black 1holes and thermodynamics could be extended to a complete correspondence since a black b hole in free space was shown to by LL radiate thermally with a temperature T given 1 t ra.uid.ie m e r m a n y WILII a t e m p e r a t u r e l g i v e n uy

*-s K is the surface gravity. As a consequence one should where K sh be able to assigi assign an entropy S to W a « . black U 1 0 V . A hole XXKJX^ given £ ji » t n by uj

SH-SH ==

\A4HA»>,

where A is the surface surface area of the black hole. Shortly afterwards the same result was rederived, still within the context of field theory on a fixed background, using a path integral formulation forn [7] which showed for the first time how the thermal character of the emission emissic could be understood in terms of the complex geometry of the Schwarzschild metric. meti In particular, in order that the path integral be better defined, it was found coi convenient to pass to a pure imaginary time coordinate r — it. The The regularity regularity of of the the resulting res = it. positive definite, or Euclidean as opposed to Lorentzian metric, demanded that th r be identified modulo ix

8wM = 1/TH- It was soon realized that this periodicity in imaginary time was related to the thermal character of the resulting Green's functions [8, 9] and was not restricted to the Schwarzschild case but could be extended quite generally to cover all time-independent horizons including what has turned out to be of considerable importance in connection with inflationary cosmology — the case of cosmological event horizons [9]. These early advances provided the needed clue on how to incorporate the quan­ tized gravitational field. A formalism was required which allowed one both to exploit to the full the topological and geometrical character of Einstein's general relativity and to perform the calculations efficiently. It had, by that time, become clear that the path integral method, despite its considerable mathematical problems, provided the most direct way to quantize the Yang-Mills theory. It was found that impor­ tant nonperturbative information could be extracted, as well as the structure of the divergences of the theory, by considering Yang-Mills connections defined on flat Eu­ clidean four-space. Nonsingular and nontrivial solutions of the classical equations of motion, called "instantons" allowed one to calculate nonperturbative amplitudes. The extension of these Euclidean methods to quantum gravity was initiated in [11] where it was shown that, about the topologically trivial case of iZ4, the coef­ ficients of the one-loop divergences for pure gravity vanish. The first application of these ideas to the topologically nontrivial case of black-hole backgrounds was contained in [12] where it was shown how the evaluation of the classical Einstein action with the appropriate and essential boundary term [13] allowed a derivation of the relation between entropy and event horizon surface area. There followed a period of rapid development of the general formalism of Eu­ clidean quantum gravity and since then it has been applied to a number of other important physical problems. It has also spawned a number of purely mathematical developments, for example the study of gravitational instantons which have, in turn been applied to problems in other areas of physics. The present volume is intended to provide a selection of some of the most in­ fluential and important papers in this field. We have arranged the material in five sections, each dealing with a particular aspect of the theory. We shall describe them in more detail below. We do not pretend that Euclidean quantum grav­ ity is a complete or entirely mathematically consistent quantum theory of gravity. The well-known divergences which not even supergravity theories have succeeded in eliminating suggest that a field theory of gravity like Einstein's general relativity is a purely low-energy or large-distance approximation to some more fundamental underlying theory. It is too early at present to say what such a theory might be. Despite its considerable successes in apparently eliminating divergences it is clear, for example, that superstring theory is not at present sufficiently well developed to tackle the physical questions which the Euclidean formulation addresses. Moreover it seems very likely that, whatever the true fundamental quantum theory of gravity turns out to be, it will have Einstein's theory of general relativity as its classical limit and very probably some form of path integral formulation as its semi-classical limit. Certainly we know of no other formalism capable at present of dealing with the problems raised by gravitational collapse either in black holes or in cosmology. Indeed it is interesting to note how many attempts to apply quantum mechani­ cal ideas to gravity invoke not quite explicitly the basic ideas behind Euclidean quantum gravity. x

Outline of this Volume We turn now to a more detailed description of the contents of this reprint volume. The first section is concerned with the general formalism. It begins with the impor­ tant paper by 't Hooft and Veltmann which initiated a whole series of investigations into the divergences and one-loop corrections to quantum gravity by showing how these could be calculated in a path integral formalism analogous to that used for the Yang-Mills gauge theory. The next two papers contain reviews of the first stage of the subject and the initial applications to black-hole theory and gravitational instantons, the third paper containing applications to supergravity theories. The fourth paper of Section I deals with a central problem in this approach and one which has yet to be completely resolved: that of the indefiniteness of the grav­ itational action. By means of a conformal rescaling the gravitational action can be made arbitrarily negative. Thus one cannot merely integrate over all metrics with positive definite signature subject to suitable boundary conditions; rather what ap­ pears to be needed is a complex contour in the space of all metrics. In the fourth paper the proposal was presented that, at least for asymptotically Euclidean (or ALE) metrics, this contour should run over all positive definite AE metrics with vanishing Ricci-scalar and over all conformal rescalings of such metrics, the rescalings being in the purely imaginary direction. The reasonableness of this proposal in the purely perturbative case would seem to be confirmed by the results of the seventh paper of Section I. Its correctness in the nonperturbative regime depends, at the very least, on the correctness of a conjecture concerning the Einstein action of asymptotically Euclidean metrics with vanishing Ricci scalar which generalizes the positivity of the ADM mass in classical general relativity. A proof of this im­ portant and mathematically difficult conjecture was announced in the fifth paper of Section I. The perturbative method for calculating functional integrals involves the evaluation of certain "determinants" of differential operators. An appropri­ ate method is by means of their "zeta functions" and this technique is described in the sixth paper of Section I. For certain tunneling amplitudes, the eigenvalues of these differential operators may be negative. The correct method for evaluat­ ing them, at least in flat spacetime, is described in the eighth paper of Section I. The Positive Action Theorem is a purely mathematical result about the Euclidean action of asymptotically and Euclidean four-metrics. For application to quantum cosmology and the theory of wormholes, it is more appropriate to consider com­ pact four-manifolds with no boundary. A highly relevant mathematical result is the theorem of Bishop described in the penultimate paper of Section I which provides us with a strict lower boundary for the action of a compact metric with constant Ricci scalar analogous to the result of Schoen and Yau. The last paper of Section I illustrates the relevance of this mathematical result to one of the most challenging problems confronting any quantum theory of gravity — the cosmological constant problem. The second section of this reprint volume is concerned with what is perhaps the most successful area for the application of the basic formalism, and which, as described above, stimulated the entire development — the subject of black holes. In addition to the papers whose significance has been described above, we have in­ cluded as paper five of this section an attempt to relate the Euclidean Schwarzschild solution in a tunneling calculation. The Euclidean formulation of black-hole ther­ modynamics allows the calculation of expectation values of the energy-momentum XI

tensor of quantum fields on the Schwarzschild background (T)|„). We have included as the penultimate paper of Section II, one describing a particularly useful approximation for (T^u) for a scalar field whose somewhat unexpected accuracy was confirmed by detailed numerical studies described in the last paper of Section II. Perhaps the most ambitious area to which quantum gravity has been applied is the entire Universe. This is the subject of Section III. The first paper describes the basic quantum properties of De Sitter. Apart from its intrinsic interest in helping to elucidate various conceptual and technical problems, this has turned out to be of great importance for the theory of inflation. However the cosmological term that is believed to be present during inflation arises as a (classical) expectation value, and may change during a phase transition. If this change is forbidden classically it may happen by quantum tunneling. This requires an appropriate formalism and in the second and third papers of Section III it is shown how Euclidean quantum gravity may provide such a formalism. We know of no other. A different viewpoint was inaugurated in the fourth paper of Section III and further developed in paper five, where it is shown how the Euclidean path integral can provide solutions of the Wheeler-De Witt equation, wave functions for the universe. A proposal is made for the wave function of the universe which is suggested by its simplicity and naturalness in the Euclidean framework. The final paper of Section III shows how one may extract detailed cosmological predictions, such as the spectrum of fluctuations from the basic formalism together with the non-boundary boundary condition. The fourth section of this reprint collection deals with one of the most recent developments in Euclidean quantum gravity — the possible relevance of topologically nontrivial Euclidean manifolds in the path integral. It has always been a major goal of Euclidean quantum gravity to incorporate the nonperturbative effects of quantum fluctuations of such metrics. Of particular interest are metrics in which apparently distant flat regions of the manifold are connected by short "wofmholes." Some examples and an account of the effects they might give rise to was first given in the first paper of Section IV. If one considers asymptotically Euclidean vacuum metrics, the results of Schoen and Yau and Witten show that the only stationary point of the classical action is flat space. The inclusion of particular types of matter can alter this fact. In the second paper of Section IV, it is shown how axion fields such as those occurring in string theory coupled to general relativity allow the existence of simple wormhole solutions. In the third paper of Section IV, it is shown how such wormholes may have preferred consequences for our understanding of the origin of the values of the observed coupling constants of nature. The most important example is the cosmological constant but the last three papers of Section IV show that even elementary particle coupling constants such as 0QCD may be affected as well. This offers the prospect of a much more intimate connection between Planck scale physics and observable physics than had hitherto been envisaged. The last section of this reprint collection is more mathematical in nature, and is concerned with complete nonsingular Ricci-flat four-metrics or "gravitational instantons." As mentioned above there are no asymptotically Euclidean (AE) metrics of this sort but Eguchi and Hanson (in the first paper of Section V) provided the first explicit nonsingular example of what turned out to be an asymptotically locally Euclidean (or ALE) metric. In the second paper of Section V, a generalization to a multi-instanton metric which is also asymptotically locally Euclidean is made. XII

More information about asymptotically Euclidean metrics is contained in the third paper of Section V. The multi-instanton metrics of paper two of this section were rederived using Twistor techniques by Hitchin in the fourth paper of Section V and the existence of other examples related to each of the discrete sub-groups of SU(2) was conjectured. The confirmation of this conjecture by means of an explicit construction is provided by Kronheimer's work in the fifth paper of Section V. The programme of using techniques from classical general relativity to find complete nonsingular Einstein metrics (with cosmological constant) gave rise to a construction by Page of the first, and to date only, explicitly or implicitly known inhomogeneous Einstein metric with positive Ricci curvature. This is given in the sixth paper of Section V. In the seventh paper, a description of the geometry and possible physical significance of most of the known gravitational instantons is provided. Finally we have included what might at first sight appear to be a completely unrelated paper on the low-energy scattering of non-Abelian monopoles. This may be represented by geodesic motion on a self-dual gravitational instanton (The existence of this so­ lution had, regrettably, been overlooked in the third paper of this section). This represents one of a number of "spin-offs" arising from the study of gravitational instantons — another notable example being their application to the Kaluza-Klein theory and Kaluza-Klein monopoles. References Items marked with an asterisk are reprinted in this volume. [1] S. W. Hawking k G. F. R. Ellis, The large scale structure of spacetime (Cam­ bridge University Press, Cambridge, 1973). [2] C. Dewitt k B. S. DeWitt eds., Black Holes (Gordon k Breach, New York, 1973). [3] J. M. Bardeen, B. Carter k S. W. Hawking, Commun. Math. Phys. 31, 181 (1973). [4] E. Schrodinger, Physica 6, 899 (1939). [5] S. W. Hawking, Nature 248, 30 (1974). [6*] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975). [7*] J. B. Hartle k S. W. Hawking, Phys. Rev. D13, 2188 (1976). [8] G. W. Gibbons k M. J. Perry, Phys. Rev. Letts. 36, 985 (1976). [9*] G. W. Gibbons k M. J. Perry, Proc. Roy. Soc. London. A358, 467 (1978). [10*] G. W. Gibbons k S. W. Hawking, Phys. Rev. D15, 2738 (1977). [11*] G. 't Hooft k M. Veltman, Ann. Inst. Henri Poincare 20, 69 (1974). [12*] G. W. Gibbons k S. W. Hawking, Phys. Rev. D15, 2752 (1977). [13] J. York, Phys. Rev. Letts. 6, 1656 (1972).

xiii

Ann. Inst. Henri Poincare.

Section A :

Vol. XX, n° 1, 1974, p. 69-94.

Physique

theohque.

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One-loop divergencies in the theory of gravitation par G. 't H O O F T (*) and M. VELTMAN (*) C. E. R. N., Geneva.

ABSTRACT. — All one-loop divergencies of pure gravity and all those of gravitation interacting with a scalar particle are calculated. In the case of pure gravity, no physically relevant divergencies remain; they can all be absorbed in a field renormalization. In case of gravitation interacting with scalar particles, divergencies in physical quantities remain, even when employing the socalled improved energy-momentum tensor.

1. INTRODUCTION The recent advances in the understanding of gauge theories make a fresh approach to the quantum theory of gravitation possible. First, we now know precisely how to obtain Feynman rules for a gauge theory [/]; secondly, the dimensional regularization scheme provides a powerful tool to handle divergencies [2]. In fact, several authors have already published work using these methods [5], [4], One may ask why one would be interested in quantum gravity. The foremost reason is that gravitation undeniably exists; but in addition we may hope that study of this gauge theory, apparantly realized in nature, gives insight that can be useful in other areas of field theory. Of course, one may entertain all kinds of speculative ideas about the role of gravitation in elementary particle physics, and several authors have amused themselves imagining elementary particles as little black holes etc. It may well be true that gravitation functions as a cut-off for other interactions; in view of the fact that it seems possible to formulate all known (*) On leave from the University of Utrecht, Netherlands. Annettes de rinstitut Henri Poincare - Section A - Vol. XX, n° 1-1974.

3

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70

ONE-LOOP DIVERGENCIES IN THE THEORY OF GRAVITATION

interactions in terms of field-theoretical models that show only logarithmic divergencies, the smallness of the gravitational coupling constant need not be an obstacle. For the time being no reasonable or convincing analysis of this type of possibilities has been presented, and in this paper we have no ambitions in that direction. Mainly, we consider the present work as a kind of finger exercise without really any further underlying motive. Our starting point is the linearized theory of gravitation. Of course, much work has been reported already in the literature [5], in particular we mention the work of B, S. Dewitt [6]. For the sake of clarity and completeness we will rederive several equations that can be found in his work. It may be noted that he also arrives at the conclusion that for pure gravitation the counterterms for one closed loop are of the form R2 or RMVRMV> this really follows from invariance considerations and an identity derived by him. This latter identity is demonstrated in a somewhat easier way in appendix B of this paper. Within the formalism of gauge theory developed in ref. 7, we must first establish a gauge that shows clearly the unitarity of the theory. This is done in section 2. The work of ref. 7, that on purpose has been formulated such as to encompass quantum gravity, assures us that the S-matrix remains invariant under a change of gauge. In section 3 we consider the one loop divergencies when the gravitational field is treated as an external field. This calculation necessitates a slight generalization of the algorithms recently reported by one of us [8]. From the result one may read o Iff the known fact that there are fewer divergencies if one employs the so-called improved energy-momentum tensor [9]. Symanzik's criticism [70] applies to higher order results, see ref. 11. In the one loop approximation we indeed find the results of Callan et al. [9]. Next we consider the quantum theory of gravity using the method of the background field [vV E>vV ^v -*—A„v

(2.2) (2.2) (2.3) (2.3)

In here the rj^ are four independent infinitesimal functions of space-time. The DM are the usual covariant derivatives. In order to define Feynman rules we must supplement the Lagrangian (2.1) with a gauge breaking term - - C * and a Faddeev-Popov ghost Lagrangian (historically, the name Feynman-DeWitt ghost Lagrangian would be more correct). In order to check unitarity and positivity of the theory we first consider the (non-covariant) Prentki gauge which is much like the Coulomb gauge in quantum-electrodynamics: 3

2JK

..,4. = °> /r=l, ...,4.

(2.4) (2.4)

1=1

In the language of ref. 7 we take correspondingly: 33

C„ = b YdA,, i= - 1

Vol. XX. XX, n° 111 -- 1974. 1974.

5

b*>->^ oo. oo.

(2.5) (2.5)

72

ONE-LOOP -LOOP DIVERGENCIES IN THE THEORY OF GRAVITATION GRAVI

With this choice1 for C the part quadratic in the /i/iMV MV is (comma denotes differentiation):

^* *Kfi^Kfi,* Kp.nKfi,* Ka^^n Koijh^ K(i,vKfi,» Ka^fifi^ h9*.fihfiw .fihfiw --" 44^ .A/u ~~~- 2 2hia.fihfiM . /A/u ++^^"aa,A/U 4 1

1 .

(2.6)

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b2b2 hh P»** K.i + 2 hfi»** fi»**hh0*.» 0*.»-~~ 22 62/l «> K.i wJwjA

1i This can be written as — transform of V is ten as - h^Y^^Jt^y, h^V^^h^. The The Fourier Fourier trans! v

1 22 2 ^ A v ) -- fc. MMA*A* * ++ VM ^ AMM ~-- fc* 2 ^^M/c«v(5Ma^.«^v = 2 fc2(