Heat Kernel and Quantum Gravity

Preface This book is aimed primarily at theoretical physicists as well as graduate theory, quantum gravity, gauge t...

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Preface

This book is aimed

primarily

at theoretical

physicists as well as graduate theory, quantum gravity, gauge theories, and, to sdme extent, general relativity and cosmology. Although it is not aimed at a mathematically rigorous level, I hope that it may also be of interest to mathematical physicists and mathematicians working in spectral geometry, spectral asymptotics of differential operators, analysis on manifolds, differential geometry and mathematical methods in quantum theory. This book will certainly be considered too abstract by some physicists, but not detailed and complete enough by most mathematicians. This means, in particular, that the material is presented at the "physical" level of rigor. So, there are no lemmas, theorems and proofs and long technical calculations are omitted. Instead, I tried to give a detailed presentation of the basic ideas, methods and results. Also, I tried to make the exposition as explicit and complete as possible, the language less abstract and have illustrated the methods and results with some examples. As is well known, "one cannot cover everything", especially in an introductory text. The approach presented in this book goes along the lines (and is a further development) of the so-called background field method of De Witt. As a consequence, I have not dealt at all with manifolds with boundary, non-Laplace type (or nonminimal) operators, Riemann-Cartan manifolds as well as with many recent developments and more advanced topics, such as Ashtekar's approach, supergravity, strings, membranes, matrix models, M-theory etc. The interested reader is referred to the corresponding literature. students

working

in quantum field

These lecture notes

versity. Although

are

based

my Ph.D. thesis at Moscow State Uni-

on

most of the results

presented here

were

published

in

a

series

of papers, this book allows for the much more detail and is easier to read. It can be used as a pedagogical introduction to quantum field theory and

quantum gravity for graduate students with

some basic knowledge of quantheory and general relativity. Based on this material, I gave a series lectures for graduate students at the University of Naples during the fall

tum field

of

semester of the 1995.

It should be noted that

completely self-consistent The bibliography reflects

no

nor

attempts have been made

to

more or

to make the book

fully comprehensive list of references. less adequately the situation of the late

give

a

VIII

Preface

original Ph.D. thesis was written. A complete update of the obviously beyond my scope and capabilities. Nevertheless, bibliography I updated some old references and added some new ones that are intimately connected to the material of this book. I apologize in advance for not quoting the work of many authors who made significant contributions in the subject over the last decade. Besides, I believe that in an introductory text such as this a comprehensive bibliography is not as important as in a research monograph or a thorough survey. 1980s, when

my

was

I would like to express my sincere appreciation to many friends and colleagues who contributed in various ways to this book. First of all, I am especially indebted to Andrei 0. Barvinsky, Vladislav R. Khalilov, Grigori

Grigori A. Vilkovisky who inspired my interest in quantum theory and quantum gravity and from whom I learned most of the material of this book. I also have learned a great deal from the pioneering works of V.A. Fock, J. Schwinger and B.S. De Witt, as well as from more recent papers of T. Branson, S.M. Christensen, J.S. Dowker, M.J. Duff, E.S. Fradkin, S. Fulling, P.B. Gilkey, S. Hawking, H. Osborn, T. Osborn, L. Parker and A. Tseytlin among others. It was also a great pleasure to collaborate with Andrei Barvinsky, Thomas Branson, Giampiero Esposito and Rainer Schimming. Over the last ten years my research has been financially supported in part by the Deutsche Akademische Austauschdienst, the Max Planck Institute for Physics and Astrophysics, the Russian Ministry for Science and Higher Education, Istituto Nazionale di Fizica Nucleare, the Alexander von Humboldt Foundation and the Deutsche Forschungsgemeinschaft. M. Vereshkov and

field

Socorro, January

2000

Ivan G. Avramidi

Contents

Introduction 1.

..................................................

Background Field Method in Quantum Field Theory 1.1 Generating Functional, Green

.................................

and Effective Action

1.3 2.

Technique

4.

14

...........

17

....................

........................

for Calculation of De Witt Coefficients

2.3

Technique

2.4

De Witt Coefficients a3 and a4 Effective Action of Massive Fields

2.5

9

..........

...................................

Covariant Expansions in Curved Space Elements of Covariant Expansions

2.2

3.

....................................

for Calculation

of De Witt Coefficients 2.1

9

Functions

Green Functions of Minimal Differential Operators Divergences, Regularization and Renormalization

1.2

1

...........

21

21

27

34

...........................

37

........................

46

Partial Summation .........................

51

....................

51

3.2

Witt Expansion Asymptotic Expansions Covariant Methods for Investigation of

3.3

Summation of First-Order Terms

3.4

Summation of Second-Order Terms

3.5

De Witt Coefficients in De Sitter

of

Schwinger-De

3.1

Summation of

of

.......................

......................

77

Field Theories

.....................

77

..................

83

4.5

Effective Potential

Conclusion

57 61 68

4.4

4.3

53

...................

Space

Gauge Quantization One-Loop Divergences in Minimal Gauge One-Loop Divergences in Arbitrary Gauge and Vilkovisky's Effective Action Renormalization. Group and Ultraviolet Asymptotics

4.2

........

.........................

Higher-Derivative Quantum Gravity 4.1

Nonlocalities

.........................

94

........

101

......................................

...................................................

108 125

X

Contents

References

Notation Index

....................................................

......................................................

.........................................................

127 141 143

Introduction

macroscopic gravitational phenomena are described very well by general relativity [178, 211]. However, general reltreated be cannot as a complete self-consistent theory in view of a ativity The classical

the classical Einstein's

number of serious difficulties that

were

not

overcome

since its creation

[156].

This concerns, first of all, the problem of space-time singularities, which are unavoidable in the solutions of the Einstein equations [178, 211, 156, 151,

vicinity of these singularities general relativity becomes incomplete predict what is coming out from the singularity. In other words, the causal structure of the space-time breaks down at the singularities [151]. Another serious problem of general relativity is the problem of the energy of the gravitational field [100, 172, 70, 71].

67]. as

In the

it cannot

theory have motivated the need to congravitation [99, 130]. Also the progress towards quantum theory the unification of all non-gravitational interactions [128] shows the need to include gravitation in a general scheme of an unified quantum field theory. The first problem in quantizing gravity is the construction of a covariant perturbation theory. Einstein's theory of gravitation is a typical non-Abelian gauge theory with the difleomorphism group as a gauge group [781. The quantization of gauge theories faces the known difficulty connected with the presence of constraints [207, 190]. This problem was successfully solved in the works of Feynman [104], De Witt [78] and Faddeev and Popov [101]. The most fruitful approach in quantum gravity is the background field method of De Witt [78], [80, 85]. This method is a generalization of the method of generating functionals in quantum field theory [50, 155, 193] to the case of non-vanishing background field. Both the gravitational field and the matter fields can have the background classical part. The basic object in the background field method is the effective action functional. The effective action encodes, in principle, all the information of the standard quantum field theory. It determines the elements of the diagrammatic technique of perturbation theory, i.e., the full (or exact) onepoint propagator and the full (or exact) vertex functions, with regard to all quantum corrections, and, hence, the perturbative S-matrix [78, 83, 223]. On The difficulties of the classical

struct

of

a

the other

hand,

the effective action

gives

at

once

the

physical amplitudes

in real external classical fields and describes all quantum effects in external

I. G. Avramidi: LNPm 64, pp. 1 - 7, 2000 © Springer-Verlag Berlin Heidelberg 2000

Introduction

fields

[81, 82] (vacuum polarization of quantized fields, particle creation etc.) [137, 42,129, 187, 120, 121]. The effective action functional is the most appro-

priate tool for investigating the structure of the physical vacuum in various models of quantum field theory with spontaneous symmetry breaking (Higgs vacuum, gluon condensation, superconductivity) [191, 167, 164, 136, 53]. The effective action makes it possible to take into account the backreaction of the quantum processes on the classical background, i.e., to obtain the effective equations for the background fields [83, 84, 223, 224, 122, 225, 113]. In this way, however, one runs into a difficulty connected with the dependence of the off-shell effective action on the gauge and the parametrization of the quantum field. In the paper [84] a gauge-invariant effective action (which still depends parametrically on the gauge fixing and the parametrization) was constructed. An explicitly reparametrization invariant functional that does not depend on the gauge fixing (so called Vilkovisky's effective action) was was

constructed in the papers [223, 224]. The "Vilkovisky's" effective action studied in the paper [114] in different models of quantum field theory

(including Einstein gravity) and in the paper of the author and Barvinsky [22] in case of higher-derivative quantum gravity. The Vilkovisky's effective action was improved further by De Witt in [86]. This effective action is called Witt effective action.

However, in many cases this modificaone-lop results, that is why we will not consider it in this book (for more details, see the original papers or the monograph [53]). Thus, the calculation of the effective action is of high interest from the point of view of the general formalism as well as for concrete applications. The only practical method for the calculation of the effective action is the perturbative expansion in the number of loops [50, 155, 193]. All the fields are split in a background classical part and quantum perturbations propagating on this background. The part of the classical action, which is quadratic in quantum fields, determines the propagators of the quantum fields in background fields, and higher-order terms reproduce the vertices of the perturbation theory [83]. At one-loop level, the contribution of the gravitational loop is of the same order as the contributions of matter fields [93, 166]. At usual energies much lower than the Planck energy, EPIanck hc'IG Pz 1019 GeV, the contributions of additional gravitational loops are highly suppressed. Therefore, a semi-classical concept applies when the quantum matter fields together with the linearized perturbations of the gravitational field interact with the background gravitational field (and, probably, with the background matter fields) [137, 42, 129, 187, 93, 166]. This approximation is known as one-loop quantum gravity [83, 150, 91, 72, 99, 130]. To evaluate the effective action it is necessary to find, first of all, the Green functions of the quantum fields in the background classical fields of different nature. The Green functions in background fields were investigated by a number of authors. Fock [105] proposed a method for solving the wave equation

Vilkovisky-De

tion has

no

effect

on

=

Introduction

background electromagnetic field by an integral transform in the proper parameter (so called fifth parameter). Schwinger [203, 204] generalized the proper time method and applied it to the calculation of the one-loop in

time

effective action. De Witt

[80, 82]

reformulated the proper time method in it to the case of background gravitational

geometrical language and applied Analogous questions for the elliptic partial differential operators were investigated by mathematicians (see the bibliography). In the papers [34, 35] the standard Schwinger-De Witt technique was generalized to the case of arbitrary differential operators satisfying the condition of causality. The proper time method gives at once the Green functions in the neighborhood of the light-cone. Therefore, it is the most suitable tool for investigation of the ultraviolet divergences (calculation of counter-terms, 0-functions and anomalies). The most essential advantage of the proper time method is that it is explicitly covariant and enables one to introduce various covariant regularizations of divergent integrals. The most popular are the analytical regularizations: dimensional regularization, (-function regularization etc. [137, 42, 97]. There are a lot of works along this line of investigation over the last two decades (see the bibliography). Although most of the papers restrict themselves to the one-loop approximation, the proper time method is applicable at higher loops too. In the papers [157, 169, 61, 36, 59] it was applied to analyze two-loop divergences in various models of quantum field theory including Einstein's quantum gravity. Another important area, where the Schwinger-De Witt proper-time method is successfully applied, is the vacuum polarization of massive quantum h/mc, corfields by background fields. When the Compton wave length A characteristic the than much smaller is length field the mass to m, responding the method time the immediately the L of gives scale proper background field,

field.

=

expansion of the effective action in a series in the small parameter (A/L)2 [120, 121, 218]. The coefficients of this expansion are proportional to the so-called De Witt coefficients and are local invariants, constructed from the

background fields and their covariant derivatives. In the papers [119, 223] the general structure of the Schwinger-De Witt expansion of the effective action the limits was discussed. It was pointed out that there is a need to go beyond of the local expansion by the summation of the leading derivatives of the background fields in this expansion. In the paper [223], based on some additional assumptions concerning the convergence of the corresponding series and integrals, the leading derivatives of the background fields were summed non-local expression for the one-loop effective action in case of a was obtained. Thus, so far, effective and manifestly covariant methods for calculation of the effective action in arbitrary background fields are absent. All the calculations performed so far concern either the local structures of the effective

up and

a

massless field

action

or some

etc.) [137, 42].

specific background fields (constant fields, homogeneous spaces

Introduction

That is

why the development of general methods for covariant calculaaction, which is especially needed in quantum theory of gauge fields and gravity, is an actual and new area of research. There are many papers (see, among others, [8]-[40]), which are devoted to the development of this line of investigation. Therein an explicitly covariant and effective technique for the calculation of De Witt coefficients is elaborated. This technique is applicable in the most general case of arbitrary background fields and spaces and can be easily adopted to automated symbolic computation on computers [44]. In the papers [6, 12] the renormalized one-loop effective action for massive scalar, spinor and vector fields in background gravitational tions of the effective

field up to terms of order 0(1/m 6) is calculated. In spite of impressive progress in one-loop quantum gravity, a complete self-consistent quantum theory of gravitation does not exist at present [154]. The difficulties of quantum

fact that there is

arising

94]

gravity

are

connected,

in the first

line,

with the

consistent way to eliminate the ultraviolet divergences in perturbation theory [229, 95]. It was found [60, 212, 163, 54, 131, 66, no

that in the

one-loop approximation the pure Einstein gravity is finite on case of non-vanishing cosmological constant). However, two-loop Einstein gravity is no longer renormalizable on-shell [135]. On the other hand, the interaction with the matter fields also leads to nonrenormalizability on mass shell even in one-loop approximation [56, 58, 76, 75, 74, 184, 77, 206, 222, 33]. Among various approaches to the problem of ultraviolet divergences in quantum gravity (such as supergravity [221, 174, 103], resummation [79, 165, 197] etc. [229, 95]) an important place is occupied by the modification of the gravitational Lagrangian by adding quadratic terms in the curvature of general type (higher-derivative theory of gravitation). This theory was investigated by various authors both at the classical and at the quantum mass

shell

(or

levels

(see

the

renormalizable in

bibliography).

The main argument against higher-derivative quantum gravity is the presence of ghosts in the linearized perturbation theory on flat background, that breaks down the unitarity of the theory [185, 186, 180, 208, 209, 160, 214,

213, 198, 145, 2, 170]. There the

were

different attempts to solve this problem by propagator in the momentum

summation of radiative corrections in the

representation [214, 213, 198, 145], [108, 109, 111]. However, at present they cannot be regarded as convincing in view of causality violation, which results from the unusual analytic properties of the S-matrix. It seems that the problem of unitarity can be solved only beyond the limits of perturbation theory

[215]. Ultraviolet behavior of many papers

(see

the

higher-derivative quantum gravity bibliography). However, the one-loop

was

studied in

counter-terms

first obtained in the paper of Julve and Tonin [160]. The most detailed investigation of the ultraviolet behavior of higher-derivative quantum gravity were

was

carried out in the papers of Fradkin and

Tseytlin [108, 109, 111, 110].

In

Introduction

these papers, an inconsistency was found in the calculations of Julve and Tonin. The one-loop counter-terms were recalculated in higher-derivative

quantum gravity of general type

as

well

as

in

conformally

invariant models

supergravity [111, 110]. The main conclusion of the papers 109, 111, 110] is that higher-derivative quantum gravity is asymptotifree in the physical region of coupling constants, which is characterized

and in conformal

[108,

cally by the absence of tachyons on the flat background. The presence of reasonably arbitrary matter does not affect this conclusion. Thus, the investigation of the ultraviolet behavior of higher-derivative quantum gravity is an important and actual problem in the general program of constructing a consistent quantum gravity. It is this problem that was studied in the papers [22, 7]. Therein the off-shell one-loop divergences of higher-derivative quantum gravity in arbitrary covaxiant gauge of the quantum field were calculated. It was shown that the results of previous authors contain

a

numerical

error

in the coefficient of the

R'-divergent

term. The

radically changed the asymptotic properties of the conformal sector. the in Although the conclusion of [108, 42, 111, 110] theory about the asymptotic freedom in the tensor sector of the theory remains true, the conformal sector exhibits just the opposite "zero-charge" behavior in the physical region of coupling constants considered in all previous papers (see the bibliography). In the unphysical region of coupling constants, which corresponds to the positive definiteness of the part of the Euclidean action quadratic in curvature, the zero-charge singularities at finite energies correction of this mistake

absent. The present book is devoted to further development of the covariant methods for calculation of the effective action in quantum field theory and quanare

tum

gravity, and

to the

investigation of

the ultraviolet behavior of

derivative quantum gravity. In Chap. 1. the background field method is

presented. Sect.

higher--

1.1 contains

short functional formulation of quantum field theory in the form that is convenient for subsequent discussion. In Sect. 1.2 the standard proper time a

method with

some

extensions is

presented

in detail. Sect. 1.3 is concerned

questions connected with the problem of ultraviolet divergences, regularization, renormalization and the renormalization group. In Chap. 2 a manifestly covariant technique for the calculation of the with the

De Witt coefficients is elaborated. In Sect. 2.1 the methods of covariant

ex-

in curved space with axbitrary lineax connection covariant Taylor series and the Fourier integral are for-

pansions of arbitrary fields in the

generalized

quantities that will be needed later are calculated in form of covariant Taylor series. In Sect. 2.3, based on the method of covariant expansions, the covariant technique for the calculation of the De Witt coefficients in matrix terms is developed. The corresponding diagrammatic formulation of this technique is given. The developed technique enables one to compute explicitly the De Witt coeffimulated in the most

general

form. In Sect. 2.2 all the

Introduction

cients

well

analyze their general structure. The possibility to use the corresponding symbolic manipulations on computers is pointed out. In Sect. as

as

to

2.4 the calculation of the De Witt coefficients a3 and a4 at coinciding points presented. In Sect. 2.5. the one-loop effective action for massive scalar,

is

spinor and

vector fields in

to terms of order

In

Chap.

an

background gravitational

3 the

general

structure of the

Schwinger-De

expansion is analyzed and partial summation of various In Sect. 3.1

field is calculated up

1/m 4. Witt asymptotic

terms is carried out.

method for summation of the asymptotic series due to Borel e.g., [192], sect. 11.4) is presented and its application to quantum field theory is discussed. In Sect. 3.2. the covariant methods for investigations of a

(see,

the non-local structure of the effective action terms of first order in the

are

developed.

In Sect. 3.3 the

fields in De Witt coefficients

background are calculated and their summation is carried out. The non-local expression for the Green function at coinciding points, up to terms of second order in back-

ground fields, is obtained. The massless case is considered too. It is shown conformally invariant case the Green function at coinciding points is finite at first order in background fields. In Sect. 3.4. the De Witt coefficients at second order in background fields are calculated. The summation of the terms quadratic in background fields is carried out, and the explicitly covariant non-local expression for the one-loop effective action up to terms that in the

of third order in

background

fields is obtained. All the

formfactors,

their ul-

traviolet

asymptotics and imaginary parts in the pseudo-Euclidean region above the threshold are obtained explicitly. The massless case in four- and

two-dimensional spaces is studied too. In Sect. 3.5 all terms without covariant derivatives of the background fields in De Witt coefficients, in the case of scalar field, are picked out. It is shown that in this case the asymptotic series of the summation

covariantly constant terms diverges. By making use of the Borel procedure of the asymptotic series, the Borel sum of the cor-

responding semi-classical series is calculated. An explicit expression for the one-loop effective action, non-analytic in the background fields, is obtained up to the terms with covariant derivatives of the background fields. Chapter 4 is devoted to the investigation of higher-derivative quantum gravity. In Sect. 4.1 the standard procedure of quantizing the gauge theories as well as the formulation of the Vilkovisky's effective action is presented. In Sect. 4.2 the one-loop divergences of higher-derivative quantum gravity with the help of the methods of the generalized Schwinger-De Witt technique are calculated. The error in the coefficient of the RI-divergent term, due to previous authors, is pointed out. In Sect. 4.3 the dependence of the

divergences lyzed.

of the effective action

The off-shell

on

the gauge of the quantum field is

ana-

divergences of the standard effective action in arbitrary covariant gauge, and the divergences of the Vilkovisky's effective action, are calculated. In Sect. 4.4 the corresponding renormalization-group equations are solved and the ultraviolet asymptotics of the coupling constants are ob-

Introduction

7

theory there is no asymptotic freedom in the "physical" region of the coupling constants. The presence of the low-spin' matter fields does not change this general conclusion: higher-derivative quantum gravity necessarily goes beyond the limits of the weak conformal coupling at high energies. The physical interpretation of such ultraviolet behavior is discussed. It is shown that the asymptotic freedom both in tensor and conformal sectors is realized in the "unphysical" region of coupling constants, which corresponds to the positive-definite Eutained. It is shown that in the conformal sector of the

potential (i.e., the effective action on higher-derivative quantum gravity is calculated.

clidean action. In Sect. 4.5 the effective the De Sitter

background)

in

The determinants of the second- and fourth-order operators are calculated with the help of the technique of the generalized (-function. It is maintained that the result for the R 2-divergence obtained in Sect.

4.2,

as

well

as

the

results for the arbitrary gauge and for the "Vilkovisky's" effective action obtained in Sect. 4.3, are correct. Both the effective potential in arbitrary gauge and the

"Vilkovisky's" effective potential

are

calculated. The

"Vilkovisky's"

effective equations for the background field, i.e., for the curvature of De Sitter space, that do not depend on the gauge and the parametrization of the

quantum field, results

are

obtained. The first quantum correction to the background by the quantum effects is found. In Conclusion the main

are

curvature caused

summarized.

Background Field Method Quantum Field Theory

1.

in

1.1

Generating Functional,

Green Functions

and Effective Action Let

us

consider

an

arbitrary field o(x)

on a

n-dimensional

space-time given by

VA(X) that transform with respect to some (in representation of the diffeomorphism group, i.e. the group

its contravariant components

general, reducible) of general transformations

of the coordinates. The

field components

WA(X)

can be of both bosonic and fermionic nature. The fermionic components treated as anticommuting Grassmanian variables [41], i.e.,

OAW B

(_,)AB(PBWA

=

(-1)

where the indices in the exponent of the indices and to 1 for the fermionic ones. For the construction of metric of the

configuration

a

space

EAB, i.e.,

((P1 (P2)

=

i

that enables

one

AB

where E- 1

=

equal

are

OB EBA

VB

i

scalar

S( o)

to 0 for bosonic

one

also needs

=

a

product

(PA1 EAB (PB 2

to define the covariant fields

PA

(1. 1)

,

local action functional a

are

(1.2)

components

VA E-1

AB

(1.3)

,

is the inverse matrix E

-

1

ABEBC

=

6AC

,

EAcE

-1 CB =

jAB.

(1.4)

non-degenerate both in bose-bose and fermi-fermi satisfy the supersymmetry conditions

The metric EAB must be sectors and

EAB

=

In the

case

ghost

fields

of

(_l)A+B+AB EBA

E-1

gauge-invariant field

are

AB =

)

theories

(_,)AB E-1

we assume

included in the set of the fields

OA

BA.

that the

(1-5)

corresponding

and the.action

S(W)

is

by inclusion of the gauge fixing and the ghost terms. To reduce the writing we will follow, hereafter, the condensed notation of De Witt [80, 83] modified

I. G. Avramidi: LNPm 64, pp. 9 - 20, 2000 © Springer-Verlag Berlin Heidelberg 2000

1.

10

Background

Field Method

and substitute the mixed set of indices time

small Latin index i

point, by summation-integration should be done one

=_

(A, x), where x labels the spaceA (A, x): pi V (X). The combined =

over

the

repeated

upper and lower

small Latin indices

W,

j(p'2

= -

f

dnX WI

A

(X)WA2 (X)

(1.6)

.

Now let us single out two causally connected in- and out-regions in space-time, that lie in the past and in the future respectively relative to region, which is of interest from the dynamical standpoint. Let us define vacuum states lin, vac > and lout, vac > in these regions and consider vacuum-vacuum

transition

some

the the the

amplitude

(out, vac I in, vac) in presence of

the

=_

expiW(J) I h

background classical

sources

(1-7)

Ji vanishing in in- and

out-

regions.

amplitude (1.7) can be expressed integral (or path integral) [50, 155, 193] The

i exp

h

fd o

w (j)

in form of

M (W) exp

a

functional

formal

[S (W) + Ji (p']

h

(1-8)

functional, which should be determined by the theory [117, 223]. The integration in (1.8) should quantization be taken over all fields satisfying the boundary conditions determined by The functional W(J) is of the vacuum states I in, vac > and I out, vac > central interest in quantum field theory. It is the generating functional for the Schwinger averages where M (W) is

a measure

of the

canonical

.

Wik)

exp

iW(J) (h)k 6ji, i

6Lk ...

jji"

exp

h

W (J)

(1 -9)

where

(F (W)) 6L is the left functional ordering.

E

(out, vac I T (F (W)) I in, vac) (out, vac I in, vac)

derivative and 'T' is the operator of

The first derivative of the functional to the tradition

we

(1.10)

will call it the

W(J) gives the background field)

V(J) the second derivative determines the

6L =

6ii

W (J)

mean

,

one-point propagator

chronological

field

(according

1.1

Generating Functional,

(ViVk) gik (j) and the

higher

derivatives

9ti

...

Green Functions and Effective Action

pi pk

=

hgik

+

(1.12)

i

=

6ji6jk

W (J)

give the many-point Green functions zk

6kL

V)

(1.13)

_W(j) 6ii"

...

the effective

generating functional for the vertex functions, called F(fl, is defined by the functional Legendre transform:

The action

-V(4i) where the

by

=

W(J)

-

JiV,

(1.14)

expressed in terms of the background fields, equation !P -P(J), (1.11). derivative of the effective action gives the sources

sources are

inversion of the functional

The first

6R

11

J

=

J(fl,

=

F(fl

-li(!P)

=

-Ji(fl

(1.15)

,

the second derivative determines the one-point propagator

6LJR 60i 64jk

-V (fl

Dik (0)

=

Dik

,

Dik gkn where 6R is the

==

_6in

=

(-1) i+k+ik Dki

,

(1.16)

,

6A J(x, x'), and 6(x, x') right functional derivative, 6, higher derivatives determine the vertex functions n

=

is

the delta-function. The

k

ri Rom the definition

functional

exp

equation

6R=.

ik

(1.14)

p(!p)

(1.17)

equation (1.8)

it is easy to obtain the

&Pil and the

...

j4izk

for the effective action

WiF(flj f dWM( p)

exp

_i [S(V) h

-

_1 j(4i%o'

-

V)]

(1-18)

Differentiating the equation (L 15) with respect to the sources one can express all the many-point Green functions (1.13) in terms of the vertex functions (L 17) and the one-point propagator (1. 12). If one uses the diagrammatic technique, where the propagator is represented by a line and the vertex functions by vertexes, then each differentiation with respect to the sources adds a new line in previous diagrams in all possible ways. Therefore, a many-point Green function is represented by all kinds of tree diagrams with a given number of external lines.

12

1.

Background

Thus when

S-matrix

Field Method

using the effective

(when it exists)

corrections determined

one

action functional for the construction of the

needs

only the tree diagrams, since all quantum are already included in the full one-point functions. Therefore, the effective equations

by the loops

propagator and the full

vertex

0

(1.19)

,

(in absence of classical sources, J 0) describe the dynamics of the background fields with regard to all quantum corrections. The possibility to work directly with the effective action is an obvious advantage. First, the effective action contains all the information needed to construct the standard S-matrix [78, 161, 223]. Second, it gives the effective =

equations (1.19) that enable one to take into account the influence of the quantum effects on the classical configurations of the background fields [122,

225]. In

practice, the following difficulty appears on this way. The background fields, as well as all other Green functions, are not Vilkovisky'sly defined objects. They depend on the parametrization of the quantum field [223, 224]. Accordingly, the effective action is not Vilkovisky's too. It depends essentially on the parametrization of the quantum field off mass shell, i.e., for background fields that do not satisfy the classical equations of motion

S'i(4 ) On

0.

=

(1.20)

shell, (1.20), the effective action is a well defined quantity and leads [78, 161, 138, 155]. A possible way to solve this difficulty was proposed in the papers [223, 224], where an effective action functional was constructed, that is explicitly invariant with respect to local reparametrizations of quantum fields (so called Vilkovisky's effective action). This was done by introducing a metric and a connection in the configuration space. Therein, [223, 224], the "Vilkovisky's" mass

to the correct S-matrix

effective action for the gauge field theories was constructed too. We will study the consequences of such a definition of the effective action in Chap. 5 when

investigating the higher-derivative quantum gravity. This aproach was improved further by De Witt [86]. The formal scheme of quantum field theory, described above, begins to take on a concrete meaning in the framework of perturbation theory in the number of loops [50, 155, 193] (i.e., in the Planck constant h):

_V(,p)

=

S( p)

+

E hk r(k) (C

(1.21)

k>1

the expansion

in (1.18), shifting the integration variable pi +,,Ah-hi, expanding the action S(W) and the measure M (W) in quantum fields h' and equating the coefficients at equal powers of h, we get the recurrence relations that uniquely define all the coef-

Substituting

in the functional

ficients

T(k).

integral Wi

(1.21) =

All the functional

integrals

are

Gaussian and

can

be calculated

1.1

Generating Functional,

Green Functions and Effective Action

13

in the standard way [191]. As the result the diagrammatic technique for the effective action is reproduced. The elements of this technique are the bare

one-point propagator, i.e., the Green function of the differential operator

Aik (W)

`

jW i Wk

and the local vertexes, determined sure M (W). In

particular,

the

SM

(1.22)

1

by the classical

one-loop effective

2i

S(W)

and the

mea-

action has the form

sdet A

1

T(j) (fl

action

log

(1.23)

M2

where sdet A is the functional Berezin

str F

exp

superdeterminant [41], and

=

(- 1)'F'i

is the functional supertrace. The local functional measure

perdeterminant

(1.24)

(str log A)

=

M(V)

of the metric of the

M

f

dnX

(_l)A FAA (X)

can

be taken in the form of the

configuration

(sdet Eik (V))

=

(1.25)

su-

space

1/2

(1.26)

where

Eik(W) In this

case

=

EAB(W(X))J(X,XI)

configuration space that

is the volume element of the

dWM (W)

(1.27)

-

point transformations of the fields: W(x) -+ F(W(x)). the multiplicativity of the superdeterminant [41], the one-loop effective

is invariant under the

Using

action with the

measure

(1.26)

can

r(l) (C

be rewritten

in

the form

1log sdot. i

(1.28)

2i

with sdet The local

measure

M (W)

can

in the

3ik

E-1

in

be also chosen in such

theory, proportional divergences coinciding points 6(0), vanish [115, 117]. ultraviolet

(1.29)

Ank a

way, that the

leading

to the delta-function in

14

1.

Background

Field Method

1.2 Green Functions of Minimal Differential

Operators

The construction of Green functions of arbitrary differential operators (1.22), (1.29) can be reduced finally to the construction of the Green functions of the "minimal" differential operators of second order [35] that have the form "-I' k where 1:1 ant

=

=

JjAB (E] _M2) + QAB (X)J g112(X)j(X, XI)

g/"V,,V,

derivative,

is the covariant D'Alambert

defined

by

V/, o g"' (x)

is the

is the

of

means

A

alVA

=

metric of the

some

operator, V. is the covari-

background

+

connection

A,,(x),

AABIAP B,

(1-31)

background space-time,

parameter of the quantum field and matrix-valued function (potential term). m

(1-30)

,

mass

g (x)

QAB (x)

det g,,, (x), is

an

arbitrary

The Green functions GA (x, x') of the differential operator (1.30) are twoB point objects, which transform as the field oA(X) under the transformations of coordinates at the point x, and as the current JB, (x') under the coordinate transformations at the point x'. The indices, belonging to the tangent space at the point x', are labeled with a prime. We will construct solutions of the equation for the Green functions ,

f jAC (E] _,rn2) with

+

QAC I GCB (x, x1) ,

=

_6AB9

-

112(X)j(X,Xl)

(1-32)

,

appropriate boundary conditions, by means of the Fock-Schwinger[105, 203, 204, 80, 82] in form of a contour

De Witt proper time method

integral

over an

auxiliary variable G

f

s,

ids

exp(_iSrn2)U(S)

(1-33)

C

where the "evolution function"

(or

the heat

kernel) U(s)

=

UA,(Slx,x B

satisfies the equation

is U(s) with the

boundary

El

boundary

The evolution equation tial

,

6AB,

(1-34)

condition

UA, B (SIX, Xi) where OC is the

+Q) U(s)

lac

=

-

6AB9 _11'(X)6(X, X')

(1.35)

,

of the contour C.

(1.34)

is

as

difficult to solve

exactly

as

the ini-

equation (1.32). However, the representation of the Green functions in form of the contour integrals over the proper time, (1.33), is more convenient to use for the construction of the asymptotic expansion of the Green

1.2 Green Functions of Differential

15

Operators

study of the behavior light-cone, x -+ x1, as regularization and renormalization of the divergent vacuum

functions in inverse powers of the mass and for the of the Green functions and their derivatives on the well

for the

as

expectation values of local variables (such as the energy-momentum tensor, one-loop effective action etc.). Deforming the contour of integration, C, over s in (1.33) we can get different Green functions for the Green function

same

evolution function. To obtain the causal

(Feynman propagator)

has to integrate

one

infinitesimal negative imaginary part to the this contour that we mean hereafter. and add

oo

an

single out in the evolution function reproduces the initial condition (1.35) at s Let

us

U(s) where x

and

2

from 0 to

[80, 82].

It is

rapidly oscillating factor that

0:

i(47rs) -n/2 A1/2 exp

=

is half the square of the

a(x, x')

a

over s m

2is

geodesic

) P 0 (s)

(1.36)

,

distance between the

points

x',

,A(X, X')

=

-g-1 /2 (x) det (_Vtl, VVa(X, X1)) g-1/2 (XI)

(1.37)

pAB (X, XI) is the parallel disis, the Van Fleck-Morette determinant, P from the point x, to the the field the of geodesic along placement operator is there one that connecting the points x We assume x. geodesic only point =

,

being not conjugate, and suppose the two-point -pAB (X, xi) to be single-valued differentiable x') (x, x'), functions of the coordinates of the points x and x'. When the points x and x1 are close enough to each other this will be always the case [211, 80, 82]. and

x',

the

functions

points

x

and x'

and

A (x,

a

,

The introduced "transfer function" f2 (S) scalar at the

primed).

point

x

and

This function is

as a

regular

in

at

s

flA" B (01X, Xt) independently If

on

S?A" B (S I X, X')

=

transforms

point x' (both the point s 0, i.e.,

matrix at the

its indices

as a

are

=

1X1X1

_-:

6A1BI

(1.38)

the way how x -+ x'. that there are no boundary surfaces in

one assumes

space-time (that

close to the point x = x, for any s, i.e., there exist finite coincidence limits of the and its derivatives x' that do not depend on the way how x approaches x'. at x we

will do

hereafter),

then the is

analytic

also in

x

=

Using

the equations for the introduced functions

[80, 82],

1 a

=

-9 Ila

a/, VIY

=

Up VtJ

0

,

log 'Al/2

(1.39)

VA07,

01A

2

pAB (Xi, Xt) ,

=

6A'BI

,

(1.40)

1 =

2

(n

-

El

a)

,

(1.41)

1.

16

obtain from

we

a ais If

(1.34)

and

(1.36)

the transfer

1 +

S?(S)

_U

is

p-1

=

equation for the function S?(s):

(i,/A-1/2 E] A1/2

solves the transfer equation

one

variable

Field Method

Background

(1.42)

+

Q P J? (s)

in form of

(1.42)

power series in the

a

s

f2(s)

(1.43)

k! k>O

then from

(1.38)

(1.42)

and

al'V,,ao 1

kcAV,.

+

)

0

=

=P

ak

gets the

one

recurrence

A" (X i, X i)

aO B

,

-1/2 E]

-1

A1/2

=

+

relations for the ak

6A'B1

1

Q) Pak-1

(1.44)

(1.45)

The coefficients ak (X, x') are widely known under the name "heat kernel coefficients", or "HMDS (Hadamard-Minakshisundaram-De Witt-Seeley) coefficients", according to the names of the people who made major contributions to the study of these objects (see [148, 176, 78, 205]). The significance of these coefficients in theoretical and mathematical physics is difficult to overestimate. In this book, following the tradition of the physical literature, we

call these coefficients "De Witt coefficients".

Rom the

equations (1.44)

it is easy to find the zeroth coefficient

A" (X, Xi)

ao B

The other coefficients

(1.45)

and

taking

are

calculated

=

6A'B1

(1.46)

-

usually by differentiating the relations

the coincidence limits

[80, 82].

However such method of

calculations is very cumbersome and non-effective. In this way only the coefficients a, and a2 at coinciding points were calculated [80, 62, 63]. The same

coefficients

as

well

in the paper [132] The coefficient a4 papers

[12, 11, 9]

as

by

was

in

computed completely independently in [4] and in our a manifestly covariant method for calculation of the

where

De Witt coefficients

computed

the coefficient a3 at coinciding points were calculated of a completely different non-covariant method.

means

[219].

was

elaborated. The coefficient a5 in the flat space was computation of the coef-

Reviews of different methods for

ficients ak

along with historical comments are presented in [24, 202, 21]. An interesting approach for calculating the heat kernel coefficients was developed

in recent papers [188, 189]. In the Chap. 2 we develop

and effective

technique

coefficients ak

as

reformulated in and

can

be

well

as

a

manifestly

covariant and very convenient explicitly arbitrary De Witt

that enables to calculate to

analyze their general structure. This technique was the elaborated technique is very algorithmic

[220]. Moreover,

easily realized

on

computers

[44].

1.3

Divergences, Regularization

and Renormalization

17

Let us stress that the expansion (1.43) is asymptotic and does not reflect possible nontrivial analytical properties of the transfer function, which are very important when doing the contour integration in (1.33). The expansion in the power series in proper time, (1.43), corresponds physically to the expansion in the dimensionless parameter that is equal to the ratio of the Compton wave length, A h/mc, to the characteristic scale of variation of the background fields, L. That means that it corresponds to the expansion =

in the Planck constant h in usual units

[120, 121].

This is the usual semi-

classical

approximation of quantum mechanics. This approximation is good for the study of the light-cone singularities of the Green functions, for

enough the regularization

and renormalization of the

divergent

coincidence limits of

well

the Green functions and their derivatives at

a

the calculation of the

of the massive fields in the

space-time point,

vacuum polarization Compton wave length A is much smaller length scale L, A/L h/(mcL) < 1. At the same time the expansion in powers of the

when the

as

as

for

case

than the characteristic

=

not contain any information about all effects that

proper time

(1.43)

does

depend non-analytically

(such as the particle creation and the vacuum of massless polarization fields) [82, 121]. Such effects can be described only of the summation by asymptotic expansion (1.43). The exact summation in on

the Planck constant h

is, obviously, impossible. One can, however, pick up the leading approximation and sum them up in the first line. Such partial summation of the asymptotic (in general, divergent) series is possible only by employing additional physical assumptions about the analytical structure of the exact expression and corresponding analytical continuation. In the Chap. 3 we will carry out the partial summation of the terms that a,re linear and quadratic in background fields as well as the terms without the covariant derivatives of the background fields.

general

case

terms in

1.3

some

Divergences, Regularization

and Renormalization

problem of quantum field theory is the presence of the ultravidivergences that appear in practical calculations in perturbation theory. They are exhibited by the divergence of many integrals (over coordinates and momentums) because of the singular behavior of the Green functions at small distances. The Green functions are, generally speaking, distributions, A well known

olet

i.e., linear functionals defined on smooth finite functions [50, 155]. Therefore, numerous products of Green functions appeared in perturbation theory cannot be defined

correctly.

A consistent scheme for

eliminating the ultraviolet divergences and obtaining finite results is the theory of renormalizations [50, 155], that can be carried out consequently in renormalizable field theories. First of all, one has to introduce an intermediate regularization to give, in some way, the finite values to the formal divergent expressions. Then one should single out the

18

1.

Background

Field Method

divergent part and include the counter-terms in the classical action that compensate the corresponding divergences. In renormalizable field theories one introduces the counter-terms that have the structure of the individual terms of the classical action. They are interpreted in terms of renormalizations of the

fields, the masses and the coupling constants. By the regularization some new parameters are introduced: a dimensionless regularizing parameter r and a dimensional renormalization parameter p. After subtracting the divergences and going to the limit r -+ 0 the regularizing parameter disappears, but the renormalization parameter p remains and enters the finite renormalized expressions. In renormalized quantum field theories the change of this parameter is compensated by the change of the coupling constants of the renormalized action, gj(p), that are defined at the renormalization point characterized by the energy scale P. The physical quantities do not depend on the choice of the renormalization point /-I where the couplings are defined, i.e., they are renormalization. invariant. The transformation of the renormalization parameter p and the compensating transformations of the parameters of the renormalized action gi(p) form the group of renormalization transformations [50, 226, 229]. The infinitesimal form of

these transformations determines the differential equations of the renormalization group that are used for investigating the scaling properties (i.e., the behavior under the

homogeneous

scale

transformation)

of the renormalized

coupling parameters gj(p), many-point Green functions and other quantities. In particular, the equations for renormalized coupling constants have the

the form

[229] d P

d1i

Pi(A)

=

M90,O)

(1.47)

where gi (p) = /'L-di gj(p) are the dimensionless coupling parameters (di is the dimension of the coupling gi), and Pj(g) are the Gell-Mann-Low #-function. Let

us

note, that among the parameters

couplings [229] (like are

gi(l-i)

there

are

also non-essential

the renormalization constants of the fields

not invariant under the redefinition of the fields. The

renormalization constants

Z,(p)

have

more

Z,(P))

that

equations for the

simple form [229]

d P

dp

Z' 0")

=

-/, (9 W) Z' (/')

,

(1.48)

where y, (g) are the anomalous dimensions. The physical quantities (such as the matrix elements of the S-matrix the

shell)

do not

depend on the details of the field definition and, couplings. On the other hand, the off mass shell Green functions depend on all coupling constants including the non-essential ones. We will apply the renormalization. group equations for the investigation of the ultraviolet behavior of the higher-derivative quantum gravity in Chap. on

mass

therefore,

4.

on

the non-essential

1.3

Let

illustrate the

us

19

Divergences, Regularization and Renormalization

procedure

of

the ultraviolet

eliminating

divergences

by the example of the Green function of the minimal differential operator (1.30) at coinciding points, G(x, x), and the corresponding one-loop effective action

F(j), (1.28). Making use of the Schwinger-De (1.33) we have

Witt

representation

for

the Green function

00

G(x, x)

=

f

i ds

i(47ris) -n/2 exp(_iSrn2)fl('9jX'X)

(1.49)

,

0

0')

d SI'9

I

(4,,i,) -n/2 exp (_iSrn2)

d nX

g1/2 str S? (s I x, x)

(1.50)

0

It is clear that in four-dimensional

space-time (n

=

4)

the

integrals

over

the proper time in (1.49) and (1.50) diverge at the lower limit. Therefore, they should be regularized. To do this one can introduce in the proper time integral a

regularizing function p(isp 2; r)

that

depends

on

the

regularizing parameter

and the renormalization parameter IL. In the limit r -+ 0 the regularizing function must tend to unity, and for r 0 0 it must ensure the convergence of r

approach zero sufficiently rapidly at by a polynomial). The concrete form of the function p does not matter. In practice, one uses the cut-off regularization, the Pauli-Villars one, the analytical one, the dimensional one, the (-function regularization and others [42, 50, 155]. The dimensional regularization is one of the most convenient for the prac-

the proper time integrals s -+ 0 and be bounded at

(i.e.,

it must

s -+ oo

(especially in massless and gauge theories) as well as for general investigations [207, 193, 42], [60, 212, 163, 54]. The theory is formu-

tical calculations

lated in the space of arbitrary dimension n while the topology and the metric 4 dimensions can be arbitrary. To preserve the physical of the additional n -

dimension of all

quantities

in the n-dimensional

space-time

it is necessary to

introduce the dimensional parameter p. All integrals are calculated in that region of the complex plane of n where they converge. It is obvious that for Re n < C, with some constant C, the integrals (1.49) and (1.50) converge and

analytic functions of the dimension n. The analytical continuation of these functions to the neighborhood of the physical dimension leads to sin4. After subtracting these singularities we obtain gularities at the point n the in vicinity of the physical dimension, the value of this analytical functions 4 defines the finite value of the initial expression. function at the point n define

=

=

Let

lytical

us

make

some

remarks

on

of the dimension

n

is not

regularization. The anatheory to the complex plane

the dimensional

continuation of all the relations of the

single-valued, since the values of integer values of the argument

variable at discrete

a

function of

do not define

complex the unique analytical function [210]. There is also an arbitrariness connected with the subtraction of the divergences. Together with the poles in (n 4) -

one can

also subtract

some

finite terms

(non-minimal renormalization).

It is

20

1.

Background

Field Method

also not necessary to take into account the dependence on the dimension of some quantities (such as the volume element d' x g'/2 (x), background fields,

quantities

On the other

etc.).

curvatures

on

the additional

and then calculate the to

hand,

n

-

integrals

specify

one can

4 coordinates in

over

the

additional factor that will

n

-

the

some

dependence of all special explicit way

4 dimensions. This would lead

when

give, expanding in n 4, additional finite terms. This uncertainty affects only the finite renormalization terms that should be determined from the experiment. an

Using and

-

the asymptotic expansion (1.43) we obtain in this way from (1.49) the Green function at coinciding points and the one-loop effective

(1.50)

action in dimensional

G(x,x)

i =

regularization M2

I G-4 +C+ log Ii-r-p7 ) (M 2

_F47rF

2 _

a,

(x, x))

-

m

21

(1.51)

+Gren (X, X) Gren (X) X)

FM

1 ":::

=

2

-2(-4F7r7

2

n-4

ak

1)2

(41r

+ C +

k>2

k(k M2

log T-Irk-7

-

(x, x) 1),M2(k-1)

) (M4

-

2M2AI

(1.52)

+

A2)

(1.53) +IM4 Ao 4 1

-1'(l)ren where C

;z

':_

2(47r)',

M2A,

E k(k

k>3

r(l)ren

+

Ak -

1)(k

-

2)M2(k-2)

(1-54)

0.577 is the Euler constant and

Ak

=-

f

dnX g1/2 str ak (X,

X)

-

(1.55)

Here all the coefficients ak and Ak are n-dimensional. However in that part, analytical in n 4, one can treat them as 4-dimensional.

which is

-

2.

for Calculation

Technique

of De Witt Coefficients

2.1 Covariant Let

us

single

out

a

connect any other

where

small

point

x

Chap. 1,

is

Space

in the space, fix

regular region

with the point x'

affine parameter. The world function a (x, x') -r

in Curved

Expansions

by

a

a

geodesic x

point x' =

in

it, and

x(-r), x(O)

=

x',

an

of

(in terminology

[211]),

introduced in the

has the form 0'

(X, X')

=

1Ir

2

b2 (,r)

2

(2.1)

,

where d

(2.2) The first derivatives of the function

proportional

to the

tangent

u(x, x')

vectors to the

-rV, (-r)

with respect to coordinates are at the points x and x'

geodesic

a,'

,

--rV'(0)

,

(2.3)

where

VI'a From

here,

it

follows,

in

particular,

VI" a

0'/,,

,

the basic

.

identity (1.39), that the function

a(x, x') satisfies, (D

-

2)u

0

=

D

,

a"VA

,

(2.4)

the coincidence limits

[0]

=

[a,]

V (X, X')]

=

=

[a" I1

0

(2.5)

,

lim, f (X, X')

,

(2.6)

X-+X

and the relation between the tangent vectors UIA

A =

-9

V

,

OVP

(2.7)

g'V,(x,x') is the parallel displacement operator of vectors along the geodesic from the point x' to the point x. The non-primed (primed) indices are lowered and risen by the metric tensor in the point x (xi). By differentiating the basic identity (2.4) we obtain the relations

where

I. G. Avramidi: LNPm 64, pp. 21 - 49, 2000 © Springer-Verlag Berlin Heidelberg 2000

2. Calculation of De Witt Coefficients

22

(D

1)o,"

-

(D

-

=

1)u"

0,

a"

0,

a"

=

61'

a

(2.8)

,

V

77'"' a,

(2.9)

where

V"O"'

(2.10)

77

Therefrom the coincidence limits follow

(2.11)

V

lvu' Let

us

...

V4")Uvl

consider

a

(k > 2). lv('- vxou" 11 0 OA (x) and an affine connection A,, o

(2.12)

=

...

field

=

that defines the covariant derivative

(1.31)

AAB W, t'

and the commutator of covariant

derivatives,

[V1" VVhO

=

Ri,v =,91,Av -,OA4 Let

(2.13)

72-11VV +

[Al, A,]

(2.14)

define the

us parallel displacement operator of the field V along the -pAB (x, x'), to be the solution geodesic from the point x' to the point X, p of the equation of parallel transport, =

D'P

,

(2.15)

0,

=

with the initial condition

['P1 ='P(x,x) Rom here

=

i

(2.16)

-

obtain the coincidence limits

one can

[V(J' In

...

VJ") P]

=

0

(k

,

>

1)

(2.17)

particular, when V V11 is a vector field, and the connection A,, F',4,6 is connection, the equations (2.15) and (2.16) define the paxallel displacement operator of the vectors: P g" (x, x'). Let us transport the field p parallel along the geodesic to the point x' =

=

the Christoffel

=

V

0_ where P-1

=

=

0_

C'

Pc'(x, x) A

(X)

is the

posite path (from the point

x

'P

=

'(Xi, X) VA (X)

A

The obtained

considering oTaylor series

as a

,

parallel displacement operator along point x' along the geodesic):

object o-, (2.18), point

(P

(2.18) the op-

to the

P'P-1

mations at the

,

is

a

=

i

(2.19)

-

scalar under the coordinate transfor-

x, since it does not have any

non-prime indices. By us expand it in the

function of the affine parameter r, let

2.1 Covariant

Expansions

Space

23

d

1

1 ,_Tk 1

E k! 7k

(2.20)

('0

r=o

k>O

Noting that dldr

in Curved

EAo9,,, 0,, o- V,, o- and using the equation of the geodesic, 0, and the equations (2.3), (2.9) and (2.18), we obtain =

=

I

W=P E

UAI

...

k!

aAk WA ,..11,

(2.21)

VA.)

(2.22)

1

k

k>O

where

(2.21)

The equation field with Let

us

arbitrary

is the

covariant

generalized

affine connection in

(2.21)

show that the series

Taylor

series for

a curved space. is the expansion in

a

arbitrary

complete

set

,

eigenfunctions of the operator D, (2.4). The vectors a/' and a" are the eigenfunctions of the operator D with the eigenvalues equal to 1 (see (2.8) of

and

(2.9)). Therefore,

construct the

one can

eigenfunctions

with

arbitrary

positive integer eigenvalues:

10 In

>-=

Ii/1...Vn'

1,

>=-

Or-1

>=

...

U'n'

(n

n!

DIn

nIn

>=

>

(2.23)

1)

(2.24)

>

We have n

In,

> (9

...

&

Ink

(nl,... nk) In

>=

(2.25)

>

where

n!

n

ni!

nl,..., nk

Let

(u)' In

us

...

note that there exist

> with

D

(o)'In

>=

(n

=

nj +

+ nk

general eigenfunctions

more

arbitrary eigenvalues (n

n

'

nk!

of the form

2z)

+

2z)(a)'In

+

(2.26)

.

(2.27)

>

However, for non-integer or negative z these functions are not analytic in coordinates of the point x in the vicinity of the point x. For positive integer z they reduce to the linear combinations of the functions (2.23). Therefore, we restrict ourselves to the functions (2.23) having in mind to study only regular fields near the point x. Let

us

<

MI

introduce the'dual functions =<

and the scalar

JL'1

-

-

-

A' I M

product

=

(-l)'gl,'

-

-

-

gl,-V(",

-

-

-

V,,-)J(X, X')

(2.28)

2. Calculation of De Witt Coefficients

24

<

mIn

>=

f

d nX <

Using

the coincidence limits

of the

eigenfunctions (2.23) and (2.28)

mIn

<

>=

(2.11)

6mnl(n)

IL,1

is

=

it is easy to prove that the set

orthonormal. V1-Vn

6111

*

Mko

Therefore,

the covariant

>=

Taylor

lv(,.

...

6VI (A

...

6V-

(2.30)

An)

to the coincidence limit of the

V/,m) 01

(2.21)

series

=

-/An

The introduced scalar product (2.29) reduces symmetrized covariant derivatives, <

(2.29)

>

...

(2.12)

and

1(n)

I

ILMIV,1 Vn'

...

can

(2-31)

be rewritten in

a

compact

form

IW

In

>

><

n

IW

>

(2.32)

.

n>O

From here it follows the condition of

functions

(2.23)

completeness of the set of the eigenregular in the vicinity of the

in the space of scalar functions

point x'

In

><

nj

(2.33)

,

n>O

or,

more

precisely,

5 (X,

Y)

E n! uAl (X, XI)

=

-

...

OrAn

(X, Xl)gUl (y, X') A

go, (y, XI) An n

...

I

n>O

X

Let

us

note

VY

that,

*

,

"

Vyn) J(Y, X')

since the

(2-34)

.

parallel displacement operator P

is

an

eigen-

function of the operator D with zero eigenvalue, (2.15), one can also introduce 1. a complete orthonormal set of "isotopic" eigenfunctions P I n > and < n I P -

employed present an complete eigenfunctions (2.23) can arbitrary linear differential operator F defined on the fields p in the form set of

The

be

PIm ><,mjP-1F'Pjn

F

><

to

n

IP-1

(2-35)

m,n>O

where

<

are

mIP-'FPln

>=

IV(",

Vi,_)P-'FP

the "matrix elements" of the operator F

(2.36)

are

expressed finally

avi n1

(2.35).

...

a

V.,

(2.36)

The matrix elements

in terms of the coincidence limits of the deriva-

tives of the coefficient functions of the

operator P and the world function

a.

operator F, the parallel displacement

Covahant Expansions in Curved

2.1

Space

25

For calculation of the matrix elements of differential operators (2.36) as as for constructing the covariant Fourier integral it is convenient to make

well a

change

of the variables X/I

to consider

i.e.,

a" (x, x)

a

X1,

=

(a"

X'\ I)

,

function of the coordinates xO

and the coordinates

(2.37)

1

as

the function of the vectors

x

The derivatives and the differentials in old and nected

by

'90 dxl'

=

=

09"

77""9 "

7

dav'

y",dav'

=

V

where

a,,

new

variables

are

con-

the relations

=

ala.4", 7v,,

alax,4,

V

lqi'

?7v'dx"

is defined in

(2.38)

,

(2.10), -y"v,

are

the

elements of the inverse matrix, =

ly

and 77 is a matrix with elements From the coincidence limits

(2-39)

77

77v" it follows that for close

(2.11)

points

x

and

XI det 77

0

det-y 54

,

0

(2.40)

,

change of variables (2.37) is admissible. corresponding covariant derivatives are connected by analogous

and, therefore, The

:A

the

rela-

tions

V1,

V/ =

77

A

VV,

VV,

Rom the definition of the matrices q, relations

V]

vt"

0

=

-Y"V, V A

and 7,

(2. 10),

Vt" -Y A,

,

V

(A-1

77 1,V

=

g1/2 (XIW 1/2 (x) det(-77)

=

det(-q)

=

(2.39),

one can

=0,

get the

(2.42) (2.43)

0

where A is the Van Fleck-Morette determinant

' A(X' XI)

(2.41)

I

(1.37)

g1/2 (X I)g-1/2 (x) det(--y) -' (2.44)

and q, y- and X

are

=

(det(-ry))-'

(det X)112

=

matrices with elements

q'V,

=

X/'V'

gVV' 77"V X AIVI

77

/I/

IL9

ItV

=

g'"JI-Y'VI

V/

77

V

.

(2.45) (2.46)

2. Calculation of De Witt Coefficients

26

Let

us

expressed

note that the dual

in terms of the

<

Mi

eigenfunctions (2.28)

I

I

=< Al

...

-

(2.32)

and

can

V('U'

PM I

Therefore, the coefficients of the

of the operator D

V/11 )J(X'X 1)

...

1

covariant

MJW

>=

be

(-J)M

The commutator of the operators operator P, has the form

lv(t"

V,

(2.47)

.

(2.21), (2.22), (2.31)

series

Taylor

V:

be also written in terms of the operators

<

can

(2.41),

operators

(2.48)

-

when

acting

on

the

parallel displace-

ment

[Vt" V" IP

=

R/" " P

(2.49)

,

where

(2.50) V/,, A,,,

-

V,,, Al,, +

[AA1, A,,,]

,

(2.51)

Al,, The

here satisfies the

quantity (2.51) introduced

o,," A,,

=

0

equation (2.15),

(2-52)

-

On the other hand, when acting on the objects D, (2.18), that non-primed indices, the operators V commute with each other

lvw VVIbo,

Thus the vectors

o,"'

and the operators

to that of usual coordinates and the

space at the

point x'. In

=

0

-

do not have

(2.53)

V,,,, (2.41), play the role analogous

operator of differentiation in the tangent

particular,

[VJ"

(2.54)

Therefore, one can construct the covariant Fourier integral in the tangent space at the point x' in the usual way using the variables al". So that for the fields D,

(2.18),

we

D- (k)

have

=f d'xg'l'(x).A(x,x')exp(ik,,,ui") o-(x) d1kA'

0- W

Note,

f (27r)n gll'(x') exp(-ik.,al") p-(k)

that the standard

rule,

(2.55)

2.2 Elements of Covaxiant

d'ki"

VA, P W takes

=

f (27r)n

9

1/2

27

Expansions

(x')exp(-ik,,,oi")(-ik,,,) o-(k)

(2.56)

,

place and the covariant momentum representation of the delta-function

has the form

dnkl"

6(x, Y)

f (27r)n 9112(Xt)g112 (X),A (X, X')

=

x

exp

IikA, (a"' (y, x)

2.2 Elements of Covariant Let

calculate the

us

By differentiating derivatives we get

o,-"'(x, x')) I

(2.57)

.

Expansions

71, -y and X introduced in

quantities

(2.8)

the equations

D6

-

6(6

+

and

i)

-

D77 +,q(6

(2.9)

+ S

=

and

previous section. commuting the covariant

(2.58)

0

(2-59)

0

-

where

S

6'V V

(2-60)

a

substituting the solution By solving (2.59) with respect to the matrix obtain the linear equation for the account and into we (2.39) taking (2.58)

matrix

(2.45)

ji (D g"'A gv

where

V

V

2

+

D)

One as a

can

solve the equation

=

-1

(2.61)

0,

+

S"V , with the boundary

,

PY-1

condition, (2.39), (2. 11),

(2.62)

-

(2.61) perturbatively, treating

the matrix

perturbation. Supposing

(2.63)

+ ly

'ZY we

PV

RIv,3a'o,,8

S" in

S'V

=

obtain

ji (D Rom here

we

(D

have 2

+

2

+

D)

0

+

(2.64)

.

formally

D)

+

9}_1 3

=

E(-1) +' I (D k>1

2

+

D) -lg I

k .

i

(2.65)

28

Calculation of De Witt Coefficients

2.

The formal expression in the

eigenfunctions

(2.65)

becomes

meaningful in terms of the expansion D, (2.23). The inverse operator

of the operator

1

(D2

D)-1

+

1: n(n + 1) In >< nj

=

(2-66)

n>O

0. Expanding > acting on the matrix S, is well defined, since < 0 S in the covariant Taylor matrix according to (2.21) and (2-32),

when

=

the matrix

E where

K(n)

is the matrix with entries K

K

v,

K

(n)

K",Vill-A. we

2)!

(n

n>2

...

(2-67)

1

(n)

01

V,tt'1

V(j'l

V,

K(n)

411

...

a

Vgn-2R/,A-11VIAn)

(2-68)

I

obtain

E

+

n!

'Y(n)

(2-69)

7

n>2

^/(n)

'YulI

AI

...

U

Pn

n

(_,)k+l (2k)!

n

(n) 2k

ni

-

,.k>2 nl,nl+---+nk=n

1
-

2,.

-

2k .,nk

K(nk)

K(n,-l)

n(n+l)

(nl+...+nk-1)(nl+-.-+nk-1+1)

-

2)

X

K(n2)

K(ni)

(2.70)

-

X**'Tn-,+n2)(nl+n2+1) ni(nl+l)'' where n!

n

ni,

with

nj +

n

Let

us

..

ni!

Ink

...

+ nk for natural numbers nj,

write down

some

'Y

first coefficients

16 A

1

P

a

7 P/JlA2,03

2

.

.

.

I

nk

.

(2.70):

21R%111,61.42) 3

I

nk!

I

V (/,, R' U2101,U3)

I

(2.71)

2.2 Elements of Covariant

29

1

3 a

'Y 0111112A3A4

Expansions

V(Al VA2 RamA3 1,8 1 A4)

5

Using the solution (2.69)

one can

-

5

find all other

R" (A1171A2

Rl.,M31,6114)

quantities. The

inverse matrix

has the form

+E

-1

(2.72)

n1

n>2

77tt,

77(n)

...

An' 07

AI

CAn

...

n

(nl,.. nk) Some first coefficients

(2.73) equal

1R

77a

3

77,81,11142A3

2

-

5

Using (2.64) and (2.72)

7R"(Al 171A2 RI1-131,61A4)

V(Al VA2 R'.031,81/14)

we

X tlY

(2.74)

V (j, R" A21,61A3)

3 Ci

77PA1112A3A4

(2.73)

nl,--.,nk>2 nl+-"+nk=n

1
15

find the matrix X:

=

9

AIV/

-

X

+

n!

AIVI (n)

(2.75)

I

n>2

XA

IV,

(n)

X/-"V" GIV) 'I

OrAl

I

...

a

ILI

n

An

Al

(n)

(n),,(,u,c,(n-k)77

+

k

2
(2-76)

The lowest order coefficients

2

3Rlu(/"

A1 A2

A I U2A3

XpV AlA21L3144

Finally,

we

V

A3

V

14)

P,2)

1

V(,,,R

6V( ',VA2R" 5 "

(2.76)

(k)

2

read

XJLV

X/.IV

v' )a

t,

V

(2.77)

A2 A 3

8

RI

+

5

(Al JaJA2

find the Van Fleck-Morette determinant

R' /-13

v

A4)

(2.44):

2. Calculation of De Witt Coefficients

30

,A

exp(2()

=

E

(2.78)

((n)

n!

n>2

((n)

OrAl

CAI

041n

...

n

1

E

nj,

nl,.--,nl,>2 nl+---+nk=n

1
The first coefficients

n

E

Tk

-

-

-,

nk)

tr

(7(ni)

(2-79)

(2.79) equal (Al/A2

-

6

(IA1112/A3

4

RA11A2

V(j, Rtl2t13)

(2-80)

3

(Al/A2,03A4 Let

(2.52)

10

us

calculate the

and

using (2.50)

V(/,Zl V/A2RA314)

J'YJA2 R" TR,(,Al 5

JU3

A,,,, (2.51).

quantity we

+

A4)

By differentiating the equation

obtain

(D

1)

+

(2.81)

where

(2.82)

V

Rom here

we

have

(D

At,, where the inverse operator

(D

1)-1

+

1)

+

(2.83)

Lv,

is defined

by

1

(D

+

1)-1

=

1:

In>< nj

.

+

n>O

Expanding

(2.82),

the vector

Liz

,

=

in the covariant

I.: n>1

Taylor

Rw (n)

(n

,

series

(2.21), (2.32)

(2.85)

where

lZjil (n)

RAI, we

obtain

tt'I -111. =

1... t"r,

07",

O'A",

--

V(,"

-

-

-

RA

(2-86)

2.2 Elements of Covaxiant

E

AuI (n)

n!

Expansions

31

(2.87)

1

n>1

Al.t, (n)

071L'

Ali, ILI

071",

n

(nk

9 -'ILI (n)

W-+1

C,

a, (n- k)

ILI (k)

(2-88)

2
In

particular, the first coefficients (2.88) have the form

AAIL,

1'1*2

=

2

ILIL,

2

AAA, A2

3

` V (A' 'I'21141/12)

3

AILAlA2.03 Thus

we

series:

4

(2.69), (2.72), (2.75), (2.78)

particular neglected,

-1

V(A1V/42" 'jjAjA3)

have obtained all the needed

In

be

=

4

R,(,,, JA1,021 "'IA3)

quantities

and

(2.89)

1

in form of covariant

Taylor

(2.87).

case, when all the derivatives of the curvature tensors

V,,R',3,y6

0

=

VA lz"o

,

=

0

can

(2.90)

,

solve

exactly the equations (2.61), (2.81) or, equivalently, sum up Taylor series, which do not contain the derivatives of the curvature tensors. In this case the Taylor series (2.69) is a power series in the matrix S. It is an eigenmatrix of the operator D, one can

all terms in the

D9

and, therefore, when acting

=

23,

the series

on

(2.91)

(2.69),

one can

present the operator

D in the form

d D

=

2S

(2-92)

d9 and treat the matrix

(2.61),

we

obtain

an

S

as

ordinary d2

-jt-2 that has the

an

2 d + t

usual scalar variable.

Substituting (2.92)

second order differential

Tt

+i

ry=o,

t

=-

in

equation

V/S=

,

(2-93)

general solution

(Cl sin v/S=

+

C2

COS

where C, and C2 are the integration constants. (2.62), we have finally C, = -1, C2 = 0, i.e.,

N/S)

Using

I

(2.94)

the initial condition

2. Calculation of De Witt Coefficients

32

sin

(2.95)

Vr In any

case

this formal

to converge for close

expression makes

points

Using (2.95), (2.39)

x

(2.46)

and

sense as

the series

(2.69). It is certain

and x'.

find

we

sin

easily the

matrices

and X

vf

S X sin

2

A, (2.44),

and the Van Fleck-Morette determinant

'A

The vector

(2.87),

matrix H,"

that does not

in the

det

(

V/S sin

(2.90)

case

depend

(2-96)

V/

is

presented

as

the

and the vector

R,,,,

on

(2-97)

v/

product of some

Z,,,, (2.82), (2.98)

Substituting (2.98)

in

(2.81)

and

using

D,Cl,, we

the

equation

(2.99)

LI,,

=

get the equation for the matrix H

(D Substituting (2.92)

and

(2.95)

+

2)HI",

in

(2.100)

V

V

(2.100),

we

obtain

an

ordinary first

order

differential equation d

( Tt +i t

The solution of this

sin t .2

H=

equation has the form H

Cos

where C3 is the we

find

C3

=

(2.101)

t

t

integration 0, i.e., finally

constant.

H

(i

Vs= + ci')

Using the

-

cos

VS_9

,

initial condition

(2.102) 0,

2.2 Elements of Covahant

Ap, Let

us

=

IS-' (i ,-

V-SO I

cos

EV,

33

Expansions

(2.103)

-

A/ VI

apply the formulas (2.95)-(2.97) and (2.103)

to the case of the

De Sitter space with the curvature

R",,v 18

A(6vMg,,8

=

a

The matrix

9, (2.60), (2.61),

-

6,89av)

and the

A

I

=

const

vectorC,,,, (2.82),

(2.104)

.

have in this

case

the

form

2Auff-L"',

S",

till

V

V

-Rp/VIOV,

=

(2.105)

where

Olt'I01VI

I

HI

6

V,

a,". Using (2.106)

we

f(S)

=

0',

11" V" A

=

0

,

vector

obtain for

E Ck 3k

=

(2.106)

H_L

hypersurface that is orthogonal to the an analytic function of the matrix S

the

on

=

2a

a,-' .U, V" A 17i_ being the projector

2

_L

-

V,

f(o)(i

=

17-L)

-

+

f (2Au)11_L

(2.107)

.

k>O

Thus the formal

expressions (2.95)-(2.97) and (2.103) take the

04,1

1

_.UI =

V,

_J/',V,!P +

VI

2a

a/" av, +

V

V

XM IVI=

9/11

I

H",

=

J

2o,

IV,

(A

1)

-

(45-1

P-2+Ort'l OrVI (1 0'

UI

v,T1

'a/

Orv,

+

(2

-

2o,

(2.108)

-

_

2o,

V

concrete form

0-2)

T1

where sin

-,F2-Aa

1

-

,/2_Aa Using (2.105)

neglect quantity A,,:

one can

and obtain the

cos

,F2Au

2Au

the

longitudinal

terms in the matrix H

(2.108)

(2.109) In the

case

(2.90)

it is transverse,

Vm'A,, Let

us

=

stress that all formulas obtained in

spaces of any dimension and

signature.

(2.110)

0

present section

are

valid for

34

Calculation of De Witt Coefficients

2.

2.3 Let

Technique for Calculation of us

apply

De Witt Coefficients

the method of covariant expansions

developed

above to the

calculation of the De Witt coefficients ak, i.e., the coefficients of the asymptotic expansion of the transfer function (1.43). Below we follow our papers

[6, 12, 11, 9]. From the

recurrence

1 1 +

ak

k

relations

1+ k

where the operator D is defined

obtain the formal solution

we

-1

1

D)_1F F

(1.45)

1

-

D)

F

(1

+

D)-'F,

by (2.4) and the operator

p-l(j' A-1/2 [:]' Al/2

=

...

+

(2.111)

F has the form

Qp

(2.112)

we develop a convenient covariant and effective gives a practical meaning to the formal expression (2.111). It will suffice, in particular, to calculate the coincidence limits of the De Witt

In the present section

method that

coefficients ak and their derivatives. First of all, we suppose that there exist finite coincidence limits of the De Witt coefficients

[akI

lim, ak (X

i

X1 )

(2.113)

)

X---) X

that do not

depend

on

the way how the points

and x'

x

approach

each

other, i.e., the De Witt coefficients ak(X,x') are analytical functions of the coordinates of the point x near the point x'. Otherwise, i.e., if the limit

(2.113)

is

singular,

the operator

operator, since there exist the

equal

to

(1 + .1k D)

does not have

eigenfunctions

a

single-valued inverse eigenvalues

of this operator with

zero

Da-,/2 in

>=

1 +

0,

1D)

-(k+n)/2

k

In

>= 0

.

(2.114)

Therefore, the zeroth coefficient ao (x, x') is defined in general up to an is arbitrary function f (a/" /V,'a-; x'), and the inverse operator (1 + 1D)-1 k defined up to an arbitrary function U-kl2fk (0,ji'1V1a;X1). Using the covariant Taylor series (2.21) and (2.32) for the De Witt coef-

ficients

In

ak

><

njak

(2.115)

>

n>O

defining the inverse operator (1 expansion (2.35), and

1 +

+

1 k

in form of the

D)

eigenfunctions

k

1D)

In

i

+ n>O

n

><

nj

,

(2.116)

Technique for Calculation of

2.3

obtain from

we

35

(2.111) k

k

njak

<

De Witt Coefficients

>

-

1

1 ...

.k

k+n

X

<

njFjnk-1

><

1 + nj

1 + nk-1

-

nk-11FInk-2

>

...

<

njjFjO

>

,

(2.117)

where <

mIFIn

>=

IV(,"

-

-

-

...O,"n'1

0",

Vl,_)F

n1

(2.118)

the matrix elements (2.36) of the operator F, (2.112). Since the operator F, (2.112), is a differential operator of second order, its matrix elements < mIFIn >, (2.118), do not vanish only for n < M + 2. Therefore, the sum (2.117) always contains a finite number of terms, i.e., the are

summation

over

ni is

limited from above

(i

: nj+j + 2,

ni

: 0,

ni

Thus

we

reduced the

I,-, k

=

-

1;

nk =-

n)

.

(2.119)

of calculation of the De Witt coefficients

problem

(2.118) of the operator F, (2.112). (2.118) it is convenient to write operators V,,,, (2.41). Using (2.112),

to the calculation of the matrix elements

For the calculation of the matrix elements the operator

(2.4l)-(2.46)

F, (2.112), and

(2.51)

in terms of the we

obtain

p-1A1/2Vjj1 /A-1XjuYVv1 Al/2,p +

F

I i (VM, =

-

("')

+

i X", v',7,, V,

+

At" I X/1, Y", V"

V,

ji (V,,

+ z

+

(V') +A,, I

+

(2.120)

,

where

P-,QP, (1', Y"' z

=

Xte" (AIA, A,,

Now

(2.5)

one can

and

(2.48)

-

=

=

i

V", (.= V1', log'Al/2

Vv,X"11'

i (I,, (,,,)

easily calculate we

obtain from

<

mIFIm

+

Vv,

2X"'vAv,

Xl" v'

(i (I,,

(2.122) +

the matrix elements

A,,,) I

i 6(v'***v-gV-+1V-+2) Jul -A-

mIFIm +

1 >=

0,

+

(2.123)

(2.118). Using (2.54),

(2.118)

+ 2 >=

<

+

(2.121)

36

Calculation of De Witt Coefficients

2.

M

<

mIFIn

>

(n)

=

+

(n

6VI***Vn Z

/Ln+l"*

(nTn 2)

-2

(AI-41-2

XVn

-

1

M

6(V1*-Vn-1 yVn) An"'IL-)

Vn) An-l*-A-)

(2.124)

where

Zlll***An

=

YV

IL

A

These In

(2.124)

or n

Thus,

...

1

I

17/2

n

X V1 V2

if k < 0

[V (A VA, ) ZI (-l)n [V W1 V1410y"] 1) 1 174U, )XV1 (-l)n

(2.125)

1

(n) k

it is meant that the binomial coefficient

is

equal

to

zero

< k.

to calculate the matrix elements

(2.118)

it is sufficient to have the

coincidence limits of the

symmetrized covariant derivatives (2.125) of the coefficient functions Xl"', Y11' and Z, (2.122), (2.123), i.e., the coefficients of their Taylor expansions (2.4 ), that are expressed in terms of the Taylor coefficients of the quantities XP v', (2.76), A,,,, (2.88), and C, (2.78), (2.79), found in the Sect. 2.2. These expressions are computed explicitly in [9, 11, 12]. Rom the dimensional arguments it is obvious that for m = n the matrix mIFIn >, (2.124), (2.125) are expressed in terms of the curvature

elements < tensors R

and the matrix

6,Y6,

quantities VR,

VR and

VQ;

for

m

Q; =

n

for + 2

m -

n

=

+ 1

-

in terms of the

in terms of the

quantities of

the form R 2 , VVR etc. In the calculation of the

De Witt coefficients

by

means

of the matrix

algorithm (2.117) "diagrammatic" technique, i.e., graphic method for the different of the terms turns out to be very consum enumerating (2.117), venient and pictorial. The matrix elements < mIFIn >, (2.118), are presented by some blocks with m lines coming in from the left and n lines going out to the right (Fig. 1), a

a

rn

n

Fig. and the

product

blocks connected

1

of the matrix elements <

by k intermediate

M

lines

mIFIk (Fig. 2),

><

kIFIn

n

>

-

by

two

37

2.4 De Witt Coefficients a3 and a4

Fig.

2

that represents the contractions of corresponding tensor indices. To obtain the coefficient < njak >, (2.117), one should draw all -

diagrams

possible

possible ways by any number of should keep in mind that the number

with k blocks connected in all

doing this, one block, cannot be greater than the number of the lines, going than two and by exactly one (see (2.124)). Then more lines, coming in, by one should sum up all diagrams with the weight determined for each diagram by the number of intermediate lines from (2.117). Drawing of such diagrams is of no difficulties. Therefore, the main problem is reduced to the calculation

intermediate lines. When

out of any

of the

of

some

standard blocks.

technique does not depend at all on the dimenthe signature of the metric and enables one space-time to obtain results in most general case. It is also very algorithmic and can be easily adopted for symbolic computer calculations [44]. Note that the elaborated

and

sion of the

on

2.4 De Witt Coefficients a3 and a4

Using the developed technique one can calculate the coincidence limits (2.113) of the De Witt coefficients

[a3]

and

The coefficients

[a4l.

[a,]

and

[a2l,

that

one-loop divergences (1.51), (1.53), have been calculated long The coefficient ago by De Witt [80], Christensen [62, 63] and Gilkey [132]. [a3l, that describes the vacuum polarization of massive quantum fields in the lowest non-vanishing approximation 1/m 2 (1.54), was calculated in general form first by Gilkey [132]. The coefficient [a4l in general form has been calculated in the papers [12, 11, 9, 4]. The papers [24, 21]) provide determine the

-

,

comprehensive reviews on the calculation of the coefficients ak as well as and more complete bibliography and historical comments as to what was computed where for the first time. The diagrams for the De Witt coefficients [a3l and [a4] have the form,

(2.117), 1

[a3l

---:

0 0 0

0

+

= 0

4 2 +4

1

2

2 +

+

-

4

1 -

-

3

0_=

+

2

COD-O

2

1

-

-

4

5

(:):J:g o

(2.126)

38

2. Calculation of De Witt Coefficients

1

0 0 0 0

[a4l

+

0 0 CEO

-

3

2

0 CEO 0

*

4

3 +

5

= 0 0

1.2 0 CE(:) C) 2.1 +4 2 0 CE3D--O 4 -

*

5

2

3 *

.

5

*

3

2

CrC -O 0

+

3

3

-

5

6

2 .

4

5

2

+-

1

4.3 0 CECM 3

3.1 5

+

CECK D 0

3

2

1

5

4

2

3

2

1

5

3

4

Cl= 0

3 2 1 +---.5 3 2 3

*

2

1

+

5

3

*

+

3

2

1

5

4

5

3

2

1

5 3

*

+

5 3

*

+

5

6

4

3

2

1

5

6

7

5

3

CE:CE=

2-1 CM9D-0

62

2-1 cEr-N=

63

2-1 CE:09D 2D

65

(2.127)

*

Substituting the matrix elements (2.124) the coefficients

[a3l

[a4l

=

[a3]

and

[a4l

in terms of the

in

(2.126)

and

(2.127)

we

quantities X,

Y and

Z, (2.125),

=p3+1 [P; Z(2)1+ + 1B"ZI, + 1Z(4) + 63) 2 2 10 p4

3 +

5

[p27 Z(2)]+

2 *

5

B"PZI,

2 *

5

-

+

5

(2.128)

P 4[P, 4PZ(2) B,'Z,,]+ 5 5 +

2Bp YVJ.IZV + 1Z(2) Z(2) + 2B"Zti(2) 5 3 5

1

Cl, ZA+

express

4

[P7 Z(4)

+ +

15

1

I)ljvZAV +

35

Z(6)

+

J4

7

(2.129)

2.4 De Witt

Coefficients

a3

and a4

39

where

1u,"VZ1.,v

63

(2.130)

6

=

T5 PUlfw Zttv

Ultw 1-0 ZI, 10

+

3

YO,

-

10

1 +

5 Here the

M"A

U2k

following

Z.

14

'YCI

-

1'

5

4 zP VA +

Z.,

v

P +

10

xO,,I "Zoo 14

Z(4)

Z(6)

=

50

Zov(2)

=

A,

=

C1,

=

(2.132)

gAlA2 Z/11A2

g/lIA2g,03/14 41-1.14

Zkt(2)

=

Zttv(2)

=

ZI-t

=

ZA

2)

19 AIIL2YA

Uf"

-2YO")

U2["IA

_Y([tVIX)

3k"""3

/16

(2.133)

7

I

(2.134)

1

(2.135)

AIA2

9A11,12 9A3 A4 YA 1

9 AlIA2Y(AV)A1A2 +

U1

7

...

glAlA2 ZAVA11t2

-

4

1

gUIA2 ZIJAlIL2

2

[A, B]+

7 +

(2.131)

gA1,42 9A3/4 9A5A6 ZI,

=

BA

and

zi, zv

notation is introduced:

Z(2)

ZAV

10

ZAVCIO

U3k

P

D"',

P ZI, v +

10

+

A"O"3

25

3

1

3

4

64

(2.136)

ig A1A29 IA31-t4XIAV Al***,U4

(2.137)

+

i gAl IL2 XILV

(2.138)

+

jgA1A2.X(ILV, AlA2

4

ILIA2

I

X(,4 VCO)

=

(2.139)

(2.140)

denotes the anti-commutator of the matrices A and B.

Using the formulas (2.132)-(2.140), (2.125), (2.122) and (2.123) and the quantities X,"", A,,, and C, (2.70)-(2.89), and omitting cumbersome computations we obtain P

YAv

=

Q

1iR,

+

Rltv

(2.141)

6 +

3

RIv

(2.142)

2. Calculation of De Witt Coefficients

40

Ulttv

U3ttvap

=

0

(2.143)

JA

(2.144)

=

1

VAP

ZA

=

Bit

=

t&VIX

1 0

Q

+

iR

IJA

(2.145)

1

3

(2.146)

0,

=

1i

+

5

3

V/-tP +

U2

Z(2)

-

RovR"v )

30

+

1RmvRAv

2

(2.147) zmv

=

D1,

=

WI.,v

-

2

V (" J,,)

(2.148)

V (A J

(2.149)

1

Wtt,

Z/,t(2)

=

C1,

+

2

Vt&

V1,

+

-

GIL

')

(2.150)

1

(2.151)

G ,

where

(2-152)

it, 3

W/,V

Q

V(IVV)

+

+

R

1

10 Ruv

+i

iR

20

20

+

30'

15

TO R,,8R"1,3v LV

I

Rj.,aRv

(2.153)

+

2

1,

VIL

Qu (2)

=

3 V,,

15

=

3

2

1

+i

QIL(2)

+

OR +

1R'VvR +

1

15

9

R,,6VtR'0

[JV, Rvt,]+

R,,,076V4R'I3'f6

(2.154)

I

gI-&1A2V(jLVUj VIL2) Q V/,

0

Q

+

VvQ]

+

1[J,,,, Q] +2RvVvQ, 3

3

/'

(2.155)

2.4 De Witt Coefficients a3 and a4

2

1 M

G1,

5

J/,&

2

R 15

+

15

""VaRo'.

2VcRotR13c' 5

The result for the coefficient

Z(4)

=

Q(4)

+

tR"'t" J,,,,

_

1[R.,6, V"Ro"']

_

10

2R,,,,-yVcR,3^1

+

7Rl,, Ja

-

45

15

1V"RRcj,

-

(2.156)

15

has

Z(4)

2[RI", ViJ,]+

more

8j

+

9

complicated 4

A

41

JA + 3

form

V/'RC"3v'UIZC"3

10

3 3

+i

E]2R+

14

4 +

7

1Ri"V,,V,R 7

R',8 VcVaR," " '

2

R"' 0

-

21

1

4 +

63

V/-&RV"R

-

42

Rjjv

VuRaeV"R"O 2

1VtR,,6V'RI3" + 283 VjR,,8,y6V/R'I3^16 + 189 Rao R1,3y R,',t, 0

TI

2

2

-

63 R,,,#Ri"R"6 4

v

9

R,,,,8"vRttvaPRP 189

ce

R,,,,3R'A ARIOI"A

+

16 -

-f

88

aa

R' 0 RI4

'

P

189

R"

P

Ct

(2.157)

where

Q(4)

=

gAlA29[13 /A4

=

E]2Q_

Q

1

+

equal

to zero,

(2.130), (2.131),

IRAV IRMI Q11 I

quantities

(2.143), (2.146),

are

I

-2[J/', VIQ] 3

1 2R,"vV,,V,,Q + VIRV,4Q 3 3

From the fact that the are

2

equal

U1ILV' U2/vA

and

(2.158)

U31'", (2.138)-(2.140),

it follows that the

quantities 63 and 64,

to zero too:

63

=

64

=

0

(2.159)

42

Calculation of De Witt Coefficients

2.

To calculate the De Witt coefficient

[a3l, (2.128),

it is sufficient to have the

formulas listed above. This coefficient is presented explicitly in the paper [132]. (Let us note, to avoid misunderstanding, that our De Witt coefficients ak,

(1.43),

63, 132] by Let

us

differ from the coefficients dk used a factor: ak M&k-)

by

the other authors

[80, 62,

=

calculate the De Witt coefficient

A3, (1.55),

that determines the

renormalized

one-loop effective action (1.54) in the four-dimensional physical space-time in the lowest non-vanishing approximation J/M2 [120, 121]. By integrating by parts and omitting the total derivatives we obtain from -

(2.128), (2.141), (2-144), (2.145), (2.147) A3

=

f

d'x g 1/2 str

1

P3+

and

(2.157)-(2.159)

(R,,,,,,3 RA"13

P

30

RA,RA'

-

+ 0R

.

I +

PRA'RMV 2

+

1

1POP2

10

JA JA

1 +

30

+i

(27ZA

R'a R'

V

[_

The formula

1

ROR+

lowing

R

AV

'v

140

630

0 RAv + '

7560(_ 64RtRvR," V

A

6RAvRA,,O'y Rv',O-t

+

V

(x,3R,,,,30'IR,

28RIa

Av -

P

R'0' 13 R'

v

0

P

P

A

(2.160)

V

is valid for any dimension of the space and for any fields. explicit expression for the coefficient [a4l one has to sub-

(2.141)-(2.159)

(2.133).

+

(2.160)

To obtain the stitute

V

1

+48R'"R,,'3 R"A +17R

2RARA,R"

M

(2.129)

in

To write down this

well

as

quantity

in

as

a

to calculate the

compact way

we

quantity

Z(6),

define the fol-

tensors constructed from the covariant derivatives of the curvature

tensor: m)3

=

I

71-ti."'Mn

K',3

-

A 1 "M n '

A

MA

1

...

An

and denote the contracted

...

V(A1 to

13

A.)

VAn-2R"/Ln-I

symmetrized

(analogously

VAn-,R' 1-YI 13

V(AI

V(111

/11-An

ber in the brackets

...

V(A' =

=

VOII

VMn, R,Un)

M.)

(2.161)

1

VAn_2R/In-IPn)

7

VAn-I 7ZAIIA.) covariant derivatives

(2.132)-(2.134).

For

just by example,

a num-

2.4 De Witt Coefficients a3 and

_Ta,8

K"6j-tv(4) L

A (4)

gAl A2gA3A4 K'13/IV/11-14

=

9Al/A2 9113/14LaAlzl-14

gAlA2

=

)ZIL

'

=

a

M(8)

9 PI/12ja,3'YAPIA2

-

yp (2)

=

v(4)

...

9A7A8 Mll

i

...

43

a4

7

(2.162)

'

A8

91"l A2gA3/L4)Z/LV111-1,14

etc.

In terms of introduced

quantity

quantities, (2.161), and the

Z(6) (2.133), 5

Z(6) Here

Z(M,)

notation

(2.162)

the

takes the form

Z(M6)

=

'

+

S Z(6)

(2.163)

is the matrix contribution

zM

(6)

32

5

Q(6)

+

2IIZAV, RAv(4)1+ + -T [Rlzva, IZpva(2)1+

-8

5[P,1ZjL(4)1+ + 9'RAVaj6RMV 2

*

27,guz/(2)Ruv(2)

5R" [ TZ.va, RjLa(2)1+ +5 Rt, Val, 1R'l^l v

4

2

I

R.)3v-yl +

44

15

M'vO'l3[lZAv,'Ra,6(2)1+ +

*

8

22 + -Kiv)3'y [Rtty,

5

16

R"V

:F5

-

15

+

45

5

17

K uv(4 +

+(24 Kttva,8(2)

40

17

51 *

80

RjjvopR)3a

R,,,OyRv

'13t

R"vJjJv 17 +

60

ap

R,,,,R') V

Rl"RvOl

17

17 +

5

30

TzVa 'Yl+

R"v'O [V,3R,,v, J,,,]+

45

+(6

,

32

R

*

[R _Kjuva(2) 5

+

256 +

15 Vi,'R,,,v ['R'"a' jv.]+ 22

Rvao]+

64 +

'3 +

a

Ri,,yR'y0 40

+ Va

40

RvyRatto

17 +

60

RjjaapR'vp'8

ROO,pRvaP IZU)31zva

(2.164)

44

2.

Calculation of De Witt Coefficients

where

gdU1A2gI-13A4g/-15A6V(tj

Q(6) an d

7

(S6)

9.'S

(6)

...

vt'r))Q

is the scalar contribution

7

5

+ R"'

-M(8) 18 20 + 21

5

( T8 Ktiv(6)

R""13KIwao(4)

-

-Lllv(4) 6

2VARLtt(4) 5

-

(35K""(2)

+

12

+M/.Iva

KItva(4)

5

6

KIv

+

25

4)Kjzv(4)

20

M,"Llivee(2)

-

+

9

27MAv (2)Mttv(2)

-

K""137Kuva,8,y(2)

48 +

25

25

K"13 2)Kuva)3(2)

16 18Mtva)3MAvceO + 25 KAv'I37PKAv,,07p

25

+

(

68

101

450

R"R" 5a

+

25

'RPO,8'

1575

1088 *

1575

P -

a

R"O'f

R`Okavo) Kuv(4)

R'13RI a

25

'

8

)Muv(2)

IROvRa,876I,,,7mv,65

2588

Lv,3,p

R1'RP'vI6 P

962 1575

5KlAa,3(2) 6

+

4Ru"PR" 13 0-

25

p

RI",, p Rv,6'P KjAva,6(2)

Ri"',3'yRv' PK,,v,,,,y,p 0

+

6 +

R X R)""

+

Rl"'PRv0'13 + p

(10 MA,,3

5

3

1575

7

*

-

3

p

1048

8,y

1

R`13'y

1RA'v,8

-

(-52k"13R'

+(_65RP

RI,

-

(-61ROvR"16

*

"137

525

6

+(_2 +

R"

RAR" + '

4 + RI'v

K ,

`3

(2)

-:T5 V"RIuvo,(2)

1KP u(2)Iopv(2)

3

2.4 De Witt Coefficients

-2 lorppvcx ITP 3

"

5Kl,,,, 6,\7Kv'I3' 7 9

+

(2)

16 _

Ice)3

4

Ia,8

15

20

40

,\,Y

+FKMP

2)

1

O R,,,,R'Rl'

V

R'AP

R Ov Rc,,8RO'

16

Rv Rl' 4725

p

1'

391 +

,P

4725

2243

-

R'P'P

1 -

ap

R,A P

75 7

RA

7

-

300

RjvR"\c,,8R",80,pR'P 675 32

Dtt

4725"

v

c,3

X -Y

299

R 4725

'P

To K',OILUP

K"3

p(2)

10

3 _

10

V'RK ,,

'8

(2)

RP

P

RRgRI' 1'

+

R" 0''\ Rvo"YP P

R"3PR1Aa,\-fRv,3\7

" or

P

R'V P

'6

RavR"13ROvpRcP6 a

RivR",,v3 R",A P R,3,\O'P ,

R`3

1?"v

+

^1

,

RAcvaR) 4725

+

VA

R,3 \ YR'\p'YORP1I'v

32

+

-

247

8

+

-

v

a

V

"

300 R,,,,,RaR"X

9450

9

-

P,3

Kv

817

-

9

a02)

KP

"0 V

21

RIRvR,'R ,, + 450 v

Ipi,,,,,6KP

3

45

a4

+40 KI, PO'06Kvpa(2)

P3

7

+

7 10

-8 VPRImvpa,8 45

MI, VP

p (2)

T, K1,P 0'(2) K,,

+

(_

MV OP +

'Pi,

15

and

2 -

7KtOP"3 5

4K1,jv),ap MAaP 1 + R"(,,,v,3) 15

a3

150-5- R"c, )37

R,"v 4725

,V

Rv'O'YRtt,\orpRv,\'P

232 -

R\'y Orp R'P

'P

R"13PR,,,\,-yRA

7 V

16

(2-165)

expression can be simplified a little bit to reduce the number of (see [9, 11, 12]). Rirther transformations of the expression for the quantity Z(6), (2.163)-(2.165), in general form appear to be pointless since they are very cumbersome and there are very many independent invariants This

invariants

46

2.

Calculation of De Witt Coefficients

constructed from the curvatures and their covariant derivatives. The set of invariants and the form of the result should be chosen in accordance with the

specific

character of the considered

for the De Witt coefficient and

problem.

[a4] by the

That is

why we present the result

set of formulas

(2.129), (2.141)-(2.159)

(2.161)-(2.165) (an

alternative reduced form is presented in [9, 11, 12, 4]. One should note that notation and the normalization of De Witt coefficients is slightly different in the cited references.

2.5 Effective Action of Massive Fields Let

illustrate the elaborated methods for the calculation of the

one-loop example of real scalar, spinor and vector massive quantum matter fields on a classical gravitational field background in the four-dimensional physical space'-time. In this section we follow our papers us

effective action

on

-the

[6, 12]. The operator

(1.22), (1.29) -

11

in this

+ R +

+

case

m

has the form

2)

(j

M

1_Jd +,rn 2 where

m

and j

are

the

mass

y01-y1)

co-differentiation

gl", on

0)

,

(j

1/2)

(j

1)

(2.166)

,

and the

is the coupling spin of the field, gravitational field, -y" are the Dirac

constant of the scalar field -with the

matrices,

=

d is the exterior derivative and 6 is the operator of

forms:

The commutator of covariant derivatives

(3-13) (2.14)

has the form

0 i

(2.167)

47

R'

01AV

Using

the equations 1 0

R

-

4

(d6

+

6d)

(61' d2

one can

0

V

=

62

=

-R") o' V

0

express the Green functions of the

(2.168)

,

operator

the Green functions of the minimal operator,

(1.30),

(2.166),

in terms of

2.5 Effective Action of Massive Fields

0

_Q

M2)

+

47

-1

(2.169)

where

- R M

1

-

Q

-Y"Vj"

-1iR. 4

=

-R'

d6

-

(2.170)

0

Using the Schwinger-De Witt representation, (1.33), 00

1

sde;t

log

F(j)

2i

1

and the

exp(is.A)

sdet

(2.171)

s

equations tr

exp(is7l'V,)

exp(is6d)

=

=

tr

exp(-is-yI'V,,)

explis(d6 + 6d)}

exp(isd6)

tr

we

i

ds

Ij=1

exp(isd6)

-

=tr exp (is

0)

1 +

log det (- dd +

m2)

=

log det

-

Thus

we

i(_

=2log det

m)

+

(2.172)

,

6(0),

1R+m 4

a

E]

-

D+

(- 0 60

log det (-

+ 1

Ij=O,

obtain up to non-essential infinite contributions

log det (-y" Vt,

,

R,6 a

+

2) 1

,

M2j,3)

+M2) Ij=O

a

(2.173)

-

have reduced the functional determinants of the operators (2.166) operators of the form (1.30). By

to the functional determinants of the minimal

making

use

of the formulas

(2.171)-(2.173), (1.50)

and

asymptotic expansion of the one-loop effective action of the

(1.54)

we

obtain the

in the inverse powers

mass

1

I'Mren

2(47r)2

E k(k k>3

Bk -

1)(k

-

2)M2(k-2)

(2.174)

where

Ak Bk

'Ak 2 Ak

(2.175) Ak

I

48

2.

Calculation of De Witt Coefficients

and the coefficients Let a

us

factor

Ak

given by (1.55). Ak, (1.55), for the spinor field contain addition to the usual trace over the spinor indices according are

stress that the coefficients

(-1)

in

to the definition of the

Substituting using (2.175)

we

(1.25). R,, (2.167),

supertrace

the matrices

obtain the first

and

coefficient, B3,

Q, (2.170),

in the

in

(2.160)

and

asymptotic expansion

(2.174) B3

f

d 4X g112

cjR 0 R

+

0 RAv +

C2Rgv

C3R

+C5RRj,,,,6R1"'13 + C6R,RRA A it V

3

+

+

c4RRAvR"v

C7R"vR,,OR'/.I 13

V

+C8Rt,vR",X,8'YR""0'Y + c9R,,a13R,,,30"R,P1'v +cloR"

v

a

'8

R'ar13 R'

J

P

IL

V

P

(2.176)

I

where the coefficients ci are given in the Table 2.1. The renormalized effective action (2.174) can be used to obtain the

renor-

malized matrix elements of the energy-momentum tensor of the quantum matter fields in the background gravitational field [120, 121]

out, vac IT,,,,, (x) I in, vac

In

)

h2

g-1/2 6F(l)ren

+

6gAV (X)

ren

0(h2)

(2.177)

Such problems were intensively investigated (see, for example, [137, 42]). particular, in the papers [120, 121] the vacuum polarization of the quantum

fields in the

background gravitational field

of the black holes

was

investigated.

In these papers an expression for the renormalized one-loop effective action was obtained that is similar to (2.174)-(2.176) but does not take into account the

terms, that do

not contribute to the effective

vacuum

energy-momentum

(2.177) when the background metric satisfies the vacuum Einstein 0. Our result (2.176) is valid, however, in general case of equations, R11v tensor

=

arbitrary background

Moreover, using the results of Sect. 2.4 for the calculate the coefficient A4 and, therefore, [a4l the next term, B4 in the asymptotic expansion of the effective action (2.174) of order 1/m 4. The technique for the calculation of the De Witt coefficients developed in this section is very algorithmic and can be realized on computers De Witt coefficient

(all

space. ,

one can

the needed information is contained in the first three sections of the

present

chapter) (see [44]). (2.174).

In this

case one can

also calculate the next terms

of the expansion

However,

the effective action functional

local and contains verse

powers of the

an

imaginary part.

mass

(2.174)

in general, essentially nonasymptotic expansion in the inreflect these properties. It describes

F(j) is,

The

does not

2.5 Effective Action of Massive Fields

49

Table 2.1. B3 for matter fields

Scalax field

Cj

1 2

1

C2

_L

C4

1

3

27

280

280

1

1

2_

140

28

28

U

k 30

1

C8

C9

0

_;F2

1

2-1

180

60

1

7 1440

I -

T-0

8

25

52

945

756

63

2

47

315

1260

1

19

61

1260

1260

14-0

17

29

67

7560

7560

2520

19-

-

I

CIO

5 -

6-4

6

(

C6

C7

-

6

30

Vector field

56

Q -0

C3

C5

+

'9

2

Spinor field

-

1

_L

108

18

-

270

105

only in weak gravitational fields (R < M2). in strong gravitational fields (R > M2) as well as for the massless matter fields, the asymptotic expansion (2.174) becomes meaningless. In this case it is necessary either to sum up some leading (in some approximation) terms or to use from the very beginning the non-local methods for the Green function and well the effective action

,

the effective action.

3. Partial Summation

of

Schwinger-De

3.1 Summation of The solution of the

Witt

Expansion

Asymptotic Expansions equation

wave

the proper time method, tigation of many general

(1.33),

in

background fields, (1.32), by

means

of

turns out to be very convenient for inves-

problems of the quantum field theory, especially analysis of the ultraviolet behavior of Green functions, regularization and renormalization. However, in practical calculations of concrete effects one fails to use the proper time method directly and one is forced to use model for the

non-covariant methods.

advantages of the covariant proper time method it s necessary to sum up the asymptotic series (1.43) for the evolution function. In general case the exact summation is impossible. Therefore, one can try to carry out the partial summation, i.e., to single out the leading (in some In order to

use

approximation) one can

the

terms and

limit oneself to

a

them up in the first line. On the one hand given order in background fields and sum up all sum

derivatives, on the other hand powers of background fields.

one can

neglect the derivatives and

sum

up all

In this way we come across a certain difficulty. The point is that the asymptotic series do not converge, in general. Therefore, in the paper [46] it

give up the Schwinger-De Witt representation (1.33), (1.49), only as an auxiliary tool for the separation of the ultraviolet It is stated there that the Schwinger-De Witt representation, divergences. for exists a small class of spaces only when the semi-classical solu(1.33),

is

proposed

to

and treat it

-

tion is exact.

However, the divergence of the asymptotic series (1.43) does not mean one must give up the Schwinger-De Witt representation (1.33). 0. Therefore, it is The point is, the Q(s) is not analytical at the point s at all that

=

natural that the direct summation of the power series in s, (1.43), leads to divergences. In spite of this one can get a certain useful information from the

asymptotic (divergent) series. a physical quantity G (a) which is defined by expansion of the perturbation theory in a parameter a

structure of the

Let

us

consider

G(a)=

1: Ckak k>O

I. G. Avramidi: LNPm 64, pp. 51 - 76, 2000 © Springer-Verlag Berlin Heidelberg 2000

an

asymptotic

(3-1)

3. Summation of

52

Schwinger-De

Witt

The radius of convergence of the series

R

If R

i4

the series

physical

(

=

0 then in the disc

Expansion

(3.1)

lim SUP I Ck k

Jai

is

given by the expression [210]

jilk

(3.2)

+oo

of the

< R

complex plane of the parameter

a

(3.1)

converges and defines an analytical function. If the considered quantity G(a) is taken to be analytical function, then outside the

disc of convergence of the series (3.1), Jai > R, it should be defined by analytical continuation. The function G(a) obtained in this way certainly have

R. The analytical singularities, the first one lying on the circle Jai through the boundary of the disc of convergence is impossible if all the points of the boundary (i.e., the circle jai R) are singular. In this case the physical quantity G(a) appears to be meaningless for Jai > R. If R 0, then the series (3.1) diverges for any a, i.e., the function G(a) is not analytic in the point a 0. In this case it is impossible to carry out the summation and the analytical continuation. Nevertheless, one can gain an impression of t he exact quantity G(a) by making use of the Borel procedure for summation of asymptotic (in general, divergent) series [192]. The idea consists in the following. One constructs a new series with better convergence properties which reproduces the initial series by an integral =

continuation

=

=

=

transform. Let

us

define the Borel function

B(z)

Ck

1: _P(pk + v) Zk,

=

(3-3)

k>0

where /-t and v are some complex numbers (Re /t, Re v > 0). The radius of convergence J of the series (3.3) equals

f?

=

Ck

lim

SUP

k-+oo

Thus, when

exp(Mk log k),

r(pk

+

v)

11k)

(3.4)

the coefficients Ck of the series (3.1) rise not faster than const, then one can always choose p in such way, Re M !

M

=

M, that the radius of convergence of the series (3.3) will be not equal R 0 0, i.e., the Borel function B(z) will be analytical at the point Outside the disc of convergence,

analytical Let

us

jzj

>

A?,

to z

zero =

the Borel function is defined

0.

by

continuation.

define

dttv-'e-tB(at1')

G(a)

,

(3.5)

C

where the integration contour C starts at the zero point and goes to infinity in the right half-plane (Re t -4 +oo). The asymptotic expansion of the function

G(a)

for

a

-4

0 has the form

(3.1). Therefore,

the function

6(a), (3.5),

3.2 Covariant Methods for

(which

is called the Borel

sum

Investigation

(3.1))

of the series

of Nonlocalities

can

53

be considered

as

the

physical quantity G(a). The analytical properties of the Borel function B(z) determine the convergence properties of the initial series (3.1). So, if the initial series (3.1) has a finite radius of convergence R 0 0, then from (3.4) it follows that the Borel series (3.3) has an infinite radius of convergence j 00 and, therefore,

true

=

B(z) is an entire function (analytical in any finite part of the complex plane). In this case the function G(a), (3.5), is equal to the sum of the initial series (3.1) for jal < R and determines its analytical continuation outside the disc of convergence jal > R. If the Borel function B(z) the Borel function

has

in the finite

singularities

from

(3.4)

part of the complex plane, i.e., f?

it follows that the series

(3.1)

< 00, then

radius of convergence equal quantity G(a) is not analytic at

has

a

R 0, and, therefore, the physical 0. At the -same time there always exist a region in the complex point a plane of the variable a where the Borel sum O(a) is still well defined and can be used for the analytical continuation to physical values of a. In this way different integration contours will give different functions G(a). In this case one should choose the contour of integration from some additional physical assumptions concerning the analytical properties of the exact function G(a).

to

zero

=

the

=

3.2 Covariant Methods for

Investigation

of Nonlocalities

-2k where L The De Witt coefficients ak have the background dimension L , Witt standard the is the length unit. Therefore, expansion, Schwinger-De

expansion in the background dimen-2k both the a given background dimension L k their well R V2(k-')R, of the as as derivatives, background fields, powers are taken into account. In order to investigate the nonlocalities it is convenient to reconstruct the local Schwinger-De Witt expansion in such a way that the expansion is carried out in the background fields but their derivatives are taken into account exactly from the very beginning. Doing this one can preserve the manifest covariance by using the methods developed in the Chap. 2.

(1.43), (1.52) sion [34, 35].

(1.54), is,

and

in

fact,

an

order in the

In

,

Let

(1-30),

A introduce instead of the Green function G B (x,y) of the operator whose upper index belongs to the tangent space in the point x and us

the lower that

one

depends

g(X, YJXI)

-

in the

on some

=

point

y,

a

three-point Green function gA" B (X, Y jX%

additional fixed point

-p-1 (X, Xl).

x',

A-112(X' x')G(x, Y). A-1/2 (Y' XI),p(y, XI)

(3.6)

This Green function is scalar at the points x and y and a matrix at the point x'. In the following we will not exhibit the dependence of all quantities on the fixed

point x'.

The equation for the Green function

9(x, y), (3.6),

has the form

(1.32)

54

3. Summation of

(F

_

X

Schwinger-De

jM2) g(X,Y)

=

Witt

Expansion

_jg-112(X),A-1(X)j(X,Y)

(3.7)

,

where Fx is the operator (2.112) and 6AB* Let us single out in the operator Fx the free part that is of the background fields. Using (2.120) we have

Fx

i Ox

=

+

P,

zero

order in

(3-8)

,

where

OX

PX

ikt,' ", (X)

=

.k,"' and the operators

formulas

V,,,

X

X

(X,)VA, VVI VX"

,

+

X,"', (X)

=

Y", (X) -

(2.41), (2.46), (2.122)

+ Z (X)

(3.10)

,

(3.11)

gp',,' (X') Y"'

and the quantities XA and

(3-9)

(2.123).

The

and Z

background fields and can be considered By introducing the free Green function go (x, y),

and

writing the equation (3.7)

9(X,Z) we

go (X, Y)

obtain from

9(X, Z)

=

90(X,Z)

=

+

(3.13) by

90 (X, Z)

=

in the

f d'y

means

g

as a

integral

by the

is of the

perturbation.

i "A-1 (X)g-112 (X)6(X, Y)

(3.12)

form

112(Y),A(Y)gO(X,Y)Pyg(y, Z)

(3-13)

,

of direct iterations

E f d'yj g 1/2(yl),A(yl)

+

defined

operator.P, (3.10),

first order in the

(_OX +,M2)

are

...

d'Yk g1/2 (Yk), A(Yk)

k>1

X

90 (X

YI)-Pyj 90 (Y1 Y2)

"

"

7

Pyk 90 (Yk, Z)

(3.14)

-

Using the covariant, Fourier integral (2.55) and the equations (2.56) and (2.57) obtain from (3.12) and (3.14) the momentum representations for the free

we

Green function

d'kA'

90(X,Y)

f (27r)n 91/2 (x')

exp

ik,,,

(ag'(y)

-

1

oA'(x) m

2

+k 2

1

(3-15) and for the full Green function

dnk"'

9 (X, Y)

f (27r)n x

exp

9

112(XI)

(27r)n

I ip4, a," (y)

-

9

1/2

(XI)

1

ikj,, o,"' (x) 9 (k, p)

,

(3-16)

3.2 Covariant Methods for

Investigation

of Nonlocalities

55

where

9(k,p)

=

(2-7r)'g- 1/2 (XI)

J(kl"

f

+

x.P(k, qj) M2 1" " (k

k2 and

A"' (q), Y11' (q)

of the coefficients

and

X11"',

1

Z q)

=

+

k2)(M2

9

1/2

(XI)

d'n q.

Vi_

...

(27r)n

+

-

2 q1

-

1/2

(XI)

-

F(qi-l, qj ; -22 F(qi, p) + qi

iY1" (k

-

g1,,,,,,(x')k1"V

are

9

1

-

-

(3.17)

P

+

u'

a

2 7r)n

p)pp, p,,,

-

(M2

1

i>1

(k, p)

+

M2 + V

n

H(k,p) =.P(k,p)

U (k, p)

py

-

p)pi,,

+ Z (k

-

,

p)

(3.18) (3.19)

,

the covariant Fourier components,

(2.55),

Z, (3.11), (2.46), (2.122), (2.123). The formulas (3.15)-(3.19) reproduce the covariant generalization of the usual diagrammatic technique. Therefore, one can apply well elaborated methods of the Feynman momentum integrals. The Fourier components of the coefficient functions A"' (q), YA'(q) and Z(q), can be expressed in terms of the Fourier components of the background fields, R, and Q, using R vc"O) the formulas obtained in Chap. 2. As usual [50, 155, 193, 42] one should choose the contour of integration over ko in the momentum integrals (3.15)-(3.18). Different ways of integration correspond to different Green functions. For the YA

and

Ul/

causal

(Feynman)

Green function

one

should either

assume

0

-4

k2

-

ie

or

go to the

Euclidean sector of the space-time [50, 155, 193]. Similarly, one can construct the kernels of any non-local operators of general form, f (1:1), where f (z) is some function. In the zeroth approximation in

background fields

we

f (n)(X, Y) d

f P) (X, Y)

obtain

=

-p(X),A112(X)j(M)(X, Y)A1/2(Y),p-1(Y)

k"'

=

(27r)n

9

1/2

(XI ) exp ik.,

(a"' (y) au'(x)) I f (-k 2) -

.

(3.20) An

important method for the investigation of the nonlocalities is the analysis of the De Witt coefficients ak and the partial summation of the asymptotic series (1.43). In this case one should limit oneself to some order in background fields and sum up all derivatives of background fields. In order to get an effective expansion in background fields it is convenient to change a little the "diagrammatic" technique for the De Witt coefficients developed in Sect. 2.3. Although all the terms in the sum (2.117) have equal background dimension, Ln-2k they are of different order in background fields. From the formula (2.117) it is not seen immediately what order in background ,

3. Summation of

56

Schwinger-De

fields has each term of the

Witt

Expansion

(2.117), i.e., a single diagram, since all the njak > have k blocks. However, among these elements < mIFIm + 2 >, there are dimensionless sum

for the coefficient <

diagrams blocks, i.e.,

the matrix

blocks that do not have any background dimension and are of in background fields. These are the blocks (matrix elements <

zero

order

mIFIM

+

outgoing lines equal to the number of the incoming lines plus 2 (c.f. (2.124)). Therefore, one can order all the diagrams for the De Witt coefficients (i.e., different terms of the sum (2.117)) in the following way. The first diagram contains only one dimensional block, all others being dimensionless. The second class contains all diagrams with two dimensional three etc. The last diagram contains k dimensional blocks, the third one blocks. To obtain the De Witt coefficients in the first order in background fields it is sufficient to restrict oneself to the first diagram. To get the De Witt coefficients in the second order in background fields it is sufficient to restrict oneself to the first diagram and the set of diagrams with two dimensional blocks etc. This method is completely analogous to the separation of the free part of the operator F, (3.8). The dimensional matrix elements < MIFIn >, (with m > n), of the operator F, (2.120), are equal to the matrix elements of the operator F, (3.10). When calculating the matrix elements (2.124) and (2.125) one can also neglect the terms that do not contribute in the given order in background fields. After such reconstruction (and making use of (2.124)) the formula (2.117) 2

>)

with the number of

-

for the De Witt coefficients <

<

njak

njak

> takes the form

( 2ii+nj-l ) ij-1 > :

I
*

*

< n;

iN-1

-

i2

< n2;

...

where the < n;

k

-

>-=

il

-

g"'2 io

and the summation ements n

are

+ 2k >

11FIn,

>< nN-1;

>< ni;

ii

iN-1

11FIO

-

(2ij+nj-l)

1<j
iN-2

-

11FInN-2

>

(3.21)

>

notation is introduced

following

kIFIm

11FInN-1

-

ni

...

=-

over

9

V2k-IV2k

0,

ni is

dimensional, (i.e.,

iN

Vn+2k IFIIL,

< V1 =-

k,

...

JL"' >

(3.22)

nN =- n,

carried out in such limits that all matrix el-

kIFIm

for each < n;

>,

(3.22),

there holds:

m):

nj+2(ij-1) ! 0, n2+2(i2-il-1) !

ni

-

,

-

-

,

n+2(k-iN-1-1)

nN-1

-

The formula (3.21) also enables one to use the diagrammatic technique. However, for analyzing the general structure of the De Witt coefficients and for the partial summation it is no longer effective. Therefore, one should use the analytic expression (3.21).

3.3 Summation of First-Order Terms

57

3.3 Summation of First-Order Terms Let

us

0jak

calculate the coincidence limit of the De Witt coefficients

> in the

first order in

background

fields.

Using the

formula

[ak] (3.21)

we

obtain 1

[ak]

--":

I)

0; k

<

-

11FIO

+O(R 2)

>

(3.23)

k

O(R 2 ) denotes all omitted terms of second order in background fields. Using (3.22), (2.121)-(2.125) and (2.46) and the formulas of the Sect. 2.2 obtain from (3.23) up to quadratic terms

where

we

[ak]

o

:---

(2k

-

Q

1)!

The expression

first order in

k

k-1

(3.24)

can

+ is

_(2k + 1)

iR

+O(R 2)

(k

,

be used for the calculation of the

fields.

background

Q(sjx, x)

+

Substituting (3.24)

in

(fl (is D)Q + i f2(is D)R)

+

>

1)

.

(3.24)

f2(slx, x) in (1.43), we obtain O(R 2)

the

(3.25)

where

fl(z)

k!

E

=

k>O

f2 (Z)

=

(k

E

k z

(2k

1)!

+

(3.26)

'

1)!(k + 1) k z (2k + 3)!

+

k>O

(3.27)

.

The power series (3.26) and (3-27) converge for any finite z and hence define entire functions. One can sum up the series of the type (3.26) and (3.27)

using the general formula

(k (2k

+

11

1)!

+ 21 +

1)!

d

(21)!

1

21

(

T

1

2

-

4

)

,

(3,28)

0

that is

easily obtained from the definition of the Euler beta-function [98]. Substituting (3.28) in (3.26) and (3.27) and summing over k we obtain the integral representations of the functions f, (z) and f2 (z) 1

fi(z)

=

fd

exp

11 (1

_

4

2)

Z

1

(3.29)

1

0

f2 (Z)

=

fd

1

4

(1

_

2 ) exp

1

1

(1

_

62)

Z.

(3.30)

0

The kernels of the non-local operators in terms of covariant momentum

f, (is 1:1), f2 (is 1:1) should be understood expansions (3.20).

3. Summation of

58

Schwinger-De

Witt

Expansion

Using the obtained (3.25), one can easily obtain the Green function at coinciding points, G(x, x), in the first order in background fields. Substituting (3.25) in (1.49) and supposing Im m' < 0 (for the causal Green function), we obtain after the integration over the proper time in the n-dimensional space ,

G(x, x)

=

i(4-7r) -n/2

+1'

Jr (1

_

n) i Mn-2 2

E3

(2 n) rnn-4

M2

2

0

Q+i-P2

_

-

4M2

) R] (3.31)

+O(R 2), where

F(z)

is the Euler

gamma-function,

and

1

AW

fd

=

[1 + (1

2)Z]

_

(n-4)/2

(3.32)

1

0

A (Z)

=

1

fd

4

(1

_

2) [I

+

(1

2),](n-4)/2

_

(3-33)

.

0

4 By expanding in the dimension n in the neighborhood of the point n and subtracting the pole 1/(n-4) we obtain the renormalized Green function, Gren (X, X), in the physical four-dimensional space-time (1.51), (1.52), up to terms of second order in background fields =

0

(4-7r )2 IF, (- iw ) 1

Gren (X, X)

=

Q

+

=

2

iF2

+O(R

E3

(- 2). R) 4m

2)

,

(3-34)

where

FI(z) 1

F2 (z)

5-

=

18

-

1

3)

-

-

6

Z

(3.35)

J(z) I

(1

-

2z

) J(Z)

,

(3.36)

1

J(z)

=

2(1

+

z)

f

d 1 +

(1

-

(3-37)

2)Z

0

The formfactors F, (z),

(3-35), and F2 (z), (3.36),

are

normalized

according

the conditions

Fj(O)

=

F2(0)

=

0

-

(3.38)

integral (3.37) determines an analytical single-valued function in the complex plane z with a cut along the negative part of the real axis from -1

The

to

-00:

3.3 Summation of First-Order Terms

J(z)

J(x

21

=

iE)

=

21

+

!log

(vFz-+l+vqz

arg(z

+

i7r j

+

1)1

<

59

(3.39)

7r

z

1

(vl---x-1 + V'---x)

log

+ X

1

(X

,

<

-1)

.

X

(3.40) Thus

inciding

(1.52),

obtained the non-local expression for the Green function at copoints, (3.34). It reproduces the local Schwinger-De Witt expansion, we

up to

quadratic

terms in eternal fields

by expanding in inverse powers determining the formfactors F, (z), (3.35), and 1 there is a threshold F2 (z), region I z I < 1. For z the branching point. Outside the circle IzI : 1 the formfactors singularity are defined by the analytical continuation. The boundary conditions for the formfactors fix uniquely the ambiguity in the Green function (3-34). For the causal Green function (Imm 2 < 0) the lower bank of the cut is the physical one. Therefore, with account of (3.40), the imaginary parts of the formfactors (3.35) and (3.36) in the pseudo-Euclidean region above the threshold z x ie, x < -1, equal of the

The power series (3.36), converge in the

mass.

=

-

-

=

-

ImFj (x

i,-)

-

7r

ImF2 (x The ultraviolet

-

i6)

asymptotics IzI

=

6 -+

=

(1

oo

, ;1 +

7r

i -

2x

(3.41)

1

) , +T

(3.42)

.

X

of the formfactors

(3.35)

and

(3.36)

have the form

F, (z)

Jzj-+00

log (4z)

(Z z)

+ 2 + 0

(3.43)

log

1

1

(-log (4z) +5)+o (Z

F2 (z)

log

z

(3.44)

IZI-+00 Let 2 M

-+

consider the

us

0 in

(3.3 1)

case

of the massless

for Re n > 2

field at

coinciding points background fields

we

=

i(4,)-n/2

X

m

2 =

0.

Taking

the limit

obtain the Green function of the massless

in the n-dimensional space in the first order in

F

G(x, x)

field,

(2

(_ E]) (n-4) /2

-

11) 2 2-- (1, (n r(n

(Q

-

_

1))2

2)

n

-

2

+

4(n

-

1)

i

R)

+

O(R2)

(3.46)

3. Summation of

60

The formula in the

determines the

(3.46)

points

n

4,6,8,.

=

-

Expansion

analytic function of the dimension n analytical continuation there appear

2 < Re n < 4. After the

region in the

poles

Witt

Schwinger-De

that reflect the ultraviolet

.,

the

2

reflecting pole point two-dimensional space (3.46) gives and

in the

a

n

i

G(x, x)

=

(

-2

i-7r

=

2 n

infrared

+C+Iog

2

TI A_

In the

1

1Q-i

-

-

divergences,

divergence.

R

+

O(R 2)

.

(3.47) For

even

dimensions,

n

(N (2N

1

G(x, x)

2N, (N

=

=

F4, A

-2TI(2N

-

-

-

2)

2)! 3)!

(3.46)

we

obtain

2 + n

-

2N

N-2

47r[12

N-2

E] -

from

2),

1-(

log

+

1

_ (2N _2

>

R

+

1)2

O(R2)

TI(N

-

1)

N

Q+

-

1 1R

2(2N

-

1)

(3.48)

,

where

TI(k)

=

(3.49)

-C + 1<1
In from

particular,

in the

physical four-dimensional space-time,

4,

n

we

have

(3.48)

_(

1

G(x,x)

=

(4,7r)

2

1

2 n

_C3

-

iR

18

4

+C-2+log

4.7rp2

) (Q+ 1iR) 6

+O(R2)

(3.50)

The renormalized Green function of the massless field

can

be obtained

by using the ultraviolet asymptotics of the formfactors (3.43) and substituting the renormalization parameter instead of the mass, This reduces

Let

us

simply to

a

change

(3.44) 2 m

of the normalization of the formfactors

stress that in the massless

and

-+ IL

2 .

(3.38).

divergence of the coincidence background fields, (3.48), coefficient [a(n-2) /21 (3.24). Therefore, case

the

limit of the Green function in the first order in

(3-50), in the

proportional to the De Witt conformally invariant case [42, 137], is

1

n-2

Q

(n

1R

1)

(3.51)

3.4 Summation of Second-Order Terms

61

the linear part of the coefficient [a(n-2)/217 (3.24), is equal to zero and the Green function at coinciding points is finite in the first order in background

fields, (3.47)-(3.50).

(3.31), (3.46),

In odd

dimensions,

n

=

2N +

1, the Green function,

is finite.

3.4 Summation of Second-Order Terms Let

us

calculate the coincidence limit of the De Witt coefficients

background fields. background fields

in the second order in up to cubic terms in

[ak]

<

=

(2k 1)

O;k- 11FIO

Rom the formula

[ak] =< 0jak > (3.21) we have

>

2k

i-1)

I
x

where

O(RI)

<

0; k

-

i

-

11F.Ini

1) (2i+ni-1) ( 2k-1) k i

>< ni; i

-

11FIO

>

denotes all omitted terms of third order in

+O(R3)

,

background

(3-52) fields.

The total number of terms, that are quadratic in background fields and contain arbitrary number of derivatives, is infinite. Therefore, we will also

neglect use

(3.52) assuming to Ak, (1.55), and the one-

the total derivatives and the trace-free terms in

the results for the calculation of the coefficients

loop effective action, (1.50), (1.54). The number of the terms quadratic in background fields, in the coefficients Ak is finite. Let us write the general form of the coefficients Ak up to the terms of the third order in background fields

f

Ak

dnX g1/2 str

2

+i

where

4

-1

akQ

(6kR,,, E]

Ej

k-2Q

k-2

-

20kJjt

RA' + EkR E]

[:]

k-3

k-2

ja + Yk Q r-1 k-2R

R)

,

+

O(R 3)

(3.53)

= *

quadratic invariants can be reduced to those written above up to third order terms using the integration by parts and the Bianci identity. For example, All other

f

d nX g 1/2 str R

.

E]

kRILv

=

-2

f

d nX g 1/2 str

k

-

1

p +

O(R3)}

3. Summation of

62

f

Schwinger-De

d'x g 1/2 Rm,,,,o El

Witt Expansion

f

kRM"0

d'x g 1/2

1

+O(R3) Using (3.22), (2.121)-(2.125)

(2.46)

and

14Rttv

n k R,"v

RE]

-

k

(3-54)

-

and the formulas of the Sect. 2.2

calculate the coefficients ak, flk) ^fk) 6k and -k from (3.52) and Omitting the intermediate computations we present the result

one can

k! (k an

(2k

fl'-

2(k

'Yk

2)! 3)!

-

k! (k

(2k

-

1)! 1)!

(2k

-

(1.55).

(3.55)

(3-56)

'

k! (k

1)

-

R

1)! 1)!

(3-57)

k!k!

6k

6k

These coefficients

were

=

(k

(3.58)

2

(2k

+

1)! k!k!

2

k

-

1)

(2k

(3-59)

1)!

+

computed completely independently

in the papers

[8, 10, 12, 52]. Using the

the obtained

coefficients, (3.55)-(3.59),

we

calculate the trace of

fl(s), (1.43), k

f

d'x g 1/2 str S? (s I x,

x)

=

1: k1-) Ak

(3.60)

k>O

Substituting (3.53)

f

and

d'x g 1/2 str

(is) 2 +

2

(3.55)-(3-59)

fl(six, x)

=

f

in

(3.60)

d'x g 1/2 str

i+is

[Qfj(is0)Q+2Jjj3(is0)

+i (R,,f4(is 0)RI"

Rf5(is

+

obtain

we

JA +

D)R)]

+

O(R 3)

1

Q+

iR

6

2Qf2 (is D)R

(3-61)

where

f3 (Z)

=

E k>O

k z

(2k

+

3)!

,

(3-62)

3.4 Summation of Second-Order Terms

f4(z)

2(2k + 5)!z

k>O

A (Z)

=

E k>O

and

fj (z)

and

(k + 2)! (k2 (2k + 5)!

k

63

(3-63)

+ 3k +

1)Zk,

(3.64)

f2 (z)

are given by the formulas (3.26), (3.27), (3.29) and (3.30). (3.62)-(3.64) converge for any finite z and define in the same manner as (3.26) and (3.27) entire functions. The summation of the series (3-62) and (3.63) can be performed by means of the formula (3.28). Substituting (3.28) in (3.62) and (3.63) and summing over k we obtain the integral representation of the functions f3 (z) and f4 (z):

The series

1

f3 (Z)

=

1 2 exp

fd

2

1

4

(1

_

2)Z

(3-65)

(1

_

2)Z

(3-65)

0

h (Z)

=

1

fd

4 exp

6

1 4

0

Noting

that from

(3.64)

it follows

A (Z) we

=

-

16

obtain from the formulas

A (Z)

-f3 (Z)

-

(3-29), (3.65) 1

A (Z)

f d -1 (2

=

_

8

2

-

-

4

8

and

h (Z)

(3.66)

1 4 ) exp 141 (1

_

(3.67)

7

62)Z

_

6

1

(3.68)

-

0

Using (3.61) one can calculate the one-loop effective action up to cubic terms background fields. Substituting (3.61) in (1.50) and assuming IMM2 < 0, after integration over the proper time we obtain in the n-dimensional space

in

I

2(4,,)n/2

+

1r (2

f

dnx g1/2 str

_

2

+2JI,

+i

,,(_n)jMn+r

i-

2

n) Mn-4

Q-A

4M2

2

1-F_ C3)

RtP4

3

jM-2

J" +

0

(- ) 4M2

R-uv +

2

I +

6

i

R)

)Q

2Q_P2

R_P5

n) Mn-2 (Q

0

R

4m2

E3

4M2

) R) I

+

O(R 3)

I

(3.69)

1 Summation of

64

Witt

Schwinger-De

Expansion

where

162 [1 + (1

fd

F3 (Z)

_

62)Z]

_

2)Z]

2

(n-4)/2

(3.70)

0

A (Z)

1

fd

=

4 [I

6

(1

+

(n-4)/2

(3-71)

7

0

1

.P5 (Z)

`

1

1 d6 (2

62

_

8

164

_

[1 + (1

6

2),]

_

(n-4)/2

(3-72)

7

0

and

P, (z)

and

P2 (z) the

Subtracting

given by the formulas (3-32) and (3.33).

are

pole

in the dimension

1/(n

-

4),

we

obtain the

renor-

physical four-dimensional space-time, (1.53), of third order in background fields

malized effective action in the

(1.54),

up to terms

1

F(l)ren

_(47r_T f

d4X

+i

(- ) 4,rn2

2

1 + 2 JI,

3

g112 str1QF, 0

(-

A

(- EI)

0

R,,,,-P4

P +

4m 2

4m

2)

2QP2

4m

0

RA5

R"' +

Q

4m

2)

R

2) R)l

O(R3)

+

(3.73) where 4

F3 (z)

9

F4 (z)

=

+

225

1

2

1

+

-

(3-75)

-

,

Z

-

1

3

1

1

_360z

(3-74)

,

Z

45z

-

-

9-00

+

6

U

-

37

1

F5 (z)

1

-

1

23 =

+1

(1 1) J(Z) 7+ ) J(Z) To- (1 i5Z2 ( jZ2 ) J(z) 12OZ2

=-

-

1

-

_

60

,

(3-76)

Z

J(z) are given by the formulas (3.35)-(3.37). F3(z), (3.74), F4(z), (3.75), and F5(z), (3.76),

and F, (z), F2 (z) and

The formfactors malized

are nor-

by the conditions

F3(0)

=

F4(0)

=

F5(0)

=

The normalization conditions of the formfactors to the normalization of the

effective action

(3.77)

0.

(3.38)

and

(3.77) correspond

3.4 Summation of Second-Order Terms

F(l)ren

IM2_+00

(3.78)

0.

__':

65

Thus, by means of the partial summation of the local Schwinger-De Witt expansion, (1.43), (3.60), we obtained a non-local expression for the effective action up to terms of third order in background fields, (3.73). Although the power series, that define the formfactors, i.e., the power series for the func-1 being the tion J(z), (3.37), converge only in the region IzI < 1, z threshold branching point, the expressions (3.35)-(3.37) and (3.74)-(3.76) are valid for any z. That means that the proper time method automatically does the analytical continuation in the ultraviolet region IzI -+ 00. All the ambiguity, which arises by the partial summation of the asymptotic expansion (3.60), reduces to the arbitrariness in the boundary conditions for the formfactors. Specifying the causal boundary conditions leads to the single2 valued expression for the effective action. Using the prescription M _+ M2_ie and the equation (3.40) we obtain the imaginary parts of the formfactors (3-74)-(3.76) in the pseudo-Euclidean region (z x i,-) above the threshold =

=

(X

-

< 7r

Im F3 (x

-

i,-)

,

6

X

X

2

Im F4 (x

-

ie)

=

-

ie)

-

30 *7r

Im F5 (x

=

60

(3.79)

_J_

=

(

1

(1+ 1)

1 1+ +

-

3 1

-

4x 2

X

(3.80)

,

X

X

V-1; 1 +

.

(3.81)

X

The imaginary parts of all formfactors, (3.41), (3.42), (3.79)-(3.81), positive. This ensures the fulfillment of the important condition IM F(l)ren > 0 The ultraviolet

asymptotics IzI

-4 oo

are

(3.82)

-

of the formfactors

(3.74)-(3-76)

have

the form

F3 (Z)

F4 (z)

F5 (z) Let in

us

(3.69)

I

I

I

lzl-+oo 1 30

IZI-+00

1 =

00

consider the

in the

region

(-log (4z) +8)+o (Z (- (4z) 46) (Z log (- log (4z) )

-

60

case

log

3

6

=

IZI --

1

I

'::-

z

(3-83)

1

log

+0

+

(3.84)

z

(3.85)

1

1

+

-

+ 0

15

we

obtain the

-

log

z

2

Taking the limit m -+0 one-loop effective action for

of the massless field.

Re n > 2

z

15

the massless field in the n-dimensional space up to terms of third order in background fields

3. Summation of

66

F(1)

Schwinger-De

1n/2 fd 2(47r)

=

1

Xstr

n

+

+

8

2

1)

-

Q (_

Expansion

11) (.1, (a2 2

-

F(n

1Q(_E])(n-4)/2Q+ 2

-

4(n

(2

F

nX g1/2

Witt

-

1

2(n

im (_

1)

-

[:])(n-6)/2 jM

[:])(n-4)/2 R E])(n-4)/2 Rl"

(n 2-

E])(n-4)/2 RI

-

1))2

2)

((n-21- 1) i [R,,, 2n

_

4)R

+

O(R 3)

(3.86)

analytical function of the dimension n in analytical continuation leads to the poles in region dimensions the expression (3.86) is finite In odd the points n 2,4,6,.... and directly defines the effective action. In even dimensions the expression (3.86) is proportional to the De Witt coefficient An/21 (3.53). Separating the pole 1/(n 4) in (3.86) we obtain the effective action for the massless field in the physical four-dimensional space-time up to terms The formula

defines

(3.86)

an

2 < Re n < 4. The

the

=

-

O(R 3) 1

F(1)

-1

dnX g1/2 str

=

-(4-7

2

r

2

1 +

+

6il. (- EI)

lQ (_

6

+i

C +

4

R1"'

R 120

(

8-log 3

log

2

1

1

_

3

-

n

-

C +

-

n-4

46- log

1

2 -

711

47riL2)

15

4

C +

-

15

-

C+2-log

-

4

5 -

-

160 (_

+

4

2 n

C +

-

n

-

2

Q(

log

-13)

0)

47rp 2

Q

J1,

47rp2 R

47r/z2

-O)R]

4jr/L2

+O(R 3)

(3.87)

analogous expression was obtained in the paper [223]. However, in that paper the coefficients ak, Ok, 'Yki Jk, and 6k, (3.55)-(3.59), in the De Witt coefficient Ak, (3.53), were not calculated. That is why in the paper [223] An

3.4 Summation of Second-Order Terms

it

was

assumed

additionally

67

that the power series in the proper time for the

A(z), (3.26), (3.27) and (3-62)-(3.64), converge as well as the proper time integral (1.50) at the upper limit does. Besides, in [223] only divergent and logarithmic, log(- 0), terms in (3.87) were calculated explicitly. The complete result (3.87) together with the finite constants is obtained in present

functions

work. The renormalized effective action for the massless field

can

be obtained

using the ultraviolet asymptotics of the formfactors (3.43), (3.44) and (3.83)(3-85) and substituting the renormalization parameter instead of the mass, M2 _+ p2 This reduces just to a renormalization of the formfactors (3.38) and (3.77). Although in the massless case the finite terms in (3.87) are absorbed by the renormalization ambiguity, the finite terms in the ultraviolet asymptotics .

M2

(3.78)

are

1/(n

essential.

consider the massless field in the two-dimensional space. In this there are no ultraviolet divergences; instead, the pole in the dimension

Let case

effective action

0 of the formfactors with fixed normalization of the

-+

-

us

2)

in

(3.86)

reflects the non-local infrared divergences 1

47r(n

2)

-

1 *

-f

f

dnX g112 str

d 2X

2(47r)

-

* J,,

log

+

R

0

47rL2

1

(- EI)

12

where

tically

we

made

use

the Einstein

g112 str

R+O(R 3)

VIR,,,

0, Q

Q

Q

(_

(log T77

J/.' 2

(- ri)

Q

1 2

R

P+Q

(3-88)

I

equations

0,

2

(3-89)

givR

conformally

invariant

case

(3.51), i.e., J,'

the effective action of the massless field in the two-

1

=

J/'

+C

1 (:]

dimensional space is finite up to cubic terms in and has the form

F(1)

+

of the fact that any two-dimensional space satisfies iden-

For the scalar field in the =

Q

+C-2)

R1,,

=

(

24(47r) f

d 2Xg112

background fields, (3.88),

1

R (_ 1:1)

R+

O(R

3)1

.

(3.90)

On the other hand, any two-dimensional space is conformally flat. Therefore, any functional of the metric giv is uniquely determined by the trace

3. Summation of

68

Schwinger-De

of the functional derivative.

Hence,

Witt

Expansion

the effective action of the massless

con-

formally invariant field in the two-dimensional space can be obtained by the integration of the conformal anomaly [223, 42, 194]. The exact answer has the form

(3.90)

the

O(RI), i.e., they

without the third order terms invariant

conformally -1 divergences Q 13 Q and J,, higher orders O(R 3), (3.88), appear. infrared

-'JO

El

as

well

3.5 De Witt Coefficients in De Sitter In the Sects. 3.3 and 3.4

we

vanish in

When the conformal invariance is absent the

case.

as

the finite terms of

Space

summed up the linear and

quadratic

terms in

fields in the asymptotic expansion of the fl(s) in the powers of the proper time, (1.43). We showed that the corresponding series (3-26),

background

(3.27)

and

(3.62)-(3.64) converge for any value of the proper time (3.29), (3.30) and (3.65)-(3.68). The opposite

the entire functions

summation of those terms in the

asymptotic expansion of the

at

and define case

is the

coinciding

points, (1.43),

E L) k! 18

S2(slx, x)

=

[ak]

(3-91)

,

k>O

that do not contain the covariant derivatives of the summation of the linear and

asymptotics of the effective

ourselves,

to the scalar case,

simplicity,

The De Witt coefficients in the

general

> R 2), whereas the terms without

(11 R

low-energy asymptotics (E] R

derivatives determine its for

quadratic

action

background field. The high-energy

terms determines the

< R 2). We limit

Ruv

0. i.e., have the following coinciding points [ak] we

set

=

form

Q1 R*-

[ak) 0<1
where the

sum

contains

a

curvature tensor and the

...

R.... +O(VR)

(3.92)

k-I

finite number of local terms constructed from the

potential

term

Q, and O(VR)

denotes

a

finite

num-

ber of local terms constructed from the curvature tensor, the potential term Q and their covariant derivatives, so that in the case of covariantly constant curvature tensor and the

potential term, =

these terms

vanish, O(VR)

=

0.

Thus,

derivatives in the De Witt coefficients

oneself to the constant

V1'Q

0,

=

0,

(3.93)

to find the terms without covariant

[ak], (3.92),

potential term, Q

=

it is sufficient to restrict

const, and symmetric spaces,

(3-93). of

The problem of low-energy asymptotics is much more difficult than high-energy asymptotics. A proper treatment of this problem goes

that well

3.5 De Witt Coefficients in De Sitter

Space

69

beyond the scope of this book, since it requires the apparatus of harmonic analysis on symmetric spaces (see, for example, our recent papers [13]-[21]). Our goal in present work is to illustrate how one can employ the technique developed in Sect. 2 to get non-perturbative results. We would like to warn the reader that our result will not be exact but it will reproduce correctly the semi-classical Schwinger-De Witt expansion. That is why we restrict ourselves for simplicity to a De Sitter space, when the background curvature is characterized by only one dimensional

(2.104),

constant-the scalar curvature, R

R"

-

avo

; _(n

-

1)

( pga'3

-

5 Agav P

'

Therefore, the De Witt coefficients

)

R

I

=

const > 0

in De Sitter space

can

be

.

(3.94)

expressed only

in terms of the scalar curvature R

[ak]

E Ck,,Rk-IQI

=

(3-95)

0<1
where Ck,j are numerical coefficients. Hereafter we omit the terms O(VR). On the other hand, from the definition of the De Witt coefficients as the

coefficients of the asymptotic expansion of the transfer function, (1.43), it is easy to get the dependence of the De Witt coefficients on the potential term

Q,

(k)

[ak]

Ck-IR k-1Q1

(3.96)

0<1
where Ck =- Ck,o

are

=

R-k[ak] 1Q=0

dimensionless De Witt coefficients

computed

for

Q

=

0.

The De Witt coefficients in De Sitter space can be calculated along the lines of the method developed in the Chap. 2. However, it is more convenient

by means of the asymptotic expansion of the Green function in coinciding points. It was obtained in the paper [89] in any n-dimensional

to find them

the

De Sitter space

R

i

G(x, x)

( n(n )

=

2

F

F

Pn (0)

=

(1 n) 4,,,(,), _

2

-

i'8) (n 2 + ip) F (n 2 r(!+iP)FQ-iP) 2 2

(3.97a)

(3-97b)

where

#2

n(n

-

1)

R

n

2

M

_

_

Q

-

I

R

_

4n

(3.98)

70

of

3. Summation of

Schwinger-De

Witt

Expansion

The asymptotic expansion of the function has the form

(3-97b)

in the inverse powers

fl2

!Pn (8)

dk (n)

(3-99)

2

'

k>0

The coefficients dk (n)

the

are

gamm -function [98]

E dk (n)Zk

=

determined by using the asymptotic expansion of

from the relation

E

exp,

k>O

k>1

B2k+l

k(2k + 1)

(

1

n

2

)

zk

(3.100)

Bk (x) being the Bernoulli polynomials [98], and have the form do (n)

(_l)k+l

dk(n)

1,

=

B2kj+I

(n2l)

B2kj+1

n

( 21

...

1! 1<1
ki (2ki

kl,...,kl>l

+

1)

kj(2kj

+

1)

(3.101)

k-,+---+kl=k

On the other hand, using the Schwinger-De Witt presentation of the Green function, (1.49), the proper time expansion of the transfer function, (3.91), and the equation (3.95), we obtain the asymptotic Schwinger-De Witt

expansion for the Green function i

G(x, x)

R

I

-47r)n/2 T7r)n/2 - n(n

I:r(k+l- 11) bk(n)#2( a2-1-k)

2

1)

-

S

2

k!

k>0

(3.102) where

R

bk (n) are

=

f n(n

1)

I-k [ak] IQ=-[(n-l)/(4n)]R

the dimensionless De Witt coefficients calculated for The total De Witt coefficients for

the coefficients

bk, (3.103), according

(k)

[ak]

1

arbitrary Q

Comparing (3.97), (3.99) bk (n)

( n(n 1) ) (Q

=

and

(3.102)

we

(_j)k k! _p

To get the coefficients

R.

in terms of

k-1 n

-

+

-

0<1
4n

(3.95):

to

R

bk-I (n)

Q

expressed

are

(3-103)

4n

1R)l

obtain the coefficients

(n)

dk(n)

.

(3-104) bk(n)

(3.105)

2

bk(n) for integer values of the dimension n one has to (3.105). Thereby it is important to take into account the dependence of the coefficients dk(n), (3-101), on the dimension n. Rom the take the limit in

3.5 De Witt Coefficients in De Sitter

coefficients, (3.97b), (3.99), one ) they vanish for k > [n/2] 2,3

definition of these dimensions

(n

=

can

Space

71

show that in

integer

....

dk (n)

0,

=

n

=

2,3,4

...

[n])

; k>

(3.106)

.

2

In this consider first the case of odd dimension (n 3, 5, 7,. gamma-function in (3.105) does not have any poles and the formula (3.105) immediately gives the coefficients bk (n). From (3.106) it follows that only first (n 1) /2 coefficients bk (n) do not vanish, i.e.,

Let

case

us

=

the

-

1

n

bk (n)

=

0,

n

=

3,5,7....

; k>

2

)

(3.107)

'

This is the consequence of the finiteness of the Green function dimension.

in odd

(3.97)

n

1 the expression 2,4,6 ) and for k < 2 (3.105) is also single-valued and immediately defines the coefficients bk (n). For k > 11 there appear poles in the gamma-function that are suppressed by the 2

In

zeros

even

dimensions

of the coefficients

=

-

....

-

dk(n), (3-106). Using the

definition of the coefficients

properties of the Bernoulli polynomials [98], obtain the coefficients bk (n) in even dimension (n 2, 4, 6,. .) for k > n/2

dk(n), (3.100), (3-101), we

(n

and the

=

.

bk (n)

F

(-l)k-(n/2)k! (221) 1' (k + 1 n2

(-l)'

E

k

-

-

(1

1

-

21+21-2k ) BU-21dj(n)

,

0<1<(n/2)-1

(3.108) 2,4,6....

n

B, (0)

where B,

=

n/2

are

-

1),

Let

us

are

; k >

n) 2

the Bernoulli numbers and the coefficients di (n),

calculated by means of the formula (3.101). list the dimensionless De Witt coefficients bk(n),

=

4 in the formulas

bk (2)

in two-

(_l)k-1 (1-21 -2k )

(3.109)

=

(3.108),

=

bo (4)

=

1

bk(4)

=

(_I)k

I The

_k T

J(k

-

1) (1

(1-2 3-2k )

B2k

-

be written in

integral representation

a more

(k

)

>

0)

,

21 -2k ) B2k

B2k-21

expression for the coefficients bk(n) can

<

(3-103), (physical) space-time. Substituting n (3.101), (3-105) and (3.108) we obtain

three- and four-dimensional n

(1

in

,

(k

even

> -

(3-110)

1)

dimension for k >

convenient and compact form.

of the Bernoulli numbers

[98]

2 and

n/2,

Using the

72

3. Summation of

Schwinger-De

Witt

Expansion 00

B2k

(_l)k+l

=

4k 1

dt

f

21-2k

-

e

27rt

t2k-I

(3.111)

+

0

and the definition of the coefficients dk (n),

from

(3.97b), (3.99)-(3. 101),

we

obtain

(3.108) 00

4(-1)S-1k!

bk (n)

(R2).V (k + 1

.P

dt t

2) f

e27rt + 1

0

.1, X

( n21 +it) F ( 2 it) t2k-n -r + ( 12' it) (.1 it) n

-

2

n

Thus

we

2,4,6....

=

; k >

n) -2

(3.112)

have obtained the De Witt coefficients in De Sitter space, [ak], bk(n) are given by the formulas (3.105), (3.101)

(3.104), (3.108)-(3.112).

where the coefficients

and

Using

the obtained De Witt

coefficients, (3.104), one can calculate the coinciding points, (3.91). Substituting (3.104) in

transfer function in the

(3.91)

and

summing the

powers of the n

S?(slx, x)

=

is

exp,

Q

-

potential 1

+

4n

term

we

obtain

isR

R)

n(n

-

1)

(3-113)

where k

-zbk(n) k!

w(z)

(3-114)

k>O

Let

us

divide the series

(3.114)

in two

(Z)

W

=

W1

parts

(Z) +'W2 (Z)

(3-115)

7

where W,

(Z)

E

=

T! bk (n)

(3-116)

,

O
W2

(Z)

k!

bk (n)

(3-117)

k jn/2] The first part w, (z),

(3-116),

is

a

polynomial. Using

for coefficients bk and the definition of the coefficients can

write it in the

integral

form

the expression

(3.105)

dk, (3.97), (3.99),

one

3.5 De Witt Coefficients in De Sitter 00

w,

(z)

=

2

(-Z) T-

f

(--1

r

(-t2)

dt t exp

+ i

2

t

2

0

) F (--1 Z)

_1/_ZZ

-

i

2

2

Z

73

Space t

Nf__Z

)

Z)

=Z

(3.118) The second part W2(Z), (3.117), is an asymptotic series. In odd dimension all coefficients of this series are equal to zero, (3.107). Therefore, in this case 0 up to an arbitrary non-analytical function in the vicinity like W2(Z) =

exp(const/z). expressions (3.108)-(3.112)

Rom the obtained totics of the coefficients

bk (n)

bk(n)

(-J) 2-1

Ik-+oo

=

as

k

in

-+ oo

k!(2k-l)!

4.

even

(27r)

r(k+l-n) F(n) 2 2

one

can

get the

asymp-

dimensions

-2k

(n

2,4,6....

=

(3.119) immediately seen that in this case the asymptotic series (3.117) diverges for any z 0 0, i.e., its radius of convergence is equal to zero and the point z 0 is a singular point of the function W2 (z). This makes the asymptotic series in De Sitter space very different from the corresponding series of linear and quadratic terms in background fields, (3.26), (3.27), (3.62)-(3.64), which converge for any z and define entire functions, (3.29), Rom here it is

=

(3.30), (3-65)-(3.68). divergence of the asymptotic series (3.117), it is concluded [46] that the Schwinger-De Witt representation of the Green function is meaningless. However, the asymptotic divergent series (3.117) can be used to obtain quite certain idea about the function W2(z). To do this one has to make use of the methods for summation of asymptotic series discussed Based

on

the

in the paper

in Sect. 3. 1.

Let

us

define the function W2 (z)

(3.5)

the formulas

according to

and

(3.3):

00

W2

(Z)

=

f dyy'-1e-YB(zy")

(3.120)

,

0

where

Zk

B(z)

k! n/2

k

bk

F(pk

+

(3.121)

V)

and p and v are some complex numbers, such that Re series (3.121) for the Borel function B(z) converges in any

z

and in

formulas

(3.4)

bk, (3-112),

summation. We

=

=

Let

in

2. This

IzI

<

us

substitute the

(3.121)

-7r

and

be

can

+

v)

> 0. The

Rep > 1 for easily using the

case

seen

integral representation of change the order of integration and

get

-4.(-z) F

1 for

(3.119).

and

the coefficients

B(z)

Rep

case

(pa2

2

(!!) 2

( j 0

dtt e

27rt

+ 1

p

(n-1 r Q2

2

it) F (ayi it) H (Zt2) + it) r Q2 it) +

-

,

(3.122)

74

3. Summation of

Schwinger-De

Witt

Expansion

where Z

H(z) k>O

The series

example, for

(3.123) I.L

=

(3.123)

(pk +

kir

v

+

2

converges for any z and defines an entire function. For R it reduces to the Bessel function of zeroth = 1

1,

v

-

2

order

H(z) One

can

show that the

1/.t=l,

integral (3.122)

part of the real axis, Re z

analytical on

Im

0,

<

z

continuation of the series

the whole

the

=Jo

complex plane

(2,/--z)

(3-124)

v=l-(n/2)

can

=

converges for sure on the negative 0. Therefore, for such z it gives the

(3.121).

be obtained

The whole Borel function

by the analytic

B(z)

continuation of

integral (3.122). Let

us substitute now the expression (3.122) for the Borel function integral (3.120) and change the order of integration over t and over Integrating over y and summing over k we obtain

the

in y.

00

(Z)

W2

-4

=

F(R) f 2

r

dt t e

27rt

+ 1

(n-1 F (1 2

2

it) r (ay-1 it) + it).- q it) 2 +

-

exp,

(Zt2) .(3.125)

-

0

This

integral converges in the region Re z < 0. In the other part of the complex plane the function W2 (Z) is defined by analytical continuation. In this way we obtain a single-valued analytical function in the complex plane with a cut along the positive part of the real axis from 0 to oo. Thus the 0 is a singular (branching) point of the function W2(Z)- It is this point z fact that makes the power series over z, (3-117), to diverge for any z :A 0. In the region Re z < 0 one can obtain analogous expression for the total function w(z), (3.115). Changing the integration variable t -+ tv,--z in (3.118) and adding it to (3.125) we obtain =

00 M

w(z)

=

2

(-Z)1 7(12-1)

f

F dtttanh

(7rt)

+ it) F (n-1 (n-1 it) 2exp(zt2) r (21 + it) r it) 2 -

(12

.

-

0

(3.126) summed up the divergent asymptotic series of the terms without covariant derivatives of the background field in the transfer function, (3.113),

Thus,

we

(3.114), (3-126). Using

the obtained transfer function

,

(3.113), (3.118), (3.126),

one can

calculate the

one-loop effective action, (1.50). In order to obtain the renormalized effective action r(i)ren with the normalization condition (3.78) it is sufficient to subtract from the and

integrate

space

over

Q(s), (3.113),

the

the proper time. As the result

potentially divergent

we

terms

obtain in two-dimensional

3.5 De Witt Coefficients in De Sitter

2(47r) n=2

rn2

1

I (rn2

g1/2

d 2X

-F(l)ren

_R 6

Q

75

Space

Q

-

log M

-

18R

2

1

+Q and in

R

+

2

2RX1

-

(3.127)

-

R

8

physical four-dimensional space-time 1

F(l)ren

-

2 (47r) 2

f

1

d 4X g112

1

[M4

_

2

2rn2

(Q R) +

6

n=4

QR +

+

108-0

3

9

1M2

M2

29

1

+Q2

Q

+

i-8

2

R

)

Q

_

R2 log

+

R2

+

6

1024

48

R

M2

5Q2 + 43 QR 96

4

9)

1

41 +

9 _

R2X2

12

4

R

(3-128) where 00

(Z)

X1

=

f

P

dt t

e21rt + 1

log

(1 Z)

(3-129)

-

1

0 00

X2 (Z)

=

I

1

dt t

(t2 ) (1 Z) log

+

e27rt + 1

4

-

-

(3-130)

0

In odd dimensions the effective action is finite.

(3.113)

in

(1.50)

and

integrating

over

the proper time

Substituting (3.118), obtain, for example,

we

in three-dimensional space

F(1)

_1_

-

3 (4-7r)

3/2

f

d3X g1/2

M2

_

Q

6

R)

(3-131)

n=3

again that our study of the low-energy effective action in this book aimed only illustrative purposes. In general case the structure of symmetric spaces is much more complicated that De Sitter space. In particular, the invariants of the curvature do not reduce to the powers of the scalar curvature but also include invariants of the Ricci and the Weyl tenfor example, sors. For more accurate treatment of low-energy asymptotics see, Let

us

[13]-[21].

remind

once

3. Summation of

76

Schwinger-De

Witt

Expansion

only showed that by partial summation of some low-energy terms Schwinger-De Witt asymptotic expansion one can obtain a non-trivial expression for the effective action, (3.127)-(3.131). Although the corresponding asymptotic series diverge, the expressions (3.129) and (3.130) are defined in the whole complex plane z with the cut along the positive part of the real axis. There appears a natural arbitrariness connected with the possibility to Here

we

in the

choose the different banks of the cut.

Higher-Derivative Quantum Gravity

4.

Quantization

4.1

a

Gauge Field Theories

boson gauge field and S( O) classical action functional that is invariant with respect to local gauge

Let M be

of

=

J pij

be the

configuration

space of

a

transformations

(4.1) the gauge group G. Here ' are the group parameters and R',,,( O) the local generators of the gauge transformations that form a closed Lie

forming are

algebra

Ift"', A,31

C-0 ftY

=

(4.2)

where

R',,, C'Y,,,0

and

are

(4.3) j1p

the structure constants of the gauge group

satisfying the Jacobi

identity Ca C/, A[13 76]

(4.4)

0

=

The classical equations of motion determined have the form

--iM

=

by the

action functional

S(W) (4.5)

0,

where ej =- S,j is the "extremal" of the action. The equation (4.5) defines the "mass shell" in the quantum perturbation theory. The equations (4.5) are not independent. The gauge invariance of the action leads to the first

that

are

type constraints between the dynamical variables

expressed through

the N6ther identities

ft,,S

=

R'asi

=

0

.

(4.6)

physical dynamical variables are the group orbits (the classes of gauge equivalent field configurations), and the physical configuration space is the The

orbit space one space of orbits M = MIG. To have coordinates on the has to put some supplementary gauge conditions (a set of constraints) that isolate in the space M a subspace M', which intersects each orbit only in one point. Each orbit is represented then by the point in which it intersects the

I. G. Avramidi: LNPm 64, pp. 77 - 123, 2000 © Springer-Verlag Berlin Heidelberg 2000

78

4.

Higher-derivative Quantum Gravity

given subspace M'. If one reparametrizes the initial configuration space by new variables, M IIA, Xtt}, where JA are the physical gauge-invariant variables enumerating the orbits, M IJA} and Xm are group variables that enumerate the points on each orbit, G X,, 1, then the gauge conditions can the

=

=

,

be written in the form

X/' (,P) where on

0/-,

the

are some

constants, i.e.,

physical configuration

0/'

=

,

0. Thus

01,,j

JJA'

space M'

we

obtain the coordinates

0 0.

The group variables X1, transform under the action of the gauge group

analogously

to

(4.1) JXA

Ql.,. MV

=

(4.7)

where

Qi,,, (W)

=

X1,,j (W) R',, (W)

(det Q $ 0)

,

(4.8)

,

are the generators of gauge transformations of group variables. representation of the Lie algebra of the gauge group

[(L' (1,31

C,

=

ao

They

form

Q^'Y

a

(4.9)

where

0.

j

QMa

=

JX1,

All

physical gauge-invariant quantities (in particular, the action S( O)) expressed only in terms of the gauge-invariant variables JA and do not depend on the group variables X4 are

S(W) ='9(jA(W)). The action

S(JA)

is

quantize the theory variables the form

W'. The [223] exp

an

usual

ir(fl I

where

M(W)

is

R,,,,i

function at

a

local

and

,

integral for

the standard effective action takes

=f dWM(W)6(X,,(W)-0,,)detQ(V) x

the terms

non-gauge-invariant action. Therefore one can IA and then go back to the initial field

in the variables

functional

(4.10)

exp

measure.

Cl,,,

that

h

The are

coinciding points 6(0)

[S(W)

-

measure

(W'

-

M(W)

proportional

0)1 j(fl] is

I

I

gauge-invariant

(4.11) up to

to derivatives of the delta-

and that should vanish

by the regularby the canonical quantization of the theory. In most practically important cases 1 + 6(0)( M( O) ) [2231. Therefore, below we will simply omit the meaization. The exact form of the

=

...

sure

M (W).

measure

M(W)

must be determined

Quantization of Gauge Field Theories

4.1

(4.11)

Since the effective action

0,,,,

stants

integrate

one can

over

depend

must not

them with

the arbitrary conweight. As result we

on

Gaussian

a

79

get i exp

h

x

fd p

_V (4i)

exp

h

[S (W)

Q ( p) (det H) 1/2

det

-

2

X1, (W) H"' X, (W)

-

(o

-

(4.12) where Hill is

non-degenerate operator (det H 54 0), that does

a

not

depend

on quantum field (6H1"1JV' 0). The determinants of the operators Q and also be H in (4.12) can represented as result of integration over the anticalled Faddeev-Popov [101, 781 and Nielsen-Kallosh commuting variables, so =

[182, 162] "ghosts". Using

the

equation (4.12) F

we

find the

one-loop effective action

S + hr(1) +

=

2i

(4.13)

,

det.A

1

r(l)

0(h2)

log

det

(4.14)

H(det F )2

where

4k

=

-S,ik

+

X,,iH"vXvk F

=

Q( o) I

Xpi

=_

(4.15)

Xu,i(w)

(4.16)

*

0=0

On the mass shell, I i = 0, the standard effective action, (4.12), and therefore the S-matrix too, does not depend neither on the gauge (i.e., on the choice of arbitrary functions X1, and HAv) nor on the parametrization of the quantum field [84, 51, 223). However, off mass shell the background field

as

well

as

the Green functions and the effective action crucially depend parametrization of the quantum field.

the gauge fixing and on the Moreover, the effective action is not,

both

on

generally speaking, a gauge-invariant procedure of the gauge quantum field automatically fixes the gauge of the background

functional of the

fixing

of the

background

field because the usual

field. The gauge invariance of the effective action off mass shell can be preserved by using the De Witt's background field gauge. In this gauge the functions x1l(W) and the matrix HAv depend parametrically on the background field

[78, 84] XA (

O)

=

X,- (w,

fl

,

Xt,((P,fl

=

0

,

Hl"

=

Hl"(fl

,

4.

80

Higher-derivative Quantum Gravity

and are covariant with respect to simultaneous gauge transformations of the quantum W and background 4i fields, i.e., they form the adjoint representation

of the gauge group

R,, (W)XI, ( o, fl

R,(flHI"(fl In the

ft., (4i)X4 (W, 0)

+

-

background field

H'3'( P

C"

gauge method

+

-

one

C,X , (W,!P) 4

C'

,H1'13(4i)

=

=

0

,

(4.17)

0.

fixes the gauge of the quantum

field but not the gauge of the background field. In this case the standard effective action (4.12) will be gauge-invariant functional of the background field provided the generators of gauge transformations the

0 [84]. fields, i.e., R'a,kn(0 However, the off-shell effective

Ricj(p)

are

linear in

=

action still

depends parametrically

on

the

choice of the gauge (i.e., on the functions Xm, and H11,'), as well as on the parametrization of the quantum field W. As a matter of fact this is the same

problem, since the change of the gauge is in essence a reparametrization of physical space of orbits M' [2231. Consequently we do not have any unique effective action off mass shell. The off-shell effective action can give much more information about quantum processes than just the S-matrix. It should give the effective equations for the background field with all quantum corrections, Fli (4i) 0. A possible solution of the problem of constructing the off-shell effective action was proposed by Vilkovisky [223, 224] both for usual and the gauge field theories. It was further improved by De Witt [85, 86], but we will not discuss this improvement here. For extensive updated bibliography see, for example, [53]. The condition of the reparametrization invariance of the functional S(W) means that S(W) should be a scalar on the configuration space M which is the

=

treated action

as a

r(W)

Wi.

manifold with coordinates can

be traced to the

(Wi

the difference of coordinates

V)

The non-covariance of the effective

(Wi

term

is not

-

0)1 i(fl

in

(4.12)

as

geometric object. One can replaces this difference by a two-point function,

achieve

covariance if

ui((p,!P),

that transforms like

one

source

a

-

vector with

a

respect to transformations of the

field !P and like

a scalar with respect to transformations of the background field. This quantum quantity can be constructed by introducing a symmetric

gauge-invariant

connection

ri.,,m

M and

on

by identifying

the tangent vector at the point !P to the geodesic connecting the and 4i. More precisely, oi(W, 4i) is defined to be the solution of the k a

Vko'i

i =

with

points W equation

(4.18)

or

where

VkU with the

boundary

condition

a

=

&pk

+

I'ik,,,, (fl 0'-

,

(4.19)

Quantization of Gauge Field Theories

4.1

ail V=-P Thus

we

obtain

i exp,

h

0

(4.20)

.

unique effective action F(fl according

iP(!P)

exp

x

a

=

dW (det H (fl) 1/2

[S( o)

det

2

X4 (

to

Vilkovisky

Q (W,!P)

I -

81

o, flHl" (- 'P)X, ( o, fl

+

a'(W, flfli (fl

1

(4.21) Since the integrand in (4.21) is a scalar on M, the equation (4.21) defines a reparametrization-invariant effective action. Let us note, that such a modification corresponds to the definition of the background field -fi in a manifestly reparametrization invariant way, <

ai(w, fl

instead of the usual definition V =<

(4.22)

>= 0

p'

>.

perturbation theory is performed by the change integral, Vi -+ ai(W,fl, and by expanding covariant Taylor series

The construction of the

of the variables in the functional

all the functionals in the

S( O)

=

E

ai'

...

aill

k!

lv(i

...

vios(v)]

(4.23)

k>O

The

diagrammatic technique

for the usual

effective action results from the

substitution of the covariant functional derivatives Vi instead of the usual the terms 6(0) ones in the expression for the standard effective action (up to -

that

are

In

caused by

particular,

variables). change one-loop approximation, (4.13)-(4.16), of

the Jacobian of the

in the

det A

1

P(l)

=

-

2i

log

det H (det

F) 2

'

(4.24)

where

4k

=

-ViVkS+ XiiH"'Xvk

(4.25)

on the physical configuration space M' let of first a gauge-invariant metric Eik(V) in non-degenerate all, introduce, the initial configuration space M that satisfies the Killing equations

To construct the connection

us

E),,,Rna

+

E)nR,,,,,,

=

0,

(4.26)

where R,,,,, = EmkRk. and D,,, means the covariant derivative with Christoffel connection of the metric Eik

82

4.

Higher-derivative Quantum Gravity

Ijkl =2 E-1"(Emj,k The metric

Eik ( P)

must

Nt,, This enables

ensure

=

the

RgEikR

P'

k

(4.27)

of the matrix

(det N 0 0)

V,

Rm

Ejk,m)

-

non-degeneracy

to define the De Witt

one

EmkJ

+

(4.28)

projector [78, 83] H2J_

B'

=

JJ-_L

(4.29)

where

N-1,"'R' Ein

B ',, n

The

every orbit to

projector U1 projects

Ili_" R'o, orthogonal

to the

the

(4-31)

generators R k. in the metric EiA;

R"OEkmlli_" Therefore,

point,

one

0,

=

S

and is

(4-30)

-

=

0

(4-32)

-

subspace Hj_M can be identified with the space of orbits. physical conditions lead to the following form of the connec-

The natural tion

the

on

configuration

space

[223, 224]

mn I

F?'mn

+

Timn

(4.33)

5

where T' mn

+ B" B'3 R -2B"Vn)Rl, (m (M n)

The main property of the connection

Vnk,,

(4.33) oc

R',,

k

DkR,,

(4.34)

is

(4.35)

-

It means that the transformation of the quantity ai under the gauge transformations of both the quantum and background field is proportional to Ri,,(fl, and, therefore, ensures the gauge invariance of the term or'. i in

(4.21). is

This leads then to the fact that the

reparametrization invariant and does

Vilkovisky's effective action F(!P) depend on gauge fixing off mass

not

shell.

On the other

hand,

it is obvious that

on mass

shell the

tive action coincides with the standard one,

f;(!P) Ion-shell and leads to the standard S-matrix

=Ir(4 )Ion-shell [223].

Vilkovisky's effec-

4.2

To calculate the

Vilkovisky's effective

The result will be the

simply put

the

non-physical

TiM,,

=

ail

,

o,

group variables

=

in Minimal

equal

the

83

Gauge

choose any gauge.

one can

use

orthogonal

gauge,

i.e.,

to zero,

R'lt (fl Eik (4i) ak ( O' 4i)

=

0

,

(4-36)

The non-metric part of the connection

17j-',,al.

u'j_

where

action

It is convenient to

same.

XIL (

i.e., ai

One-Loop Divergences

satisfies the equation

H_LmkT'm ,17-_L n.3

=

0.

(4-37)

Using this equation one can show that it does not contribute to the quantity ail Therefore in the orthogonal gauge (4.36) the quantity TiMn in the connection (4.33) can be omitted. As a result we obtain for the Vilkovisky's .

effective action the equation i

(Ri,, (4i) a'(V, fl) det Q (V,!P)

dV 6

(4i)

exp

x

[S(V)

exp

+

u'(V, flf,i

(4-38)

where

Ri,, (,P) R' , (V)

(V,!P) The

change

of variables V

-+

ai(V, fl

and covariant expansions of all

functionals of the form

S(V)

=

E

[1)(i,

k!

.'Di,,)S(V)]

(4-39)

k>O

6(0)) to the standard perturbation theory with simple functional derivatives by the covariant ones with the usual of replacement Christoffel connection, Di. In particular, in the one-loop approximation we

lead

(up

to terms

-

have

det.Aj_

1

-log

F(1)

2i

(4.40)

(det N)2

where

31

3ik

=

-E)iDkS

=

-S,ik

+

Ijikj SJ

(4.41)

4.2

One-Loop Divergences in Minimal Gauge

The

theory of gravity with higher derivatives as well as the Einstein gravity theory of gravity is a non-Abelian gauge theory with

and any other metric

84

4.

Higher-derivative Quantum Gravity

the group of

diffeomorphisms G of the space-time as the gauge group. The complete configuration space M is the space of all pseudo-Riemannian metrics on the space-time, and the physical configuration space, i.e., the space of orbits A4, is the space of geometries on the space-time. We will parametrize the gravitational field by the metric tensor of the space-time g4v W

=

W

i

,

The parameters of the gauge transformation of the infinitesimal

(AV'X)

=

are

diffeomorphism, (a general XJU

'

XP

_+

'(X)

=

'W

-

=

/-'

,

(4.42)

-

the components of the vector

coordinate

transformation)

,

(p, X)

(4.43)

-

The local generators of the gauge transformations in the parametrization = (4.42) are linear in the fields, i.e., 0, and have the form

Ri,,k,-,

R',, Here and

=

2V(tgv),,J(x, y) when

below,

derivatives act

i

,

writing

(PV, X)

=_

a -=

,

(a, y)

(4.44)

-

the kernels of the differential operators, all the

the first argument of the delta-function. The local metric tensor of the configuration space, that satisfies the on

Killing

equations (4.26), has the form Eik

=

g1/2 Euv,'13J(x, y)

i

,

g '(g '6) where

r.

0

0 is

(/-IV, X)

=_

1 V -

4

(1

+

k

,

=-

(a,8, y)

,

r.)glvg, 0

(4.45)

numerical parameter. In the four-dimensional pseudoequal to the determinant of the 10 x 10 matrix:

a

Euclidean space-time it is

x

The Christoffel

(9 1/2 E"",3 )

det

=

symbols of the

metric

fjk I fl'v' i

Eik have the form

a)3

(X) Y) 6 (X) Z)

=

\P

i

1

1.1v'a'6

AP

E

(Ap, x)

j

,

JO-1 v)(aj,6)

I

('X9

P)

1 +

4

=_

(ILV, Y)

(gAvjC1,3 AP

J"' The operator

NIL,(W), (4.28),

is

a

k

,

+

51, jV

(C' 0)

ao

(4.46)

-

=-

ga,661LV AP

(a,8, z) +

,

r.-1E1",c,6g,,p (4.47)

*

second order differential operator of the

form

Nl,v(V)

=

2g'/2

f

-

9ttv 0 +

I(K

2

-

I)V/,Vv

-

RAv

1 6(x, y)

.

(4.48)

4.2

One-Loop Divergences in Minimal Gauge

85

non-degenerate in $ 0, imposes a

The condition that the operator N,,,, (4.48), should be perturbation theory on the flat background, det N

IR=O

the

constraint

the parameter of the

on

metric,

ofgravity theory following general form

The classical action ture has the

d

S( O)

where R*R*

1

4XgI12

3.

R

2

RAP, ,R`6,,,0

is the

R+2 0

V

6V2

2f2

terms in the

quadratic

C2

cR*R* +

4

54

n

with

I

topological term, estlva,3

JU

curva-

(4.49)

is the

anti-symmetric Levi-Civita tensor, C2 _= CI'v",8Ctv,,,3 is the square of the the Weyl coupling Weyl tensor, E is the topological coupling constant, f2 2- the Einstein coupling constant, v 2- the conformal coupling constant, k A 2 is the dimensionless cosmological constant. Here and constant, and A -

=

below

omit the surface terms

we

equations of

motion

nor

11 R that do not contribute neither to the

-

to the S-matrix.

The extremal of the classical action,

and the N8ther

(4.5),

identities, (4-6),

have the form

6S 6i

( O)

Jg,,V 1

+T

[2

-

3

(1

+

9

1

1/2

R"V

jF2

w)R(R"v

-

2

1 -1gl'vR) + 9"R,,3R'I3 2 4

1

1 +

3

(1

-

2w)VIVvR

-

0 RAv + 6

V1,60V where

w

Let

(4.14).

=

us

(1)

=

(1

+

4,)gtv

F1

2R"13Rava

R]

(4.50)

(4.51)

0,

f2/ (2V2).

calculate the

From

T-Tdiv "

1gl'vR + Ag"

-

(4.14) 1

=

_

2i

we

one-loop divergences of the standard effective have up to the terms

(log det A I

div -

log det H I

The second variation of the action

S,ik

(4.49)

-

action

6(0)

div -

2logdetFI

div

(4.52)

has the form

-) (0)Apa-rVAVPVqVT + VPF(2)poVo'

+F(3)pVp

+

VPF(3)p

+

P(4)

9

1126(X, Y)

(4.53)

86

4.

Higher-derivative Quantum Gravity

where

F(O)),p,, F(2)po- F(3)p and F(4) are the tensor 10 x 10 matrices `3 (F(O),x,,,,, F(10") ,\,O,,, et c. ). They satisfy the following symmetry relations 7

,

=

^T

F

=

(2)po-

F(2),oo,

^T

F

i

F(2),oo, with the

^T

F(3),o

3),o

F(4)

=

F(4)

F(2)ao

=

(4-54)

symbol 'T' meaning the transposition, and have the form 1

FI'V'Oe'a

(gP(,,,g'8),,

-

g,\Pg",

f2

1 + 4w

2

2.)JIvJ,' AP

3

h'/'tV'a'8 (2),oo,

1

(g"('g'3)v

2Ro,

=

Tf2

+2J("Rv)('J'3) (0,

-4

P)

3

4wgO,g,,,6)

(9 ""J"' g'v g'I3j"\'P"g")

3

+

I + _

+ 2

V) (a 60) (R((P'g (P or)

g

+ R

-

2J(vg")('J )gP A

lgl"ga,3

-

+

2

0

J("Rv) (P

0)

+

4go,R('II'Iv)O

g'"kaRO) (P -) I

0) Oa 60 ((PO'g (P 0')

4 *

3

(1

(

*

M

+

2 2

2 +

-gl"Jpc'

V a

F( 3)

w) [RIv

3

-

(1

+

(6p'

-

g',Ogp,)

+

R'13

w)R) (-gpogce(tt9V),8

(6p",'

+

-

gl'vgp,)]

9PU909 ttv

a)(I'Jv) g'13JPI',v + 26(3 P)) ('9

(gc',3V(,"Rv) g"'V("RpO)) -

f2 +

(1

2

-

P

2w)

(gl"VpR"3

_9(1-t(,3V-)RPv)

-

g'16VpRIv)

1 + 12

(1

+

4w)

g

ao

+

g(O("Vv)Rc) P

Jp(IVv)R

-

g"'JP("V,8)R)

'

4.2

+1 (1

-

6

+2 (2

-

3

Gauge

87

6P(agI3)6'V')R 6(Agv)(aV'3)R

2w)

w)

in Minimal

One-Loop Divergences -

P

(6P(13V')RIv 6P(I'V")R'13) -

+3 P

2

1

1

A V,Co

F(I 4)

=

-

2f2

(g'13R,,',R'-' + gOvRP'R'3,')

(g ,(,g,),

-

+

2

-

1g"g a,8 ) 1RpRP'

1(1 + 4w) 0 R _,rn2(R

-3 RP' (g"13RI' P 2

(M2

2

+

' 0,

0,

v) RI"-8 a

1

3

(1

+

w)R2

1 +

Ra( 'R),a

2(1 + w)R) (RI"g'13 + R"Iog4'

-4 (1 + w)RI"R",6

+

2

-

w

3

-4(1 3

+2 El

R(1'1'1')13

+2 0

R(,"("go)-)

+V (I'V) R'13

f2 Ik2

P

3

a

2 m2

2A)

gi"RaB ) P

+

+6g(13('R")P ') RP'

where

-

2

4R(I'

-

2

3

+

+

-

+

-

(R(' 16)(I'R)P P

+

R(I'P ')('R'3)P

(g'16VIVR + gi"V'V,6R) 2w)g(,8('Vl') V') R

2(1 + w)

V('V,6)RO'

3

9"3

El RI"

9-`

El

R"3)

(4.55)

4.

88

Higher-derivative Quantum Gravity

Next let

choose the most

us

general

linear covariant De Witt gauge

con-

[83]

dition

X1, (W, where hi

Vi

=

P)

R

=

k

(4i) Eki (fl h'

1j,

4.56

V. In usual notation this condition reads

-

Xt&

-

-2g 1/2VV

I

1

hv

-

"

4

(1

r.)6,vh

+

I

raising and lowering the indices as well as the covariant derivative by means of the background metric gj,,v and h hill. The ghost the in is the to equal operator F, (4.8), (4.16), operator N, gauge (4.56) where are

defined

=

(4.48), Fl,v

=

Nj,,

1

2g1/2

=

-

9,UV 0 +

1(r.

It is obvious that for the operator

flat

-

2

1) Vj, V,,

,A, (4.15),

it is necessary to choose the

space-time

-

R,,,,

to be

1 J(x, y)

(4.57)

.

non-degenerate

operator H

as

a

in the

second order

differential operator 1

H,uv

=

4a

where

a

and

P

29-

are

1/2

I_gfiv

E]

+#VtiVv

+ Rl" +

numerical constants, P11' is

an

PI'v} J(x, y)

(4.58)

arbitrary symmetric tensor,

e.g.

PMv

P1

i

P2 and P3

being

piRl"

=

+

gt"

(P

2R

+ P3

arbitrary numerical

some

(4-59)

T2

constants. Such form of the

operator H does not increase the order of the operator A, (4.15), and preserves its locality. Thus we obtain a very wide class of gauges. It involves six arbitrary parameters (r., a, 0, Pl) P2) P3) In particular, the harmonic -

De Donder-Fock-Landau gauge,

V,

corresponds to r. eters disappear.

=

1 and

a

=

(

1

h'

Ph

-

"

2

1'

0. For

a

=

)

=

0 the

(4.60)

0,

dependence

on

other param-

It is most convenient to choose the "minimal" gauge 2

2 a

=

ao

1

f2

,

0 =00

which makes the operator

=

3(1

3w -

2w)

A, (4.15), diagonal

-

r.

in

=

ro

=

i

,

+

(4.61)

uj

leading derivatives, i.e.,

it

takes the form 1

4k

=

2f2

jt(O)n2+VpbP,,Vo,+Vp rP+ VPVP

+ P

J(X, Y)

(4-62)

4.2

One-Loop Divergences

in Minimal

Gauge

89

where

El",'O (r.

t(o)

=

bp,

ro)

VP =VP AV,a,3I are some

P

satisfy the

tensor matrices that

=

D

=

P0,

P'UV'a'8 symmetry conditions, (4.54).

same

They have the following form

Dttv,afl P0,

1 2f2

+2

19a

2RPa

=

(1

2 2

1 +

-6jzVga'3 PC

-3 (1

+

1

1 + 4w

1-+ W

1

1 + 4w

8

1+W

1

Tf-2

1(1

3

(1 -

1 +

+

W

)2)

9

/1 V

9

a,8

g-"'R(cJ,)3)) (P

2p,6(IRv)('b'3) (P 0')

+

6(4gv)(13R") (a P) ) -

g"13gp,)

+

R"6

9P0'9Ce (.U 9V)13

(P

2R

(bpi,v

+

+ P3

-

gl'vgp,)]

gP'gIVg"I'

T2

-

g

0 P

[-2J(3g')("6v) (0,

P)

(gttv6a'8 + ga06,uv) P0,

P0,

2

)

9P0'9 jzv9ao

1 + .

2w)

(g"vVpR"3 P

1(1+4w) 12

II

!+W

-

I

4w) (g'16V(ARv)

2

-g-

+g(,3("Vv)R') +

1 + 4w

(

(P

w)R)

+

p,

+

16

W

+ 26((:g (0, C)(AjV)) P)

2

+

+

-

2 +

L

1 +

+

4

(J(a(P g0)(vRP)0')

(M

*

P

2

4(1 + w) [R" (Jp" 3

*

V,uv,ao

9

(1 4w)) (g'13R(IJv,)

P1 +

+4gp,R("I'I')13 -4

(

(p v),8

P

-

g"'V(IRIP6)

g',3VpR1")

g(A(16V)Rv) P 3

1-

1+w) (g '165("VOR p,

.

4

P

-

g"v6("VI6)R P

90

4.

Higher-derivative Quantum Gravity

(1

+

2w

-

P2

6

2

2

(2

+

3

+

2

-

)

1

pt&v,oeo

-

p

(1

_

2f 2

Jp(i'g)('VO)R)

(Jp('V'3)Ri" Jp(IV')Rlo)

w)

PI

(2

+

-

) (Jp('g'3)(IV)R

2

1 + 4w

p, +

p

Rp 'R'P

.

4

1 +

W

(gA(aga)v lguvgceo) [Rp,RP' -

-

2

+

3

(1+4w)DR- M22(R

3 2

+(2

(,_pl .1+4w ) 12

+

RP'

1+w

pl)R'(I'R")O -12p1

+6g(,6(vR,u)p ")

0-

+

-

(M2 2

+

RP'

-

2A)

+

gl"R'Rl3p) p

1(1 + w)R

2

3

1

(gcelaRl'p'o, + gl"'RO'Pl3or)

(RP(I'Rv)

+

p

RP('R'3) (I' p

4(1 + w)RIvR',o + 4R(4 v)RP(',,)3) 3 p

2(l + w)R)

3

(Mwgo'13

+ R

a,39/,ZV

-3g('(I'R")13)

+

(p2R T2T1-2) (g('("R')Io)

+ 20

+ p3

(g',aV"VvR + gl"VVIOR)

+2 (1 + W) 3

9a)3

0 Rl'v

-

gl'v

+ 2 El

0

R'13

R("I'I")3

R(I'("g'3)v) +

V('V'3)RA"

One-Loop Divergences

4.2

+V(1'V')R'16) The and

divergences

A, (4.62),

can

4

(1

-

3

in Minimal

2w)g(16('V1') V') R

-

91

Gauge

(4.63)

of the determinants of the operators H, (4.58), F, (4.56), be calculated by means of the algorithms for the non-

minimal vector operator of second order and the minimal tensor operator of forth order. These algorithms were obtained first in [107, 108, 109] and

confirmed in

generalized Schwinger-De Witt technique. In 11 R they have the form regularization up to the terms the

[34] by using

the dimensional

-

log det 1-

0 61' + V

+ R" + V

#V"V,

,,

d4X g1/2 14) (41r) 2f 2

1

_Fn

-

I

P"I

div

8

45

6

1

2

R 36

60

1

1 +

7C2+

R*R* +

(

+

6)RlvP"v

12

(

+

2)RP

+

2p2

T8

1 +

24

(2

+

6

+

(4.64)

12)PjvF1'

where

P"

P

'U

-01 and

div

log det

E]

2+ V" Djjv Vv

+ VA V1, +

V1, Vtz

d4X g1/2 tr i (4) (41r )2 f

(n 1 +

-

1

t-ip +

R*R* + 180

Rt-1-b

-

12

6

I +

t-lbt-lb

I +

24

48

1

1

2

-

+

C2

60

1

R

+

36

2)

1Rljv t-ibl-tv

6

it-' bliv t-lbliv

(4.65)

where D

=-

D" 'U

t, bl", 01 bAv

=

and

D AB11v

P

are

etc.), i

the tensor matrices

6A, &AV B '

=

ant derivatives of the tensor field

=

RABuv

(It

EAB, P-1

=

E iBl'

is the commutator of covari-

4.

92

Higher-derivative Quantum Gravity

[V,,, V,]hA 'tr'

means

In

our

the matrix trace and case,

hA

and

(4.65)

(4-66)

is the dimension of the

n

RY 5"31,,,

=

1

(n

4) (41r)

-

1

space-time.

w)

+

5 + 2w

1

w)

+

192

c2

60

(1-2w )2 1 +

p2

W

+ 80w + 61

PA v P1,1v

+

96(w + 1)2

div

2 =

7

R*R* + 45

R1,P"'

RP +

28W2

(4.61)

8

2f d4X g1/2

12(1

24(1

(4.67)

*

13 + 10w

R2

36

log det, F

J)'UV

(Y

obtain in the minimal gauge

we

2 =

-26("RO)

=

,

di

log det H

RABI,,,hB

huv7

=

Bl,,,

Using (4.64)

=

1

(n

4) (47r)

-

(4.68)

I

2f d4X g1/2

1 X

540

(20W2

+ 100w +

1

(5W2

+

135

+ 25w +

2) C2

1 +

81

di

log detzA

2 1

(n

-

4) (47r)

(5w

2f

+ 16w +

d4X

g1/2

20)R

T4

1

W2 + 20w + 367) C2 +T(4 4 1 +

162

41)R*R*

(200w2+ 334w + 107)R 2

2

(4W2

(4.69)

+ 20w +

253)R*R*

One-Loop Divergences

4.2

I

(

+

6

40w

26

-

3) f2

-

V2)

+

3

R

+

2

1 -

192

(5f4 + V4)

5 + 2w

13 + 10w

RMPl"

-

w)

+

93

1

A(14f2 T4 [4

12(1

Gauge

k2

w

1

+

in Minimal

(1-2w )2 1 +

RP

+

24(1

p2

+

28W2

w)

+ 80w + 61

PI'm PI"

_

96(w + 1)2

W

(4.70) Substituting the obtained expressions (4.68)-(4.70) in (4.52) we get divergences of the standard one-loop effective action off mass shell 1

rdiv

f

-

(n

-

4) (47r )2

d 4X g112

1

+#3 R

#4

+

8,R*R*

+

the

2C2

1

T2- (R

+ 7

j4-

-

(4.71)

4A)

where 196 45 133

02 03

04

2

(5f4 'Y

Therefrom it is

5

f4

18

4

=

+

V4)

+

5

f4

3

T2

=

5

f2

6

2

+

5

(

seen

(4.73)

+

36

2A 10t + 15f2 3

_

immediately

(4.72)

20

V2

V2

13f2 6

_

'V2 2

that the gauge fixing tensor, we will calculate the

does not enter the result. In the next section

(4.74) (4.75) PAv, (4.59) divergences

of the effective action in

arbitrary gauge and will show that the tensor P11v 0 divergences in general case too. If one puts Plv then the divergences of the operator H do not depend on the gauge fixing does not contribute in the

parameters

=

at all.

Our result for

divergences, (4.7l)-(4.75),

does not coincide with the results

of the papers [107, 108, 109, 111] in the coefficient 03, (4.73). Namely, the last term in (4.73) is equal to 5/36 instead of the incorrect value -1/36

94

4.

Higher-derivative Quantum Gravity

obtained in of

[107, 108, 109, 111].

We will check

completely independent computation

on

our

result, (4.73), by

the De Sitter

background

means

in Sect.

4.5.

4.3

and Let

One-Loop Divergences in Arbitrary Gauge Vilkovisky's Effective Action

us

study

now

the

dependence of the obtained

result for the

divergences of

the standard effective action, (4.7l)-(4.75), on the choice of the gauge. Let us consider the variation of the one-loop effective action (4.1) with respect to variation of the gauge condition

1

1 2

X,Uiz.1

ik

XukH"v

-1 ik

Xvk

-

-

H

-1

-H

=

+

-1

11v

F-1

tta

that follow from the N6ther

blql)

_,A-1

R'oF-1

x

(4.76)

ikej Rj,,,,kF-1

(4.77)

(R kly

_

O'EjRj 3,k

A-1

identity, (4.6),

1 ik

kn

e,,,R',y^f,n)

one can

F -1'Y6H-1 6V

derive from

(4.78)

(4.76)

6jRja,kF-1 "v6Xvj

1F-1'6'6jRj0,k (R

2

k

A-1

_

kn

7

e,,,, R

'Yrn)

F- 1

-YdJ(H-1 da

here, it follows, in particular, that the one-loop effective e 0, (4.5), does not depend on the gauge,

(4.79)

action

on mass

=

6F1 Since the effective action in

6HA" H-1) AV

-

the Ward identities

,A-1

Rom

HA')

R' J-' ") 6x 'i

-

(Xt','A-1 ik Xvk

+

shell.,

the functions Xt, and

(,A-1 ik XIAHI"

-

Using

(i.e.,

background

fields in the

on

=

(4-80)

0.

on-shell

the

mass

neighborhood

shell is well of the

defined,

mass

shell

it is

analytical

(4.5). Therefore,

One-Loop Divergences

4.3

in

Arbitraxy Gauge

95

be

expanded in powers of the extremal [223]. As the extremal has the background dimension (in our case, (4.50), equal to four in mass units), this expansion will be, in fact, an expansion in the background dimension. It is obvious, that to calculate the divergences of the effective action it is sufficient to limit oneself to the terms of background dimension not greater than four. Thus one can obtain the divergences by taking into account only linear terms in the extremal. Moreover, from the dimensional grounds it follows that only it

can

the trace of the extremal

E

=

gove

(4.50),

jiv

=

1

1

1

g1/2 T2_ (R-4A)- _2_ OR

1

(4.81)

1

contributes to the divergences. Therefore, only -y-coefficient, (4.75), in the divergent part of the effective action depends on the gauge parameters. The 0-functions, (4.72)-(4.74), do not depend on the gauge. So, from (4.79) we obtain the variation of the one-loop effective action with respect to the variation of the gauge

,-jRj,,

div

-1

k'F-1

"136X,3i div

k

-

2

Rom here

one can

ejk,, ,kR 13 F-1 13'rF-1 "65(H-1) Y6

obtain the

divergences

of the

(4.82)

effective action in any

form of the gauge condition with has, first, the calculate to arbitrary parameters, then, divergences of the effective action for some convenient choice of the gauge parameters and, finally, to integrate gauge. To do this

to fix

one

some

the equation (4.82) over the gauge parameters. Let us restrict ourselves to the covariant De Witt gauge, (4.56), with arbitrary gauge parameters a, P and PA'. Since the coefficient at the variation

5H-1 in

(4.82)

does not

rameters a, 3 and

Thus

we

P"'),

obtain the

divergences

r,div (r., a,,3, P) (1)

depend one can

the operator H (i.e., on the gauge paintegrate over the operator H immediately. on

of the

effective action in

1

1-div(,'0' ao,flo)

+

1f

arbitrary

d r.

Udiv

div

(r., ao, flo,

gauge

go

2

where r,div (r,0,a0,#0) is the (1) minimal gauge, (4.7l)-(4.75),

U2Ldiv(K a,fl, P)

divergent part

-

U2c

P)]

(4.83)

of the effective action in the

96

4.

Higher-derivative Quantum Gravity

UI (r.)

=

ej

Rjce,

k

'A-1

ki(n, ao,flo,P)F-1'I3(r.)RJ8 Et. i n

d

1

Ein

Ein

1/2

49

9

(4.84)

9,xlj(x, Y)

/,IV

div k

U2(r.,a`#,P)=e--Rja,k R P F

-

8,y

1

I

(n)F-1 `(r.)H-1 -YJ (a, #, P)

(4.85)

To calculate the

quantities U1, (4.84), and U2, (4.85), one has to find the F-1 for arbitrary r., a, 8 and P and ghost propagators (r.) and H-1 Y6 (a, #, P) the gravitational propagator A-' (r.,ao,flo) for arbitrary parameter r. and minimal values of other parameters, ao and Po. The whole background dimension that causes the divergences is contained in the extremal '-j. So, when

calculating

the

divergences

of the

quantities U1, (4.84), and U2, (4.85),

one

take all propagators to be free, i.e., one can neglect the background quantities, like the space-time curvature, the commutator of covariant derivatives can

etc., and the

terms. This is

so because together with any dimensional automatically a Green function 1:1 -1, which makes the whole term finite. Therefore, in particular, the gauge fixing tensor P"' does not contribute to the divergences of the effective action at all. For the minimal gauge, (4.61), we have shown this in previous section by an explicit

mass

terms there appear

calculation. the

Using and A (n,

explicit forms of the operators F(r.), (4.56), H(a, 0), (4.58),

ao,,8o), (4.15), (4.53),

we

find the free

Green functions of these

operators

F-1 "'(r.)

IT A

-1

(a, 0)

pV

' A-l

1

2

=

ik

(_

=

4a

2

K

+Pj

1

VjAVv -

(_guv

(r., ao, flo)

-

E] +

9jAi,

=

E]

_

1_9

2f2

(gl'V Va VO

r-I

3

-2g-112j(X, Y) Ei

VJ'VV

-2

9

1126(X, Y)

(4.86)

,

,

(4-87)

EC01)I-tv'O ci 2+ P2VILVvVaV)3 -4g-112j(X, Y)

+ go"O V/' VV)

(4-88)

where

E

(O)Av,a,8

E-',,, Av, )31r.=ro 1 Pi

=

3w

1 + 4w =

9A(O'g3)v

(Oj

-

1-2i, 3

+ 1 + r.

-

3

)

,

g""gc',3

I

(4-89)

(4.90)

4 P2

Let

1) (

-

3w(w

+

3 + 1 +

W

r.

(4.61),

note that in the minimal gauge,

us

'A-1

=

ik

Co1 O)Av,a,3

-2 f2E

(0)

in

One-Loop Divergences

4.3

3

-

Arbitrary Gauge

)2

(4.91) 0

P2

=

pi

97

and, therefore,

9-1126(X, y).

[:1

Substituting the free propagators, (4.86)-(4.88), divergences of the quantities U, and U2

in

(4.84)

and

(4.85)

we

obtain the

div

f2

Udiv

3

f

d 4X ettv

(P2

6p,

+

-

2r.0 ')VtVv

E]

-3g-112j(X, Y) Y=X

div

1 + Pi

W

-

3w

)

M

gjjV

-2g- 112j(X, Y)

(4.92) V=X

div d

U2" C

=

4

(3+ (r.

2 a2

d4X6jLvVjjV E]-3g-1126(X'Y) V

3)2(l

-

-

0)

Y=x (4-93)

the

Using

Green functions

of the coincidence limits of the

divergences

and their derivatives in the dimensional

regularization

div

1:1

-2

2

9-1126(X, Y)

(n

-

4) (4-7r )2

V=X

Vt,Vv

[:]

div

-3g-112j(X, Y

2 -1

(n

(4.94)

4)(47r)2 j9AV

-

Y=X

and the

explicit

form of the extremal el"',

6V2

1

Uldiv

=

i

(n

-

4)(4ir)2 (K

-

1

div

U2

a

(n

-

4) (47r)

2

3)2 (1 2(3

(4.50),

we

obtain

2

d 4X

+

(w + 1)(tz

-

3)

1

g1/2 T2(R-4A)

,

1

4

+

3)2(l

-

fd4X g112 T2 (R

-

4A) (4.95)

Substituting these expressions finally

rdi)v I

K, a,

0, P)

(4.83)

and

integrating

over

r.

we

obtain

d F ,) (Ko, ao, Oo) v

=

1

(n where

in

-

4) (47r) 2f

d 4X

gl/2,A7(n

1

T2 (R-4A),

Higher-derivative Quantum Gravity

4.

98

13

, A7 (n, a,#)

=

6

4

P

+

V2

3

2a 2

2

6v 2 (K

a

_

-

3)2(l

2

3

-

+

P)

(r.

-

-2) 3)2

*

(4.96)

divergences of the effective action in arbitrary gauge form, (4.71), where the coefficients P1, #2) 03 and 64 do not depend on the gauge and are given by the expressions (4.72)-(4.74), and the 7-coefficient reads Thus the off-shell

have the

same

5 -y (r., a,

In

f4

0) =3 _2

particular,

5 +

3

2 V

2a2

2 a

_

(K

in the harmonic gauge,

^1 (1,

6v 2(K +

_

2

6

0, fl)

3)2(l

-

(4.60), (r.

5f4 =3 T2

fl)

-

1 and

2

=

a

-2) 3)2

0),

we

2

3V

'

(4.97) have

(4.98)

.

divergences on the parametrization of the quany-coefficient. Rather than to study this dependence, let us calculate the divergences of the Vilkovisky's effective action F, (4.24), that does not depend neither on the gauge nor on the parametrizaThe

dependence of

tum field also exhibits

the

only

in the

tion of the quantum field. Rom 1

-di

F 1"

-

2i

(log det Zjdiv

(4.24)

-

have

we

log det HI

div

2

-

log det F I

di'v

(4.99)

Vilkovisky's effective action F, (4.24), (4.99), differs from the usual one, (4.14), (4.52), only by the operator Z, (4.25). It is obtained from the operator ,A, (4.15), by substituting the covariant functional derivatives instead of the

The

usual

ones:

S,ik

-+

ViVkS ,Aik

=

=

DiDkS

'Aik'

+

-

T3ik-'j

Tjik -'j

=

Aik

+

=

S,ik

-

Fjik6j

)

(4.100)

-F3ik,'j

where

AV

=

-DiDkS

+

(4.101)

X'1_jjH"Xvk

Since the non-metric part of the connection

T3,11 (4.34),

is

non-local,

the operator Z, (4.100), is an integro-differential one. The calculation of the determinants of such operators offers a serious problem. However, as the nonlocal part of the operator Z, (4.100), is proportional to the extremal 6j, it

Therefore, the calculation of the determinant of the operator 3, (4.100), can be based on the expansion in the non-local part, Vik6j. To calculate the divergences it is again sufficient to limit oneself only

exhibits

only off

mass

shell.

to linear terms

log det 3

1div

=

log det djoc

Idiv

+

Z-1 mnTimn-'i 10C

I

div

(4.102)

One-Loop Divergences

4.3

To calculate this for the

Vilkovisky's

expression

99

Arbitrary Gauge

in

choose any gauge, because the answer depend on the gauge. Let us

one can

effective action does not

(4.56),

choose the De Witt gauge,

=

Xmi

kMEki

R

FA,

,

N1,,

=

(4.103)

.

the Ward identities for the Green function of the operator 31o, in De Witt gauge we get (up to terms proportional to the extremal)

Using

(4.101),

B' ,A-l 10C

ik

B' A-l 10C

ik

=

&

Using

the

N-1 "H-1N-1 'OR k18

+

0(6)

N-1 'AH-1N-1'O

+

0(,-).

AV

i

BOk

explicit form of

-

AV

TiM, (4.34),

(4.104)

and

(4.102)

in

(4.104) we

obtain

div

div

div 1:

log det3lo,

log det Z

U3

where

U3

Finally,

one

k

RON-'

-jDkR-7

=

has to fix the

aA

'0

I

T-r

(4.105)

AV

operator H (i.e., the parameters

determine the parameter of the metric of the

configuration

0)

and to

space, r.,

(4.45),

a

and

(4-46). In the paper [223] some conditions on the metric Eik were formulated, that make it possible to fix the parameter r.. First, the metric Eik must be contained in the term with highest derivatives in the action S(V). Second, the

operator Njv,

(4.28), (4.48),

must be

non-degenerate within

the

perturbation

Eik, (4.45), i.e., the parameter r., (4.46), one should theory. consider the second variation of the action on the physical quantum fields, 0, and h , 17-Lh, that satisfy the De Witt gauge conditions, Ri.Wi identify the metric with the matrix E in the highest derivatives To find the metric

=

=

1

h' (-S,ik)h k

=

f_2

f d4x h-L

AV g

1/2 EA"O

(r.)

E]

2h'aO

(4.106)

+ terms with the curvature.

This condition leads to

quadratic equation for

a

r.

that has two solutions

w

r.,

=

3

__

r-2

-

W+1

I

=

r-2 already noted above, the value r. the operator N, (4.48), in this case is degenerate Therefore, we find finally

As

=

we

R

=-

(4.107)

3.

=

3 is on

unacceptable, since background.

the flat

(4.108)

3 W

+

4.

100

Let

Higher-derivative Quantum Gravity note that this value of

r. coincides with the minimal one, R = no, choose the (4.61). Thus, operator H in the same form, (4.58), with the minimal parameters, a ao and # = #o, (4.61), then the operator 3joc becomes a minimal operator of the form (4.62)

us

if

we

=

ZIiOkC where

AAik

The

is

-Aik

+

I iik I "i

(4.109)

)

given by the expressions (4.62) and (4.63).

Z10,, (4.109),

of the determinant of the operator direct application of the algorithm

divergences by

calculated either

(4.65)

or

by

can

means

be of

the expansion in the extremal div

div

log det Zioc

log detAl.c

+

div I

(4.110)

U4

where

A-1

U4 the formulas

Using

gences of the

mn

fimnl

(4.99), (4.105), (4.110)

Vilkovisky's effective

.qdiv

=

r(div(r

0

1

and

(4.52)

we

obtain the diver-

u4div)

(4.111)

action

00)

C O1

div c

+

2i

M

-

where rdiv (no, ao, (1)

flo) are the divergences of the effective action in the minimal gauge, (4.7l)-(4.75). The quantities U3(div and U4d'v are calculated by using the free propagators, (4.86)-(4.88), in the minimal gauge, (4.61), d iv

div

U3(

U2`

(,zo, ao, flo)

-

4f

2

f

d 4X 6,av

fa,3,po, I /.IV

div

g,3, 0 +

XV,,,V,,

1(1

-

2w)VOV,

r-1

-4g-112j(X, .) Y=X

div div

d 4X ettv

U4 where

2 f2

/IV

7-1

'C"3'-0

0

-2

9

-112j(X, Y)

Y=x

(4.112)

d U2('v (no, ao, flo)

is given by the formula (4.95) in the minimal gauge divergences of the coincidence limits of the derivatives of the Green functions, (4.94), and the Christoffel connection, (4.47), and substituting the minimal values of the parameters rs, a and fl, (4.61), we obtain

(4.61). Using

the

4

div

U3

(n

4) (47r)

div I

U4

(n

-

4) (47r) 2

2

.-(2v

2

+

3

6(V2

P)

f2)

f

f

d 4X

d4X

1

g1/2 j-2 (R

g112 j2Y (R

-

-

4A)

4A)

.

(4.113)

Ultraviolet'Asymptotics

and

Group

4.4 Renormalization

101

Thus, the off-shell divergences of the one-loop effective action, F(j), have the standard form (4.71), where the #-functions are determined by the same expressions (4.72)-(4.74) and the 7-coefficient, (4.75), has an extra contridiv and U4div, (4.113), in (4.111). It has the bution due to the quantities U3c form 5

1 no, ao,

where

-y(ro, ao, flo)

Oo)

+

3

(11f2

-

5V2)

=

3

13

f4+ 3f2

T2

_

2

V2

6

(4.114)

given by (4.75).

is

Group and Ultraviolet Asymptotics

4.4 Renormalization

divergences of the effective action, (4.71), indicates that the higher-derivative quantum gravity is renormalizable off mass shell. Thus one can apply the renormalization group methods to study the high-energy behavior of the effective (running) coupling constants [50, 155, 226, 229]. The dimensionless constants e, f2, V2 and A are the "essential" coupling constants The structure of the

2 but the Einstein dimensional constant k is "non-essential" because its variation can be compensated by a reparametrization of the quantum field,

[229]

i.e.,

up to total derivatives

OSI

(4.115)

=0.

ak2

in

on-shell

Using the one-loop divergences of the effective action, (4.71), we obtain the standard way the renormalization group equations for the coupling

[50, 2291

constants of the renormalized effective action

dE

d =

Tt

01

wt-

,

dV2= 603 V4

5 =

d

104 2

1 =

4

(V4

+

5f4)

+

d

atwhere t

=

(41r)

-2

log('a/tIO),

f4

=

+

5f2 V2

1A 10f4

(

3

k2

=

V

2(t)

and

A(t)

as

oo

+

V2

+

5V

4

(4.117)

6

15f2

_

V2

by

(4.118) (4.119)

p is the renormalization

is determined

(4.116)

yk2

fixed energy scale. The ultraviolet behavior of the essential t -+

-2#2 f4

3

dt

Tt

f2

parameter and tio is

a

E(t), f2(t), (4.72)-(4.74). They 0-function, (1.47), and do not coupling

constants

the coefficients

play the role of generalized Gell-Mann-Low depend neither on the gauge condition nor on

the

parametrization of the

102

4.

Higher-derivative Quantum Gravity

quantum field. The non-essential coupling constant k 2 (t) is, in fact, simply 4 field renormalization constant. Thus the -y-coefficient, (4-75), (4.97) and

(4.114), play in (4.119) the role of the anomalous dimension (1.48). Correspondingly, the ultraviolet behavior of the constant k 2(t) depends essentially both

on

that

the gauge and the parametrization of the quantum field. It is obvious choose the gauge condition in such a way that the coefficient -Y,

one can

(4.97),

is

equal

to zero, -y

The equations

=

0. In this

2 all, i.e., k (t)

not renormalized at

(4.116)

be

can

,E

=

k2(0)

coupling

constant is

const.

=

easily solved

(t)

f2 (t)

the Einstein

case

=

E

(0)

+

81 t,

f2 (0) 2#2f2(0)t

_

1 +

(4.120)

"

Noting that #I < 0 and #2 > 0, (4.72), we find the following. First, the topological coupling constant e(t) becomes negative in the ultraviolet limit (t -+ oo) and its absolute value grows logarithmically regaxdless of the initial value e(O). Second, the Weyl coupling constant f2 (t) is either asymptotically free (at f2 > 0) or has a "zero-charge" singularity (at f2 < 0) We limit ourselves to the first case, f2 > 0, since, on the one hand, this condition ensures the stability of the flat background under the spin-2 tensor excitations, and, on the other hand, it leads to a positive contribution of the Weyl term to the -

Euclidean action

(4.49).

The solution of the

equation (4.117)

V2 (t)

C,

can

f2p (t)

-

f2p(t)

-

=

be written in the form

f*2P f2 (t) f2P

C2

(4.121)

where C1,2

50

V22-96401 P

There

are

f*2p

1.36

=

399 also two

2

correspond

to the values

f*2p

asymptotically free but only v22(t) The behavior of the conformal on

f*2P

its initial value

2(0)

_

elf2(0)

V2 (0)

-

C2 f2

V

(4.122)

-21.87

=

(0)

f2p (0)

(4.123)

special solutions C1,2 f2 (t)

W 1 Vi,2 that

0.091

V2--96401)

(-549

v

2(o).

In the

0,

oo

(4.124)

in

(4.121).

These solutions

are

is stable in the ultraviolet limit.

coupling case

constant

V2(0)

>

V2 (t) depends essentially

Clf2(0)

we

have

f2p (0)

>

and, therefore, the function V2 (t), (4.121), has a typical "zero-charge" t* determined from f2p(t*) singularity at a finite scale t f2P:>

0

=

=

4.4 Renormalization

f42(p+l)

2 V

and Ultraviolet

Group

W

C3

0(1)

(4.125)

--21.78.

(4.126)

i__

f2p(t)

+

f.2p

-

103

Asymptotics

where

V2--96401 C3

C1

---:

-

C2

:--

25

2 opposite case, v (0) < C1 f2(0), the function V2 (t), (4.121), does have any singularities and is asymptotically free,

In the not

2 V

(t)

L*00

---:

C2

f2 (t)

C3 f*

-

2pf2(p+1)(t)

+ 0

(f2(1+2p))

(4.127)

Thus, contrary to the conclusions of the papers [107, 108, 109, 111], we 2 region v > 0 there are no asymptotically free solutions. 2 The asymptotic freedom for the conformal coupling constant V (t) can be achieved only in the negative region v(O) < 0, (4.127). The exact solution of the equation for the dimensionless cosmological

find that in the

constant,

(4.118),

has the form t

A(t)

=

0-1(O)A(O)

O(t)

+

f d-rA(r)-!V1 (-r)

(4.128)

0

where A (-r)

lp(t)

JC,f2p(t)

=

-

C2

4

(5f4 (t)

f*2p 12 1 f2p (t)

2 q

=

+

_

V4 (t))

f*2p 12/5 1 f2 (t) I

+,\/296401)

RF5 (-241

The ultraviolet behavior of the

;zz

cosmological

-q

(4.129)

0.913 constant

A(t), (4.128),

cru-

the initial values of both the conformal

coupling constant, 2 2 V (0), and the cosmological constant itself, A(O). In the region v (0) > C, f2 (0) the solution (4.128) has a "zero-charge" pole at the same scale t*, similarly 2 to the conformal coupling constant v (t), (4.125), cially depends

on

f*2(1+2p) T2P

3

A(t) In the

opposite

case,

v

2

t-+t*

14 f-'5 f2p (t)

(0)

f2 (0),

< C,

(4.130)

+ 0 (1)

-

the function

A(t), (4.128),

grows in the

ultraviolet limit

A(t) The

sign of

I

=

the constant c4 in

C4 > 0 for A (0) >

C4

f-2q(t)

+

0(f2)

(4.131)

.

t-400

A2 (0) and

(4.131) depends on the initial for A (0) > A2 (0) where

C4 < 0

,

value

A(O), i.e.,

4.

104

Higher-derivative Quantum Gravity 00

A2 (0)

-!P(O)fd-rA(,r),P-'(,r)

---

(4.132)

.

0

In the

special

the solution

A (0)

case

(4.128)

A2 (0) the

-`

constant c4 is

equal

to

zero

(C4

=

0)

and

takes the form 00

A(t)

A2(t)

=

f d-rA(7-) V(,r)

-P(t)

=

(4.133)

t

The

special solution (4.133) A2 (t)

is

It-+00

free in the ultraviolet limit

asymptotically =

C5

f2(t)

+ 0

(f2(1+p))

(4.134)

-4.75.

(4.135)

where 5 C5

2

5 +

C2. q+1

--

266

; ,,

However, the special solution (4.133) is unstable because of the presence of growing mode (4.131). Besides, it exists only in the negative region A < 0. In the positive region A > 0 the cosmological constant is not asymptotically free, (4.13 1). Our conclusions about the asymptotic behavior of the cosmological constant A(t) also differ essentially from the results of the papers [107, 108, 109, 1111 where the asymptotic freedom for the cosmological constant in the region 2

A > 0 and

v

> 0 for any initial values of

A(O)

was

established.

discuss the influence of

arbitrary low-spin matter (except for spin3/2 fields) interacting with the quadratic gravity (4.49) on the ultraviolet behavior of the theory. The system of renormalization group equations in Let

us

presence of matter involves the

equations (4.116)-(4.119) with the total

function

fli,t.t where

Amat

gences of the

coupling quadratic in the

gravitational diver-

=

-

-

360 1

02,mat

(4.71),

curvature have the form

1

=

120

,83,mat Nj

(4.136)

Oi,mat

+

and the equations for the masses and constants. The values of the first three coefficients at the

effective action,

01,mat

where

fli

is the contribution of matter fields in the

the matter terms

=

(62N1.(0)

+

63N, +

11N1/2

(12N,(O)

+

13N,

6Nj/2

72

(Ni

+

(1

-

is the number of the fields with

massless vector

fields,

[42, 129, 187]

is the

coupling

+

+

+

No)

NO)

(4.137)

6 )2NO) spin j,

Nj(0)

is the number of

constant of scalar fields with the

Group

4.4 Renormalization

gravitational

field. In the formula

(4-137)

two-component. The coefficients (4.137)

01,rnat

< 0

02,mat

7

and Ultraviolet

Asymptotics

the spinor fields

possess

> 0

taken to be

important general properties

03,mat

1

are

105

> 0

(4.138)

-

gravitational 0-function (4.72)-(4.74) obtained in previous sections have analogous properties for f2 > 0 and V2 > 0. Therefore, the total #-function, (4.136), also satisfy the conditions (4.138) for f2 > 0 and V2 > 0. The properties (4.138) are most important for the study of the ultraviolet asymptotics of the topological coupling constant e(t), the Weyl coupling constant f2 (t) The

and the conformal

2

one v

(t).

The solution of the renormalization group equations for the and Weyl coupling constants in the presence of matter have the

topological form

same

substitution,6 -+ Pt.t. Thus the presence of matter does not change qualitatively the ultraviolet asymptotics of these constants: the coupling e(t) becomes negative and grows logarithmically and the Weyl coupling constant is asymptotically free at f2 > 0. The renormalization group equation for the conformal coupling constant

(4.120)

with the

V2 (t) in the presence of the d

V2

dt

=

matter takes the form

5f4 + 5f2V2 + 121

3

Therefrom

one can

show that at

"zero-charge" singularity The other properties

at

a

(10 + N, 2 v

+

> 0 the

(1

-

6 )2 NO)

coupling

V4.

constant

(4.139) 2 v

(t)

has

a

finite scale.

theory (in particular, the behavior of the < 0) depend essentially on the particular 2 form of the matter model. However, the strong conformal coupling, v > 1, at v2 > 0 leads to singularities in the cosmological constant as well as in all coupling constants of matter fields. Thus, we conclude that the higher-derivative quantum gravity interacting with any low-spin matter necessarily goes out of the limits of weak conformal 2 coupling at high energies in the case v > 0. This conclusion is also opposite to the results of the papers [107, 108, 109, 111] where the asymptotic freedom 2 of the higher-derivative quantum gravity in the region v > 0 in the presence of rather arbitrary matter was established. conformal constant

Let

pling

us

v

of the

2(t)

at

V2

also find the ultraviolet behavior of the non-essential Einstein

constant k 2(t). The solution of the

equation (4.119)

cou-

has the form

t

k2 (t)

=

k2(0) exp,

Of

d-r-&r)

(4.140)

The explicit expression depends on the form of the function -Y and, hence, on the gauge condition and the parametrization of the quantum field, (4.97). We will list the result for two cases: for the standard effective action in the minimal gauge and the standard parametrization (4.75) and for the Vilkovisky's effective action (4.114). In both cases the solution (4.140) has the form

106

4.

Higher-derivative Quantum Gravity

k2 (t)

=

C0 (0)

k2(0) TV

(4.141)

where

TI(t)

JC,f2p(t)

C2

-

f*2pl2 jf2p(t) f*2pls jf2(t)j-r _

3

13

-g6-5 (269 +

5 S

A 296401)

;ze

3.67

(4.142)

r

a

2

+ 2,,F2-96401) 1995 (-437

5

Pze

0.653

Here and below the upper values correspond to the Vilkovisky's effective action and the lower values correspond to the standard effective action in the minimal gauge and the standard parametrization. Therefrom it is immediately seen that the Einstein grows in the ultraviolet limit

k2 (t) Let

us

it'.

(t =

C6

coupling constant

f-2r(t)

+ 0

(f 2(p-r)

(4.143)

note that the ultraviolet behavior of the dimensional

ical constant,

A(t)

=

Vilkovisky's effective

A(t)/k 2 (t),

is

essentially different

action and in the standard

A(t)

It-+00

=

k 2 (t)

oo)

-+

C7

f2a

+ 0

in the

cosmolog-

case

of the

case

(f2(a+p)

(4.144)

where 2.76 a

=

r

-

q;:z -0.26

t-2.76 , and in the second case A(t) rapidly approaches zero, like it grows like tO.26. It is well known that the functional formulation of the quantum field

In the first case

theory assumes the Euclidean action to be positive definite [155, 193, 150]. Otherwise, (what happens, for example, in the conformal sector of the Einstein gravity), one must resort to the complexification of the configuration space to achieve the convergence of the functional integral [150, 131, 66]. The Euclidean action of the higher-derivative theory of gravity differs only by sign from the action (4.49) we are considering. It is positive definite in the

case

> 0

V2

f2

>

0,

< 0

(4.145)

,

(4.146)

,

A >

-

3V 4

2

(4.147)

4.4 Renormalization

Group

and Ultraviolet

Asymptotics

107

impose the condition (4.145) if one restricts oneself to a fixed topology. However, if one includes in the functional integral of quantum gravity the topologically non-trivial metrics with large Euler characteristic, the violation of the condition (4.145) leads to the exponential growth of their weight and, therefore, to a foam-like structure of the space-time at microscales [1501. It is this situation that occurs in the ultraviolet limit, when

It is not necessary to

e(t)

(4.120). (4.146)

-+ -oo,

The condition

is

usually held

to be

"non-physical" (see the bibli2 plays the role of

point is, the conformal coupling constant v ography). the dimensionless square of the mass of the conformal mode The

ground. to the

In the

2 case v

< 0 the

instability of the flat

unstable solutions etc.

on

the flat back-

tachyonic and leads static potential, the of oscillations (i.e.,

conformal mode becomes

space

[209]).

higher-derivative quantum gravity in the region behavior in the conformal sector, "zero-charge" unsatisfactory 2 conformal of the In coupling (V > 1) one strong region (4.125), (4.130). cannot make definitive conclusions on the basis of the perturbative calculations. However, on the qualitative level it seems that the singularity in the coupling constants v 2(t) and A(t) can be interpreted as a reconstruction of the ground state of the theory (phase transition), i.e., the conformal mode As

V2

we

showed above, the

> 0 has

"freezes" and

a

conformal condensate is formed.

against the "non-physical" condition (4.146) to enough. First, the higher-derivative quantum gravity, strictly speaking, cannot be treated as a physical theory within the limits of perturbation theory because of the presence of the ghost states in the tensor sector that violate the unitarity of the theory (see the bibliography, in particular, [107, 108, 109, 111], [25]). This is not surprising in an asymptotically free theory (that always takes place in the tensor sector), since, generally speaking, the true physical asymptotic states have nothing to do with the excitations in the perturbation theory [215]. Second, the correspondence with the macroscopic gravitation is a rather fine problem that needs a special investigation of the low-energy limit of the higher-derivative quantum gravity. Third, the cosmological constant is always not asymptotically free. This means, presumably, that the expansion around the flat space in the high energy limit is not valid anymore. Hence, the solution of the unitarity problem based on this expansion by summing the radiation corrections and analyzing the position of the poles of the propagator in momentum representation is not valid too. In this case the flat background cannot present the ground state of the theory We find the arguments

be not strong

any

longer.

From this

gravity

with

standpoint, in high energy region the higher-derivative quantum positive definite Euclidean action, i.e., with an extra condition 2 V

seems

to be

more

<

C1f2

;Z

_0.091f2,

(4.148)

intriguing. Such theory has unique stable ground state that asymptoti-

minimizes the functional of the classical Euclidean action. It is

108

4.

Higher-derivative Quantum Gravity

cally free both in the tensor and conformal sectors. Besides, instead ontradictory "zero-charge" behavior the cosmological constant just logarithmically at high energies. Let

us

stress

withstanding totically free

once more

of the grows

the main conclusion of the present section. Not-

the fact that the

higher-derivative quantum gravity is asymptheory with the natural condition f' > 0, that ensures the stability of the flat space under the tensor perturbations, the condition of the conformal stability of the flat background, 0 > 0, is incompatible with the asymptotic freedom in the conformal sector. Thus, the flat background cannot present the ground state of the theory in the ultraviolet limit. The problem with the conformal mode does not appear in the conformally invariant models [118, 119]. Therefore, they are asymptotically free [107, 108, 109, 111, 110]; however, the appearance of the R 2 -divergences at higher loops leads to their non-renormalizability [112]. in the tensor sector of the

4.5 Effective Potential

Up be

to

now

the

arbitrary.

background

field

(i.e.,

the

To construct the S-matrix

space-time metric)

one

needs the

was

background

held to fields to

be the solutions of the classical mass

shell. It is

equations of motion, (4.5), i.e., to lie on obvious that the flat space does not solve the equations of

(4.5) and (4.50) for non-vanishing cosmological constant. The most simple and maximally symmetric solution of the equations of motion (4.5) motion

and

(4.50)

is the De Sitter space 1

RA V c,

R1,V

12

4

(Pgp,

-

'

givR,

R

61&g,,,)R, 16 =

const

(4.149)

,

with the condition R

=

4A.

(4.150)

On the other hand, in quantum gravity De Sitter background, (4.149), plays the role of covariantly constant field strength in gauge theories, V,R,,,6.6 0. Therefore, the effective action on De Sitter background determines, in fact, the effective potential of the higher-derivative quantum gravity. Since in this case the background field is characterized only by one constant R, the effective potential is a usual function of one variable. The symmetry of De Sitter background makes it possible to calculate the one-loop effective potential exactly. In the particular case of De Sitter background one can also check our result for the R'-divergence of the one-loop effective action in general case, (4.71), i.e., the coefficient 03, (4.73), that differs from the results of [107, 108, =

4.5 Effective Potential

109

109, 111] and radically changes the ultraviolet behavior of the theory conformal sector

Sect.

(see

in the

4.4).

practical calculation of the effective potential we go to the space-time. Let the space-time be a compact fourdimensional sphere S4 with the volume For the

Euclidean sector of the

V

=

f

2

47r

d4X g1/2

(W)

24

=

(R

,

>

(4.151)

0),

and the Euler characteristic 1

f

2-7r jr22

4Xg112 R*R*

d

=

(4.152)

2.

In present section we will always use the Euclidean action that differs only by sign from the pseudo-Euclidean one, (4.49). All the formulas of the previous sections remain valid by changing the sign of the action S, the extremal ej

=

S,j

and the effective action.P.

On De Sitter

background (4.149)

the classical Euclidean action takes the

form

S(R) where R

=

x

=

Rk 2 and A

4A, (x

=

16

24(4ir

=

+ X

Ak 2. For A > 0 it has

a

(4.153)

2A y

minimum

on

the

mass

shell,

that reads

4A), (4.150), Son-shell

=

1

4 (c

(47r)2

31

-

_2

X

(4.154)

Our aim is to obtain the effective value of the De Sitter curvature R from the full effective

equations

ar(R) =

(4.155)

0.

aR

Several problems appear on this way: the dependence of the effective action and, therefore, the effective equations on the gauge condition and the

parametrization etc.

of the quantum

field, validity of the one-loop approximation

[113]. First of

all,

we

make

a

h,Av W

=

+

new

variables,

h,,,

1gjLv o + 2V(tev)

7

(4.156)

gl`httv

7

(4.157)

4

h

-

61, where the

of field variables

change

h-L/IV

0

a

=

and

h

7

61 A

el, A

+

=

1vj'a

2

satisfy

,

(4.158)

the differential constraints

4.

110

Higher-derivative Quantum Gravity I

wh

=

I&V

hi- 9/,IV /.IV

0,

V"El

=

0

(4.159)

0

=

(4.160)

A

In the

following

we

will call the initial field variables

hj,

without any restric-

tions

and --L, which imposed on, the "unconstrained" fields and the fields h-,L,, I,V A satisfy the differential conditions (4.159) and (4.160), the "constrained" ones. When the unconstrained field is transformed under the gauge transformations with parameters

6hAv

., (4.43), (4.44),

=

I

2V(,, v)

the constrained fields transform in the

A-L

0

=

,UV

following JO

,

J61 Therefore, are

the

a are

+ V 1,

=

A

,

6a

the transverse traceless tensor field

physical gauge-invariant components non-physical ones.

way

0,

(4.162)

2 .

(4.163)

=

=

(4.161)

h-' and the conformal field ,UV

hjAv,

whereas 61 and

De Sitter

background has

of the field

pure gauge

The second variation of the Euclidean action

on

the form

S2(g

+

1 x

4f2

3

h)

=_

-1

2h'S,ikh

h-L IA2

M

2 2

f

k

R

R) ( ) IA2

+

3V2

[ O (,AO(7n2)Ao (_R)

2 -

3

0

1 +

3

2

2

mo (R

-

4A) W,,Ao (0) a

1 +

4k

2(R

-

4A)ej-Aj

-3

(_ R)

+

2

-TnO(R

4A

2

-

4A)]

hj-

(P

R

2

M

-

3

lrn2 (R

2

6

-

_V2 2V2

g112

d 4X

0(R

-

4A) aAo (0),Ao

E_L

( 2) a] (4.164)

4

where 2

M2

-

2

-f2 T2

V

2

MO

j2

and

,Aj(X)

=

-

0 +x

(4.165)

ill

4.5 Effective Potential

the constrained differential operators

are

spin j

=

0, 1,

acting

on

the constrained fields of

2.

When going to the mass shell, (4.150), the dependence of S2, (4.164), the non-physical fields E1 and a disappears and (4.164) takes the form

S2(g + h) I

on-shell

d4X g1/2

3

32V2 Rom here it

tion, (4.164),

h_L, A2 4f2

OAO (M 2)AO 0 in

follows, on

the

eigenvalues An multiplicities d,,

dn 6

(2j

+

=

M

2 2

y2

4

+

+

V2)

A2

V2

3

(32A)

h-L

(4.166) invariance, of the second varia-

particular, the gauge shell, (4.150),

mass

The

An

(

(_34A

6S2(g

their

on

P2

+

h) Ion-shell

of the constrained

Laplace operator Aj (0)

n

i

2 =n

n

=

=

j, j

12

(4.168)

+ 1

stability of the De Sitter background (4.149),

S2(g

+

h)

Ion-shell

imposes the following restrictions (for A

f2

and

R

P2

+3n-j

1)(n+ 1 -j)(n+2+j)(2n+3)

Thus the condition of

11

[1131

are

n

(4.167)

0.

=

>

0,

in the tensor sector

0):

>

-

0

72

(4.169)

0,

>

(4.170)

"

4A

and in the conformal sector 2

V

in the

1 <

0,

<

when these conditions

However,

even

five

modes of the operator Ao

zero

case

Along these conformal directions,

(-:!A) 3

V1, in the

(4.171)

V2 are

fulfilled there

in the conformal

are

sector,

still left

(4.166).

configuration space the Euclidean

action does not grow

S2(9

+

01)

Ion-shell

=0.

(4.172)

4.

112

This

Higher-derivative Quantum Gravity the De Sitter

background (4.149) does not give Positive definite Euclidean action. This can be verified by calculating the next terms in the expansion of S (g + VI). However, in the one-loop approximation these terms do not matter. means

that,

in

fact,

the absolute minimum of the

To calculate the effective action

of variables

(4.156)-(4.158). Using

f

d

4XgI12 hA21,

f

=

one

has to find the Jacobian of the

the

simple equations

d4X g1/2

12

h

we

(_ R)

+2,- IAl

4

+3U,60 (0) A0 4

f d4X gl/262 f d4Xg112

change

(_R)

1

612

h2

0,+

(4.173)

4

3

1UAO(0)a

+

(4.173)

4

obtain

dhjjv

dh' &-L do, dV (det j2 )1/2

=

deA

=

de-L do, (det j1)1/2

(4.174)

where

J2

=

Al

(-4

3

Ji Let

us

A0 (0)

A0

"AO(o)

=

calculate the effective action

.

(4.12)

in De Witt gauge the operator N,

ghost operator F, (4.16), in this gauge equals De Sitter background (4.149) it has the form

F1,,

=

NI,,

=

The operator of

2g' /2

I (_ R) 0-

91LV

"averaging

4

over

the

+

1(s

1) Vt"VV

gauges" H, (4.58),

(4.56). The (4.48). On

1 6(X, Y)

.

(4.175)

and the gauge

fixing

term have the form

1

H `

=

4a

2g-1/2

Sgauge

2

R+

9AV

D+ 4

P)

+

flVAVv

I 6(x, Y)

,

XILHA'xv

d4X

g1/2 _a2

,6_L, Ij (R P) (_R) +

4

A21

4

6_L

(4.176)

4.5 Effective Potential

+16 (I -2r,(r.

1

18) r.2 0,60(p'),60(o)(P

-

r.R ) AO(O)Or

3)WAo(P')Ao

-

3

R

3)2UAO (PI) A20

+(r.

113

3)

AO (0)

Ul

(4.177)

I

where P

(P1

+

4P2)R + 03 k2

f d4X gl/261, jgAV (-

0

+X)

P

9p,P1jV

=

PI

Using the equation

f we

d4X g1/2

6"Al (X)E'

find the determinants of the

det F

=

-yV"VV I 6V

+

1 _'Y

U'AO

+

4

det.Al

For the

=

R

(_R) i ( )

detAo

det A,

operators F and H

(4.178)

Ao (0)'a

ghost operators (4.175) and (4.176)

to be

3

-

R+ P

det H

(1-7 ) )

,

(4.179)

detAo (P')

4

positive definite

we assume

0

< 1 and

< 3.

determinants of the ghost operators, (4.179), and the change of variables, (4.174), and integrating exp (- S2 Sgauge) one-loop effective action off mass shell in De Witt gauge (4.56)

Thus using the Jacobian of the we

obtain the

with

arbitrary

-

gauge

parameters

F(1)

=

r., a,

P and

P

1(2)

I(I)

1(0)

+

+

(4.180)

1

where 1

1(2)

2

log det A2

(R) (M2 A2

2

6

+

R

+

3V2

'M2 (R

2

2

-

4A)]

,

(4.181) det

1

2

log

A,

'A (R+p) (_ R) 4 _4 1

C,2 + 1 2 k2' (R

-

(det.Al

(_R))2 detAj (R+-P) 4

4A) (4.182)

-

4

114

4.

Higher-derivative Quantum Gravity det

2

log

[ A2 ( 0

R r.-3

) '6() (pl) AO (,rn2) (A fl det, Ao (r.R 3))2 det Ao (P') + D

0

-

4

+ C

(A

-

;a) 4 2]

---

(4.183) D

=

4,rn20

M2 -3

R

,

Ti _-3)2' Ao

3

)

',10 (P,)

8a 2

k2(j

0)(m

-

3)2

-

a

C

ZAO

M

2)

0

AO

(_ R)

(4.184)

2

V

16

=

k 4(1

-

#)(r.

3 )2

The quantity 1(2) describes the contribution of two tensor fields, I(,) gives the contribution of the vector ghost and I(o) is the contribution of the scalar conformal field off

mass

shell. The contribution of the tensor fields

depend on the gauge, the contribution of the vector 1(2)7 (4.181), on the parameters a and P and the contribution depends ghost 1(,), (4.182), of the scalar field 1(0), (4.183), (4.184), depends on all gauge parameters r., a,,6 and P. The expressions for I(j) and 1(o)7 (4.182), (4.183), are simplified in some particular gauges does not

1 =

2

a=O

det

1

,(0)

-

(0)

-

-log

log det A,

[Ao (-A) -3--

2

6=--

(_R)

(4.185)

4

+ Irn2 (R (,rn2) 3 -0 det AO (-;a3)

AO

-

0

4A)] j

(4.186) det

1

'(0)

-log

1(0) -0

[,Ao (_;a) 2

2

P=--

rn20 (R + I (rn2) 0 2 detAO (_ R)

AO

-

4A)]

2

(4.187)

1(0)

1

det

1 r.=- 00

2

log

_,M2(R [,AO(0),AO (,rn2) 0 0 det

-

4A)]

6o (0)

(4.188)

us also calculate the gauge-independent and reparametrization-invaVilkovisky's effective action, (4.21), on De Sitter background in the orthogonal gauge, (4.36), (4.38). In the one-loop approximation, (4.40), it

Let

riant

differs from the standard effective action in De Witt gauge, (4.14), (4.56), (4.58), with a = 0 only by an extra term in the operator.3, (4.41), due to the

Christoffel connection of the configuration space, (4.27), (4.47). Therefore, the Vilkovisky's effective action, (4.38), (4.40), can be obtained from the standard 0 by substituting one, (4.12), (4.14), in De Witt gauge, (4.56), (4.58), for a =

4.5 Effective Potential

115

DiVkS, (4.41), for the operator S,ik, i.e., by the replacing quadratic part of the action S2 (g + h), (4.164), by

the operator

S2(g

h'

ejh

ik

+

k

4

h)

S2(g

=

(r.-I

h)

+

1) j2- (R

-

2

-

1

1) 1-2 (R

-4

+26 -L,/_A,

R

(_ )

-

'6_L

1h'

-

f

4A

4A)f

3jk I ejhk d 4X g1/2

h1,12V

4

R

0" Ao (0)'AO

3

h

4

4

2

g1/2 h-L

d 4X

3 +

4

the

)

(4.189)

0,

n is the parameter of the configuration space metric that is given, according to the paper [223]], by the formula (4.108). Let us note that in the 1 [223, 224]. Einstein gravity one obtains for the parameter r. the value r. Therefore, the additional contribution of the connection (4.189) vanishes and the Vilkovisky's effective action on De Sitter background coincides with the 0 and standard one computed in De Witt gauge, (4.56), (4.58), with a

where

=

=

r.

=

1

in the harmonic De Donder-Fock-Landau gauge

(i.e.,

(4.60)).

Taking into account the Jacobian of the change of variables (4.174) and the ghosts determinant (4.179) the functional measure in the constrained variables takes the form

dhj,tv6(Rjj,h') det N

=

*

dh-L de-L dV da 5(e 1 )

I

5

r. o

-

(r.

(

3),Ao

-

R -

3

R

) a]

det

Ao -

3

1/2 *

Integrating

exp(-92)

we

1

det A 1

(_R)

==

(_R)]

Ao

1(2)

(4.190)

3

4

obtain the

F(1)

det

one-loop Vilkovisky's effective +

I(1)

+

action

(4.191)

1(0)

where 1

1(2)

2

log det A2

(R) 2(M2 "A

2

6

+

3V2

R)

1 +

M22 (R

-

4A)] (4.192)

1log det 2

A,

(_R) 4

(4.193)

4.

116

Higher-derivative Quantum Gravity det

1(0)

=

2

The equations

/_A0

log

(

R r.-3

)

1

(M2) 0

A0

_

r.-3

for

K

4A)

-

(4.194)

R

detAo

(4.191)-(4.194)

M20 (R

3

1 do coincide indeed with the standard

=

effective action in the gauge a = 0, K = 1, (4.180), (4.181), However, to obtain the Vilkovisky's effective action in our

put in

3f2/(f2 + 2V2), (4.108). (4.150) the dependence on the

(4.191)-(4-194)

On the

mass

r.

shell

(4.185), (4-187). case one

has to

=

gauge

disappears

and

have

we

I

1

on-shell

2)

=

2

log det. A2

1

(2A)

log det.A2

+

3

2

M

2 2

y2

4

+

+

V2)

A

V2

3

(4.195) pn-shell 2

Ion-shell 0)

Rom here

of the

theory:

one sees

2

log det.,Al (-A)

(4.196)

log detA0 (mo2)

(4.197)

immediately the spectrum

massive tensor field of

of the

physical excitations

(5 degrees

of freedom), one freedom) and the Einstein graviton, i.e., the massless tensor field of spin 2 (5 3 2 degrees of freedom). Altogether the higher-derivative quantum gravity (4.49) has 5 + 2 + 1 8 degrees of one

massive scalar field

(1 degree

spin

2

of

=

-

=

freedom. To calculate the functional determinants of the will

differential operators

we

the

technique of the generalized (-function [129, 113, 150, 149, 89, 131, 66]. Let us define the (-function by the functional trace of the complex power of the differential operator of order 2k, A(k)' use

Cj

(p;'A(k)/P2k)

=-

(' A(k) /P2k Yp

tr

(4.198)

where

' A(k) Pk (x) is

a

polynomial of order k,

=

pk(-

IL is

a

(4.199)

dimensional

mass

parameter and j

denotes the spin of the field, the operator A(k) is acting on. For Rep > 2/k the (-function is determined by the convergent series the eigenvalues (4.168)

( A(k)/p2k) E dn (Pk (An )/p2k)

-P

p;

over

(4.200)

n

where the summation

runs over all modes of the Laplace operator, (4.168), positive multiplicities, dn > 0, including the negative and zero modes of the operator A(k) The zero modes give an infinite constant that should be

with

.

simply subtracted,

whereas the

negative

modes lead to

an

imaginary part

4.5 Effective Potential

117

For Rep < 2/k the analytical continuation meromorphic function with poles on the real axis. It is 0. Therefore, one important, that the (-function is analytic at the point p

indicating to the instability [113]. of

defines

(4.200)

a

=

can

define the finite values of the total

,A(k) (taking each mode k times)

number of modes of the operator and its functional determinant

k tr 1

B(,A(k))

=

(A(k)/t12k

log det

(4.201)

,

-(j, (0)

(4.202)

where

B(,A(k))

k(j(O)

=

(p)

(j, (p) Under the behaves

as

,

(4.203)

of the scale parameter p the functional determinant

change

follows

(0;'A(k)/t,2k)

=

-B

(,A(k)) log

P2

(0;,A(k)/p2k)

+(Ij

A2

Using the spectrum of the Laplace operator (4.168)

we

(4.204)

.

rewrite

(4.200)

in

the form

Cj

(

P;

A(k)

2j

/p2k)

+ 1

V(V2

-

3

V>jA2

x

,

12)

AV=1

-P

9

[P (P2 (V2

_

_

k

4

j)

(4.205)

where 1

R

P2 The

sum

(4.205)

can

=

j

12

be calculated for

+

2

Rep

> 1 k

by

means

of the Abel-

[98]

Plan summation formula

0"

E fM V :.!2

E

=

fM

1
+

f

dt

f (t)

k+C

00

+

f

dt

e27rt +

[if (k

+

e

-

it)

-

if (k

+

e

+

it)),

(4.206)

0

integer k should be chosen in such a way that for Rev > k the f (v) is analytic, i.e., it does not have any poles. The infinitesimal

where the function

4.

118

Higher-derivative Quantum Gravity

parameter Rev

=

e

> 0 shows the way how to

k. The formula

sufficiently rapidly

(4.206)

at the

get around the poles (if any)

is valid for the functions

at

f (v) that fall off

infinity:

Av) I

-IVI-q

Req

(4.207)

> 1.

IV(-+oo

When

applying the formula (4.206) to (4.205) the second integral in (4.206) gives analytic function of the variable p. All the poles of the (-function are contained in the first integral. By using the analytical continuation and integrating by parts one can calculate both C(O) and ('(0). As a result we obtain for the operator of second order, an

,Aj(X) and for the operator of forth "A 3

=

-

0 +x

(4.208)

,

order,

2) (X, y)

=

2

E]

2X 0

-

the finite values of the total number of modes

(4.209)

+Y,

(4.201)

and the determinant

(4.202) B (,Aj (X))

B

(A 2) (X, y))

(b

=

12

=2.

3

12

1 (b

(1j (0; 'Aj (X) /P2)

2

2

+

12)2

+

12)2

1

212 +

_

a2

30

F(O) (X) i

+ 1

F

=

(2)

3

(4.210)

,

30

3

3

2j

1

212 + 3

_

=

(0;'A 2) (X, y) /P4)

_

I

,

(4.211) (4.212)

-

(X, Y)

(4.213)

where

V

9

ff

=

ff

4

a2

=

-f7

Y7

X2,

x

;;2 Y

=

P4

and

FM (1)

=

-

1b2(b2+212) log b2 + 112 b2 + 2

3

b4

8

00

+2

f

dt t

e21rt +

(t2

+

12) log lb 2

_

t2l

0

V(V2

+

! :5V<j-

2

_

12) log(V2

+ b

2)

(4.214)

4.5 Effective Potential

(2) Fj' (X, Y)

=

4

(a

2 -

V

2 12 b 2)

-

log(b

4

+

a

2) 1

larctan ( b2) 7r]

2

-a(b +12)

_

-

2

a

119

3

2

b

+

4a

4

+12 b2

4

00

+2

dt t

f

e27rt +

(t2

+

12) log [(b2

t2)2

_

+

a2]

0

E

+

V(V2

_

12) log [(V2

b2)2

+

+

a

2]

(4.215)

1

V5 '<j_ 21 The introduced

functions, (4.214) and (4.215),

ko) (X)

+

3

ko) (fl)

(

F(2) i

=

3

X

are

+

related

P, ;

2

Xf)

by

the

.

equation

(4.216)

complete analogy one can obtain the functional determinants of the operators of higher orders and even non-local, i.e., integro-differential, operIn

ators.

Using the technique of the generalized (-function and separating the dependence on the renormalization parameter p we get the one-loop effective action, (4.180)-(4.184), R

1

F(i)

=

2

Bt.t log

12p 2

+

F(1)

ren

(4.217)

7

where

F(1) For the

study of

=

ren

F (1)

J'U2=p2=

the effective action

one

(4.218)

R 12

should

calculate,

first of

all,

the coefficient Bt.t. To calculate the contributions of the tensor, (4.181), and vector, (4.182), fields in Bt.t it suffices to use the formulas (4.210) and (4.211) for the operators of the second and the forth order. Although the contribution of the scalar field in arbitrary gauge, (4.183), (4.184), contains an operator of

eighth order, it is not needed to calculate the coefficient B for the operator of eighth order. Noting that on mass shell, (4.150), the contribution of the scalar field (4.197) contains only a second order operator, one can expand the contribution of the scalar field off mass shell in the extremal, i.e., in (R 4A), limiting oneself only to linear terms. One should note, that the differential change of the variables (4.156)(4.158) brings some new zero modes that were not present in the nonnumber of constrained operators. Therefore, when calculating the total

the

-

modes

(i.e.,

the coefficient

Bt0t)

modes of the Jacobian of the

one

change

should subtract the number of

of variables

(4.174):

zero

4.

120

Higher-derivative Quantum Gravity B (4)

Bt.t

Using the

number of

zero

JV(J)

-

(4.219)

modes of the operators

entering the Jacobians,

(4.174),

Jq,A0 (0))

1

=

JV

,

(_R))

' Ao

JV

5,

=

3

(' Aj (_ R)) 4

=

10

,

(4.220) we

obtain

JV(J) Thus

we

20

f4 +20

-

3 V4

+16

[3 (V4

f2 2 +

4

x

The

-

1

29.

=

(4.221)

1

634

+247

_

45

5f4)

X

+ A

(102

+

V2

15f2

_

V2 -67

1

) I T2 ,(4.222)

Rk 2 and the coefficient -y is given by the formula (4.97). Vilkovisky's effective action (4.191) has the same form, (4.217), with =

i3tot

the coefficient is

15

x

obtain the coefficient Bt.t

Bt ot

where

2

=

given by

of the form

the formula

On the other

(4.222)

but with the

change

where' Y

-y

(4.114).

hand, the coefficient Bt.t can be obtained from the general divergences of the effective action (4.71) on De Sitter

expression for the

background (4.149), 1

Bt.t

=

4#1

+

2403

+

(2304

+16

24-y X

where the coefficients

01, 03

and

04

are

6-yA

-

given by

)

1

X2

(4.223)

I

the formulas

(4.72)-(4.74),

and the coefficient -y is given by the formula (4.97) for standard effective action in arbitrary gauge and by the expression (4.114) for the Vilkovisky's

effective action.

expressions (4.222) and (4.223) we convince ourselves that our 03, (4.73), that differs from the results of other authors [107, 108, 109, 111] (see Sect. 4.2), and our results for the divergences of the effective action in arbitrary gauge, (4.97), and for the divergences of the Vilkovisky's effective action, (4.114), are correct. On mass shell (4.150) the coefficient (4.222) does not depend on the gauge and we have a single-valued expression

Comparing

the

result for the coefficient

non-shell -"tot

20

f4

3

_ 74_

=

+

+20

( 1X V2

f2 T2

+

634 -

15f2

-

45

_

V2

)

1

1

3 +

A

4

(V4

+

5f4) A2

.

(4.224)

4.5 Effective Potential

Let

us

also calculate the finite part of the effective action

121

(4.218).

Since

depends essentially on the gauge, (4.180)-(4.184), we limit ourselves to the case of the Vilkovisky's effective action, (4.191)-(4.194). Using the results

it

and

(4.202)

(4.212)-(4-215)

F(1)

ren

obtain

we

5 F(2) (Z1, Z2 2

6

+F(2) (Z3 Z4) 0

_

1

3F(0) (- 3) 1

(

F(O) 0

-2

f

2

4)

V2

(4.225)

where

Z,

6

=

2

f2

1

+3,

X+2 T2

1

Z2

=

-96(f2

+

1

2V2)A ;2 Z3

=

+ 48

y2

f2

1

6V2

_

F

k)

are

_

_2 +

2V2)A

2,

X2

above, (4.214), (4.215), and x (4.150) the effective action does not depend on form, (4.13), (4.154), (4.195)-(4.197), (4.217),

the functions introduced

3

On

-96(f2

=

(4.226)

V2

X

X

Z4

+8f +8,

V2)

+

mass

and has the

Ton-shell

=

4 (e

(47r)2

+h

f

1 2

1

3

T2_

A

-shell

A I Og

the gauge

1

-

Bton a

Rk 2.

=

shell

k2 k2

r(on-shell 1)ren

+,

+

0(h2) (4.227)

where

ron-shell 0

ren

6

(0) 5 F2

(3f2 +4f2

+5F2(0) (2) The

expression (4.227) gives the

+4

V2

A

3F,(O) (- 3) + F0(0) vacuum

action

on

(3AV2)

De Sitter

(4.228) background

with quantum corrections. It is real, since the operators in (4.195)-(4.197) do not have any negative modes provided the conditions (4.170) and (4.171) are

fulfilled.

Although

the operator

A0 (m2) 0

has

one

negative mode,

W

=

const,

122

4.

Higher-derivative Quantum Gravity

(4.171),

to the condition

it is non-physical, since it is just the zero change of variables (4.174). All other modes of the operator AO(M20 ) are positive subject to the condition (4.171) in spite of the fact that mo2 < 0. Depending on the value of De Sitter curvature R off mass shell there can appear negative modes leading to an imaginary paxt of

subject

mode of the Jacobian of the

effective action.

the

we

Differentiating the effective action, (4.13), (4.153), (4.217), (4.222), (4.225), equation for the background field, i.e., the curvature

obtain the effective

of De Sitter space, 1

ar

1

x

-

-

hk2 OR

h

24(47r)2

.

-

4A

1 +

-

T3

2x

1Bt.t (x) log 12p2k2 2

Bt.t (x)

x

+

+F')ren(x)+OY0--":o, (4.229)

where

Bt'.t (x)

aBt.t = I

[3(v

_32 X3

4

+5f4)

4

+,\(lot! +15f2

_V2

-

V2

6-y)

1

(4.230)

-247T2F 11)ren(X) The

perturbative

Rk

=

x=Rk

4A

-

2

(4.229)

has the form

A

h

3(47r)2

Orshell Bton

+

4ABt'.t (4A) log

+Mr l)ren(4A) +O(h2) It

(4.231)

ax

solution of the effective equation

A2

2

19r(l)ren

gives the corrected value of the quantum effects.

3M2 V

(4.232)

.

curvature of De Sitter space with

regard

to the

perturbation theory

near

this solution is

applicable

for A

0

0 in the

region

1, i.e., when A is of Planck mass order l1k 2, the contributions of higher loops are essential and the perturbation theory is not adequate anymore. Apart from the perturbative solution (4.232) the equation (4.229) can also 0 the non-perturbative have non-perturbative ones. In the special case \

f2

-

v'

-

A < 1. For A

-

=

4.5 Effective Potential

123

solution R : 0 means the spontaneous creation of De Sitter space from the flat space due to quantum gravitational fluctuations. Therefore, it seems quite possible that De Sitter space, needed in the inflational cosmological scenarios

of the evolution of the Universe

However, therefore,

is

[171]],

has

quantum-gravitational origin [139]. 1 and, non-perturbative solution has the order Rk' the in inapplicable one-loop approximation.

almost any

-

Conclusion

Let

us

summarize

shortly the

main results.

I. The methods for the covariant

expansions of arbitrary fields in arbitrary connection in generalized Taylor series integral in most general form are formulated.

space with

Fourier 2. A

a

curved

and the

-

manifestly

cients based

covariant

technique

for the calculation of De Witt coeffi-

the method of covariant expansions is elaborated. The diagrammatic formulation of this technique is given.

on

corresponding

3. The De Witt coefficients a3 and a4 at 4. The renormalized

coinciding points

are

calculated.

one-loop effective action for the massive scalar, spinor an background gravitational field up to the terms of

and vector fields in order

1/m

4

is calculated.

5. Covariant methods for action

are

studying

the non-local structure of the effective

developed.

6. The terms of first order in

background

fields in De Witt coefficients

calculated. The summation of these terms is carried out and covariant

expression for the Green function

terms of second order in

in the

conformally

background fields

invariant

case

is finite in the first order in the

7. The terms of second order in

at

covariant non-local

coinciding points

the Green function at

background

up to

coinciding points

fields.

background fields

in De Witt coefficients

expression for the one-loop effective

background

are

non-local

is obtained. It is shown that

calculated. The summation of these terms is carried out and terms of third order in

a

a

are

manifestly

action up to the

fields is obtained. All

formfactors, imaginary parts (for standard definition of the asymptotic regions, ground states and causal boundary conditions) are calculated. A finite effective action in the conformally invariant case of their asymptotics and

massless scalar field in two-dimensional space is obtained. 8. The De Witt coefficients for the

calculated. It is shown that the

diverges.

of scalar field in De Sitter space are corresponding Schwinger-De Witt series case

The Borel summation of the

Schwinger-De Witt expansion is explicit non-analytic expression in the background one-loop effective action is obtained.

carried out and fields for the

an

Conclusion

126

one-loop divergences of the effective action in arbitrary as well as those of the Vilkovisky's effective action in higher-derivative quantum gravity are calculated.

9. The off-shell

covariant gauge

10. The ultraviolet

asymptotics of the coupling

constants of the

higher-

derivative quantum gravity are found. It is shown that in the "physical" region of the coupling constants, that is characterized by the absence of the tachyons on the flat background, the conformal sector has "zero-

charge" behavior. Therefore, the higher-derivative quantum gravity at higher energies goes beyond the limits of weak conformal coupling. This conclusion does not depend on the presence of the matter fields of low spins. In other words, the condition of the conformal stability of the flat background, which is held usually as "physical", is incompatible with the asymptotic freedom in the conformal sector. Therefore, the flat background cannot present the ground state of the theory in the ultraviolet region. shown, that the theory of gravity with a quadratic in the curvature and positive definite Euclidean action possesses a stable non-flat ground

11. It is

asymptotically free both in the tensor sector and the conphysical interpretation of the nontrivial ground state as a

state and is

formal

one.

A

condensate of conformal excitations, that is formed

transition,

is

12. The effective

as a

result of

a

phase

proposed. potential, i.e.,

the off-shell

one-loop effective

action in arbi-

trary covariant gauge, and the Vilkovisky's effective action in the higherderivative quantum gravity on De Sitter background, is calculated. The determinants of the operators of second and forth orders are obtained by means

of the

generalized (-function.

13. The gauge- and parametrization-independent unique effective equations for the background field, i.e., for the curvature of De Sitter space, are obtained. The perturbative solution of the effective equations, that gives

the corrected value of the curvature of De Sitter to

quantum effects, is found.

background

space due