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 socalled background field method of De Witt. As a consequence, I have not dealt at all with manifolds with boundary, nonLaplace type (or nonminimal) operators, RiemannCartan manifolds as well as with many recent developments and more advanced topics, such as Ashtekar's approach, supergravity, strings, membranes, matrix models, Mtheory 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 selfconsistent 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 FirstOrder Terms
3.4
Summation of SecondOrder Terms
3.5
De Witt Coefficients in De Sitter
of
SchwingerDe
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 OneLoop Divergences in Minimal Gauge OneLoop Divergences in Arbitrary Gauge and Vilkovisky's Effective Action Renormalization. Group and Ultraviolet Asymptotics
4.2
........
.........................
HigherDerivative 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 selfconsistent 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 spacetime 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 spacetime 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 nongravitational 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 nonAbelian 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 nonvanishing 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 Smatrix [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 © SpringerVerlag 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 offshell effective action on the gauge and the parametrization of the quantum field. In the paper [84] a gaugeinvariant 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 higherderivative 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 modificaonelop 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 higherorder terms reproduce the vertices of the perturbation theory [83]. At oneloop 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 semiclassical 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 oneloop 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
VilkoviskyDe
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 oneloop 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 SchwingerDe 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 lightcone. Therefore, it is the most suitable tool for investigation of the ultraviolet divergences (calculation of counterterms, 0functions 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 oneloop approximation, the proper time method is applicable at higher loops too. In the papers [157, 169, 61, 36, 59] it was applied to analyze twoloop divergences in various models of quantum field theory including Einstein's quantum gravity. Another important area, where the SchwingerDe Witt propertime 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 socalled 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 SchwingerDe 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 nonlocal expression for the oneloop 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 oneloop 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 oneloop quantum gravity, a complete selfconsistent 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
oneloop approximation the pure Einstein gravity is finite on case of nonvanishing cosmological constant). However, twoloop Einstein gravity is no longer renormalizable onshell [135]. On the other hand, the interaction with the matter fields also leads to nonrenormalizability on mass shell even in oneloop 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 (higherderivative 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 higherderivative 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 Smatrix. 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
higherderivative quantum gravity bibliography). However, the oneloop
was
studied in
counterterms
first obtained in the paper of Julve and Tonin [160]. The most detailed investigation of the ultraviolet behavior of higherderivative 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 oneloop counterterms were recalculated in higherderivative
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 higherderivative 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 higherderivative 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 offshell oneloop divergences of higherderivative 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 "zerocharge" 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 zerocharge 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 oneloop 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
SchwingerDe
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 nonlocal 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 nonlocal 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 nonlocal expression for the oneloop 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 pseudoEuclidean region above the threshold are obtained explicitly. The massless case in four and
twodimensional 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 semiclassical series is calculated. An explicit expression for the oneloop effective action, nonanalytic 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 higherderivative 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 oneloop divergences of higherderivative quantum gravity with the help of the methods of the generalized SchwingerDe Witt technique are calculated. The error in the coefficient of the RIdivergent term, due to previous authors, is pointed out. In Sect. 4.3 the dependence of the
divergences lyzed.
of the effective action
The offshell
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 renormalizationgroup 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 lowspin' matter fields does not change this general conclusion: higherderivative 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 positivedefinite Eutained. It is shown that in the conformal sector of the
potential (i.e., the effective action on higherderivative quantum gravity is calculated.
clidean action. In Sect. 4.5 the effective the De Sitter
background)
in
The determinants of the second and fourthorder operators are calculated with the help of the technique of the generalized (function. It is maintained that the result for the R 2divergence 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
ndimensional
spacetime 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 E1
AB
(1.3)
,
is the inverse matrix E

1
ABEBC
=
6AC
,
EAcE
1 CB =
jAB.
(1.4)
nondegenerate both in bosebose and fermifermi satisfy the supersymmetry conditions
The metric EAB must be sectors and
EAB
=
In the
case
ghost
fields
of
(_l)A+B+AB EBA
E1
gaugeinvariant field
are
AB =
)
theories
(_,)AB E1
we assume
included in the set of the fields
OA
BA.
that the
(15)
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 © SpringerVerlag Berlin Heidelberg 2000
1.
10
Background
Field Method
and substitute the mixed set of indices time
small Latin index i
point, by summationintegration 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 outregions in spacetime, 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 vacuumvacuum
transition
some
the the the
amplitude
(out, vac I in, vac) in presence of
the
=_
expiW(J) I h
background classical
sources
(17)
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
(18)
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
,
onepoint 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 manypoint 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 onepoint 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 deltafunction. 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)]
(118)
Differentiating the equation (L 15) with respect to the sources one can express all the manypoint Green functions (1.13) in terms of the vertex functions (L 17) and the onepoint 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 manypoint Green function is represented by all kinds of tree diagrams with a given number of external lines.
12
1.
Background
Thus when
Smatrix
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 onepoint 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 Smatrix [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 Smatrix
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 higherderivative 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 +,,Ahhi, 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
onepoint 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
oneloop 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 oneloop 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
E1
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 deltafunction 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,
(131)
background spacetime,
parameter of the quantum field and matrixvalued function (potential term). m
(130)
,
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)
(132)
,
appropriate boundary conditions, by means of the FockSchwinger[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)
(133)
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,
(134)
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 lightcone, 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 energymomentum tensor, oneloop 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')
=
g1 /2 (x) det (_Vtl, VVa(X, X1)) g1/2 (XI)
(1.37)
pAB (X, XI) is the parallel disis, the Van FleckMorette 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 twopoint pAB (X, xi) to be singlevalued 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
spacetime (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
p1
=
equation for the function S?(s):
(i,/A1/2 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) Pak1
(1.44)
(1.45)
The coefficients ak (X, x') are widely known under the name "heat kernel coefficients", or "HMDS (HadamardMinakshisundaramDe WittSeeley) 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 noneffective. 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 noncovariant 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 lightcone 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
spacetime 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 nonanalytically
(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 counterterms in the classical action that compensate the corresponding divergences. In renormalizable field theories one introduces the counterterms 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), manypoint 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) = /'Ldi gj(p) are the dimensionless coupling parameters (di is the dimension of the coupling gi), and Pj(g) are the GellMannLow #function. Let
us
note, that among the parameters
couplings [229] (like are
gi(li)
there
are
also nonessential
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 Smatrix 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 nonessential ones. We will apply the renormalization. group equations for the investigation of the ultraviolet behavior of the higherderivative quantum gravity in Chap. on
mass
therefore,
4.
on
the nonessential
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 oneloop effective action
F(j), (1.28). Making use of the SchwingerDe (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 fourdimensional
spacetime (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 cutoff regularization, the PauliVillars 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 ndimensional
spacetime
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
singlevalued, 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
(nonminimal 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 oneloop effective
(1.50)
action in dimensional
G(x,x)
i =
regularization M2
I G4 +C+ log Iirp7 ) (M 2
_F47rF
2 _
a,
(x, x))

m
21
(1.51)
+Gren (X, X) Gren (X) X)
FM
1 ":::
=
2
2(4F7r7
2
n4
ak
1)2
(41r
+ C +
k>2
k(k M2
log TIrk7

(x, x) 1),M2(k1)
) (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(k2)
(154)
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 ndimensional. However in that part, analytical in n 4, one can treat them as 4dimensional.
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 nonprimed (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 © SpringerVerlag 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)
7211VV +
[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 P1
=
=
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'P1
mations at the
,
is
a
=
i
(2.19)

scalar under the coordinate transfor
x, since it does not have any
nonprime 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,
>=
Or1
>=
...
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 noninteger 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. V1Vn
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
(231)
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
(234)
.
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 ><,mjP1F'Pjn
F
><
to
n
IP1
(235)
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
(239)
77
77v" it follows that for close
(2.11)
points
x
and
XI det 77
0
dety 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
(A1
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 FleckMorette determinant
' A(X' XI)
(2.41)
I
(1.37)
g1/2 (X I)g1/2 (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'"JIY'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),
(252)

On the other hand, when acting on the objects D, (2.18), that nonprimed 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 deltafunction
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
(259)
0

where
S
6'V V
(260)
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
,
PY1
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
=
(266)
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 (232),
when
=
the matrix
E where
K(n)
is the matrix with entries K
K
v,
K
(n)
K",VillA. we
2)!
(n
n>2
...
(267)
1
(n)
01
V,tt'1
V(j'l
V,
K(n)
411
...
a
Vgn2R/,A11VIAn)
(268)
I
obtain
E
+
n!
'Y(n)
(269)
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+...+nk1)(nl+.+nk1+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 RI1131,61A4)
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,(nk)77
+
k
2
(276)
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 FleckMorette 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)
(279)
(2.79) equal (Al/A2

6
(IA1112/A3
4
RA11A2
V(j, Rtl2t13)
(280)
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
(286)
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)
(288)
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
(292)
d9 and treat the matrix
(2.61),
we
obtain
an
S
as
ordinary d2
jt2 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=
,
(293)
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 FleckMorette determinant
'A
The vector
(2.87),
matrix H,"
that does not
in the
det
(
V/S sin
(2.90)
case
depend
(296)
V/
is
presented
as
the
and the vector
R,,,,
on
(297)
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 ,
VSO 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
2AuffL"',
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
=
17L)

+
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)

(451
P2+Ort'l OrVI (1 0'
UI
v,T1
'a/
Orv,
+
(2

2o,
(2.108)

_
2o,
V
concrete form
02)
T1
where sin
,F2Aa
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
pl(j' A1/2 [:]' 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
singlevalued 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 Ukl2fk (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
<
njFjnk1
><
1 + nj
1 + nk1

nk11FInk2
>
...
<
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
p1A1/2Vjj1 /A1XjuYVv1 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'***vgV+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
(AI412
XVn

1
M
6(V1*Vn1 yVn) An"'IL)
Vn) Anl*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 spacetime 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
oneloop 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 nonvanishing 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
CODO
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 CE3DO 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=
21 CM9D0
62
21 cErN=
63
21 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 10 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,03/14 411.14
Zkt(2)
=
Zttv(2)
=
ZIt
=
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 IA31t4XIAV 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 anticommutator 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
(2152)
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
oneloop effective action (1.54) in the fourdimensional physical spacetime in the lowest nonvanishing approximation J/M2 [120, 121]. By integrating by parts and omitting the total derivatives we obtain from 
(2.128), (2.141), (2144), (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',Ot
+
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
71ti."'Mn
K',3

A 1 "M n '
A
MA
1
...
An
and denote the contracted
...
V(A1 to
13
A.)
VAn2R"/LnI
symmetrized
(analogously
VAn,R' 1YI 13
V(AI
V(111
/11An
ber in the brackets
...
V(A' =
=
VOII
VMn, R,Un)
M.)
(2.161)
1
VAn_2R/InIPn)
7
VAnI 7ZAIIA.) covariant derivatives
(2.132)(2.134).
For
just by example,
a num
2.4 De Witt Coefficients a3 and
_Ta,8
K"6jtv(4) L
A (4)
gAl A2gA3A4 K'13/IV/1114
=
9Al/A2 9113/14LaAlzl14
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/LV1111,14
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.)3vyl +
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
gdU1A2gI13A4g/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
1505 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
(2165)
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
oneloop example of real scalar, spinor and vector massive quantum matter fields on a classical gravitational field background in the fourdimensional 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
y01y1)
codifferentiation
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
(313) (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 SchwingerDe 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(isyI'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 nonessential 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 oneloop 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(k2)
(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 energymomentum tensor of the quantum matter fields in the background gravitational field [120, 121]
out, vac IT,,,,, (x) I in, vac
In
)
h2
g1/2 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 oneloop 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
energymomentum
(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
21
180
60
1
7 1440
I 
T0
8
25
52
945
756
63
2
47
315
1260
1
19
61
1260
1260
140
17
29
67
7560
7560
2520
19

I
CIO
5 
64
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 nonlocal methods for the Green function and well the effective action
,
the effective action.
3. Partial Summation
of
SchwingerDe
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
noncovariant 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 SchwingerDe Witt representation (1.33), (1.49), only as an auxiliary tool for the separation of the ultraviolet It is stated there that the SchwingerDe Witt representation, divergences. for exists a small class of spaces only when the semiclassical 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 SchwingerDe 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 © SpringerVerlag Berlin Heidelberg 2000
an
asymptotic
(31)
3. Summation of
52
SchwingerDe
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,
=
(33)
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'etB(at1')
G(a)
,
(3.5)
C
where the integration contour C starts at the zero point and goes to infinity in the right halfplane (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, SchwingerDe
expansion in the background dimen2k 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 SchwingerDe 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
(130),
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
threepoint Green function gA" B (X, Y jX%
additional fixed point
p1 (X, Xl).
x',
A112(X' x')G(x, Y). A1/2 (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
SchwingerDe
jM2) g(X,Y)
=
Witt
Expansion
_jg112(X),A1(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
(38)
,
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
(39)
(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 "A1 (X)g112 (X)6(X, Y)
(3.12)
form
112(Y),A(Y)gO(X,Y)Pyg(y, Z)
(313)
,
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
(315) 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)
,
(316)
3.2 Covariant Methods for
Investigation
of Nonlocalities
55
where
9(k,p)
=
(27r)'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(qil, 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 spacetime [50, 155, 193]. Similarly, one can construct the kernels of any nonlocal 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),p1(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, Ln2k 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
SchwingerDe
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+njl ) ij1 > :
I
*
*
< n;
iN1

i2
< n2;
...
where the < n;
k

>=
il

g"'2 io
and the summation ements n
are
+ 2k >
11FIn,
>< nN1;
>< ni;
ii
iN1
11FIO

(2ij+njl)
1<j
iN2

11FInN2
>
(3.21)
>
notation is introduced
following
kIFIm
11FInN1

ni
...
=
over
9
V2kIV2k
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(ij1) ! 0, n2+2(i2il1) !
ni

,


,
n+2(kiN11)
nN1

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 FirstOrder Terms
57
3.3 Summation of FirstOrder 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
k1
(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 (327) 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 betafunction [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 nonlocal operators in terms of covariant momentum
f, (is 1:1), f2 (is 1:1) should be understood expansions (3.20).
3. Summation of
58
SchwingerDe
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 ndimensional space ,
G(x, x)
=
i(47r) n/2
+1'
Jr (1
_
n) i Mn2 2
E3
(2 n) rnn4
M2
2
0
Q+iP2
_

4M2
) R] (3.31)
+O(R 2), where
F(z)
is the Euler
gammafunction,
and
1
AW
fd
=
[1 + (1
2)Z]
_
(n4)/2
(3.32)
1
0
A (Z)
=
1
fd
4
(1
_
2) [I
+
(1
2),](n4)/2
_
(333)
.
0
4 By expanding in the dimension n in the neighborhood of the point n and subtracting the pole 1/(n4) we obtain the renormalized Green function, Gren (X, X), in the physical fourdimensional spacetime (1.51), (1.52), up to terms of second order in background fields =
0
(47r )2 IF, ( iw ) 1
Gren (X, X)
=
Q
+
=
2
iF2
+O(R
E3
( 2). R) 4m
2)
,
(334)
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

(337)
2)Z
0
The formfactors F, (z),
(335), and F2 (z), (3.36),
are
normalized
according
the conditions
Fj(O)
=
F2(0)
=
0

(3.38)
integral (3.37) determines an analytical singlevalued 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 FirstOrder Terms
J(z)
J(x
21
=
iE)
=
21
+
!log
(vFz+l+vqz
arg(z
+
i7r j
+
1)1
<
59
(3.39)
7r
z
1
(vlx1 + V'x)
log
+ X
1
(X
,
<
1)
.
X
(3.40) Thus
inciding
(1.52),
obtained the nonlocal expression for the Green function at copoints, (3.34). It reproduces the local SchwingerDe 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 (334). 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 pseudoEuclidean 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 ndimensional space in the first order in
F
G(x, x)
field,
(2
(_ E]) (n4) /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
SchwingerDe
that reflect the ultraviolet
.,
the
2
reflecting pole point twodimensional space (3.46) gives and
in the
a
n
i
G(x, x)
=
(
2
i7r
=
2 n
infrared
+C+Iog
2
TI A_
In the
1
1Qi


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
N2
47r[12
N2
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 fourdimensional spacetime,
4,
n
we
have
(3.48)
_(
1
G(x,x)
=
(4,7r)
2
1
2 n
_C3

iR
18
4
+C2+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(n2) /21 (3.24). Therefore, case
the
limit of the Green function in the first order in
(350), in the
proportional to the De Witt conformally invariant case [42, 137], is
1
n2
Q
(n
1R
1)
(3.51)
3.4 Summation of SecondOrder Terms
61
the linear part of the coefficient [a(n2)/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 SecondOrder 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
i1)
I
x
where
O(RI)
<
0; k

i

11F.Ini
1) (2i+ni1) ( 2k1) k i
>< ni; i

11FIO
>
denotes all omitted terms of third order in
+O(R3)
,
background
(352) 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 tracefree 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
k2Q
k2

20kJjt
RA' + EkR E]
[:]
k3
k2
ja + Yk Q r1 k2R
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
SchwingerDe
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
(354)

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)
(356)
'
k! (k
1)

R
1)! 1)!
(357)
k!k!
6k
6k
These coefficients
were
=
(k
(3.58)
2
(2k
+
1)! k!k!
2
k

1)
(2k
(359)
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)(359)
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
(361)
where
f3 (Z)
=
E k>O
k z
(2k
+
3)!
,
(362)
3.4 Summation of SecondOrder 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
(363)
+ 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 (362) 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
(365)
(1
_
2)Z
(365)
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)

(329), (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 oneloop 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 ndimensional space
in
I
2(4,,)n/2
+
1r (2
f
dnx g1/2 str
_
2
+2JI,
+i
,,(_n)jMn+r
i
2
n) Mn4
QA
4M2
2
1F_ C3)
RtP4
3
jM2
J" +
0
( ) 4M2
Ruv +
2
I +
6
i
R)
)Q
2Q_P2
R_P5
n) Mn2 (Q
0
R
4m2
E3
4M2
) R) I
+
O(R 3)
I
(3.69)
1 Summation of
64
Witt
SchwingerDe
Expansion
where
162 [1 + (1
fd
F3 (Z)
_
62)Z]
_
2)Z]
2
(n4)/2
(3.70)
0
A (Z)
1
fd
=
4 [I
6
(1
+
(n4)/2
(371)
7
0
1
.P5 (Z)
`
1
1 d6 (2
62
_
8
164
_
[1 + (1
6
2),]
_
(n4)/2
(372)
7
0
and
P, (z)
and
P2 (z) the
Subtracting
given by the formulas (332) and (3.33).
are
pole
in the dimension
1/(n

4),
we
obtain the
renor
physical fourdimensional spacetime, (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
+

(375)

,
Z

1
3
1
1
_360z
(374)
,
Z
45z


900
+
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
,
(376)
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 SecondOrder Terms
F(l)ren
IM2_+00
(3.78)
0.
__':
65
Thus, by means of the partial summation of the local SchwingerDe Witt expansion, (1.43), (3.60), we obtained a nonlocal 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 func1 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 (374)(3.76) in the pseudoEuclidean 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
V1; 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)(376)
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
(383)
1
log
+0
+
(3.84)
z
(3.85)
1
1
+

+ 0
15
we
obtain the

log
z
2
Taking the limit m +0 oneloop effective action for
of the massless field.
Re n > 2
z
15
the massless field in the ndimensional space up to terms of third order in background fields
3. Summation of
66
F(1)
SchwingerDe
1n/2 fd 2(47r)
=
1
Xstr
n
+
+
8
2
1)

Q (_
Expansion
11) (.1, (a2 2

F(n
1Q(_E])(n4)/2Q+ 2

4(n
(2
F
nX g1/2
Witt

1
2(n
im (_
1)

[:])(n6)/2 jM
[:])(n4)/2 R E])(n4)/2 Rl"
(n 2
E])(n4)/2 RI

1))2
2)
((n21 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 fourdimensional spacetime 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
=
(47
2
r
2
1 +
+
6il. ( EI)
lQ (_
6
+i
C +
4
R1"'
R 120
(
8log 3
log
2
1
1
_
3

n

C +

n4
46 log
1
2 
711
47riL2)
15
4
C +

15

C+2log

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 SecondOrder Terms
it
was
assumed
additionally
67
that the power series in the proper time for the
A(z), (3.26), (3.27) and (362)(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)(385) 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 twodimensional 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 nonlocal 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
(388)
I
equations
0,
2
(389)
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 twodimensional space satisfies iden
For the scalar field in the =
Q
+C2)
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 twodimensional space is conformally flat. Therefore, any functional of the metric giv is uniquely determined by the trace
3. Summation of
68
SchwingerDe
of the functional derivative.
Hence,
Witt
Expansion
the effective action of the massless
con
formally invariant field in the twodimensional 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 (326),
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]
(391)
,
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
lowenergy asymptotics (E] R
derivatives determine its for
quadratic
action
background field. The highenergy
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)
kI
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,
(393). of
The problem of lowenergy asymptotics is much more difficult than highenergy 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 nonperturbative results. We would like to warn the reader that our result will not be exact but it will reproduce correctly the semiclassical SchwingerDe 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),
constantthe 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,,RkIQI
=
(395)
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]
CkIR k1Q1
(3.96)
0<1
where Ck = Ck,o
are
=
Rk[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 ndimensional
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)FQiP) 2 2
(3.97a)
(397b)
where
#2
n(n

1)
R
n
2
M
_
_
Q

I
R
_
4n
(3.98)
70
of
3. Summation of
SchwingerDe
Witt
Expansion
The asymptotic expansion of the function has the form
(397b)
in the inverse powers
fl2
!Pn (8)
dk (n)
(399)
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 SchwingerDe 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 SchwingerDe 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( a21k)
2
1)

S
2
k!
k>0
(3.102) where
R
bk (n) are
=
f n(n
1)
Ik [ak] IQ=[(nl)/(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
k1 n

+

0<1
4n
(3.95):
to
R
bkI (n)
Q
expressed
are
(3103)
4n
1R)l
obtain the coefficients
(n)
dk(n)
.
(3104) 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), (3101), 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,. gammafunction 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 singlevalued and immediately defines the coefficients bk (n). For k > 11 there appear poles in the gammafunction that are suppressed by the 2
In
zeros
even
dimensions
of the coefficients
=

....

dk(n), (3106). 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), (3101), we
(n
and the
=
.
bk (n)
F
(l)k(n/2)k! (221) 1' (k + 1 n2
(l)'
E
k


(1
1

21+212k ) BU21dj(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)k1 (121 2k )
(3.109)
=
(3.108),
=
bo (4)
=
1
bk(4)
=
(_I)k
I The
_k T
J(k

1) (1
(12 32k )
B2k

be written in
integral representation
a more
(k
)
>
0)
,
21 2k ) B2k
B2k21
expression for the coefficients bk(n) can
<
(3103), (physical) spacetime. Substituting n (3.101), (3105) and (3.108) we obtain
three and fourdimensional n
(1
in
,
(k
even
> 
(3110)
1)
dimension for k >
convenient and compact form.
of the Bernoulli numbers
[98]
2 and
n/2,
Using the
72
3. Summation of
SchwingerDe
Witt
Expansion 00
B2k
(_l)k+l
=
4k 1
dt
f
212k

e
27rt
t2kI
(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)S1k!
bk (n)
(R2).V (k + 1
.P
dt t
2) f
e27rt + 1
0
.1, X
( n21 +it) F ( 2 it) t2kn 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)
(3113)
where k
zbk(n) k!
w(z)
(3114)
k>O
Let
us
divide the series
(3.114)
in two
(Z)
W
=
W1
parts
(Z) +'W2 (Z)
(3115)
7
where W,
(Z)
E
=
T! bk (n)
(3116)
,
O
W2
(Z)
k!
bk (n)
(3117)
k jn/2] The first part w, (z),
(3116),
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 nonanalytical 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) 21
Ik+oo
=
as
k
in
+ oo
k!(2kl)!
4.
even
(27r)
r(k+ln) 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), (365)(3.68). divergence of the asymptotic series (3.117), it is concluded [46] that the SchwingerDe 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'1eYB(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, (3112),
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
(n1 r Q2
2
it) F (ayi it) H (Zt2) + it) r Q2 it) +

,
(3.122)
74
3. Summation of
SchwingerDe
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)
(3124)
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
(n1 F (1 2
2
it) r (ay1 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 singlevalued 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, (3117), 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(121)
f
F dtttanh
(7rt)
+ it) F (n1 (n1 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), (3126). Using
the obtained transfer function
,
(3.113), (3.118), (3.126),
one can
calculate the
oneloop 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 twodimensional
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 fourdimensional spacetime 1
F(l)ren

2 (47r) 2
f
1
d 4X g112
1
[M4
_
2
2rn2
(Q R) +
6
n=4
QR +
+
1080
3
9
1M2
M2
29
1
+Q2
Q
+
i8
2
R
)
Q
_
R2 log
+
R2
+
6
1024
48
R
M2
5Q2 + 43 QR 96
4
9)
1
41 +
9 _
R2X2
12
4
R
(3128) where 00
(Z)
X1
=
f
P
dt t
e21rt + 1
log
(1 Z)
(3129)

1
0 00
X2 (Z)
=
I
1
dt t
(t2 ) (1 Z) log
+
e27rt + 1
4


(3130)
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 threedimensional space
F(1)
_1_

3 (47r)
3/2
f
d3X g1/2
M2
_
Q
6
R)
(3131)
n=3
again that our study of the lowenergy 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 lowenergy asymptotics see, Let
us
[13][21].
remind
once
3. Summation of
76
SchwingerDe
Witt
Expansion
only showed that by partial summation of some lowenergy terms SchwingerDe Witt asymptotic expansion one can obtain a nontrivial 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.
HigherDerivative 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
C0 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 © SpringerVerlag Berlin Heidelberg 2000
78
4.
Higherderivative Quantum Gravity
given subspace M'. If one reparametrizes the initial configuration space by new variables, M IIA, Xtt}, where JA are the physical gaugeinvariant 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 gaugeinvariant quantities (in particular, the action S( O)) expressed only in terms of the gaugeinvariant 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
nongaugeinvariant 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
gaugeinvariant
(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
nondegenerate 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 FaddeevPopov [101, 781 and NielsenKallosh commuting variables, so =
[182, 162] "ghosts". Using
the
equation (4.12) F
we
find the
oneloop 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 Smatrix 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 gaugeinvariant 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
Higherderivative 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 gaugeinvariant functional of the background field provided the generators of gauge transformations the
0 [84]. fields, i.e., R'a,kn(0 However, the offshell 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 offshell effective action can give much more information about quantum processes than just the Smatrix. 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 offshell 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 noncovariance of the effective
(Wi
term
is not

0)1 i(fl
in
(4.12)
as
geometric object. One can replaces this difference by a twopoint 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
gaugeinvariant
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 reparametrizationinvariant 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 oneloop 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 gaugeinvariant metric Eik(V) in nondegenerate 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.
Higherderivative Quantum Gravity
Ijkl =2 E1"(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)

nondegeneracy
to define the De Witt
one
EmkJ
+
(4.28)
projector [78, 83] H2J_
B'
=
JJ_L
(4.29)
where
N1,"'R' Ein
B ',, n
The
every orbit to
projector U1 projects
Ili_" R'o, orthogonal
to the
the
(431)
generators R k. in the metric EiA;
R"OEkmlli_" Therefore,
point,
one
0,
=
S
and is
(430)

=
0
(432)

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) Ionshell and leads to the standard Smatrix
=Ir(4 )Ionshell [223].
Vilkovisky's effec
4.2
To calculate the
Vilkovisky's effective
The result will be the
simply put
the
nonphysical
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
,
(436)
The nonmetric part of the connection
17j',,al.
u'j_
where
action
It is convenient to
same.
XIL (
i.e., ai
OneLoop Divergences
satisfies the equation
H_LmkT'm ,17_L n.3
=
0.
(437)
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
(438)
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)]
(439)
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 oneloop 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
OneLoop Divergences in Minimal Gauge
The
theory of gravity with higher derivatives as well as the Einstein gravity theory of gravity is a nonAbelian gauge theory with
and any other metric
84
4.
Higherderivative Quantum Gravity
the group of
diffeomorphisms G of the spacetime as the gauge group. The complete configuration space M is the space of all pseudoRiemannian metrics on the spacetime, and the physical configuration space, i.e., the space of orbits A4, is the space of geometries on the spacetime. We will parametrize the gravitational field by the metric tensor of the spacetime 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 deltafunction. 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 fourdimensional pseudoequal to the determinant of the 10 x 10 matrix:
a
Euclidean spacetime 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
,
JO1 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
OneLoop Divergences in Minimal Gauge
85
nondegenerate 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
antisymmetric LeviCivita 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 Smatrix.
The extremal of the classical action,
and the N8ther
(4.5),
identities, (46),
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
TTdiv "
1gl'vR + Ag"

(4.14) 1
=
_
2i
we
oneloop 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)AparVAVPVqVT + VPF(2)poVo'
+F(3)pVp
+
VPF(3)p
+
P(4)
9
1126(X, Y)
(4.53)
86
4.
Higherderivative 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
=
(454)
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(1t(,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
OneLoop 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
Higherderivative 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
spacetime

R,,,,
to be
1 J(x, y)
(4.57)
.
nondegenerate
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
(459)
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 DonderFockLandau 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)
(462)
4.2
OneLoop 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
Tf2
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.
Higherderivative 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 T2T12) (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"
OneLoop 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 SchwingerDe 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
tip +
R*R* + 180
Rt1b

12
6
I +
tlbtlb
I +
24
48
1
1
2

+
C2
60
1
R
+
36
2)
1Rljv tibltv
6
it' bliv tlbliv
(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, P1
=
E iBl'
is the commutator of covari
4.
92
Higherderivative Quantum Gravity
[V,,, V,]hA 'tr'
means
In
our
the matrix trace and case,
hA
and
(4.65)
(466)
is the dimension of the
n
RY 5"31,,,
=
1
(n
4) (41r)

1
spacetime.
w)
+
5 + 2w
1
w)
+
192
c2
60
(12w )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*
OneLoop 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
(12w )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 oneloop 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.
Higherderivative 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
OneLoop 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 oneloop 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
F1
tta
that follow from the N6ther
blql)
_,A1
R'oF1
x
(4.76)
ikej Rj,,,,kF1
(4.77)
(R kly
_
O'EjRj 3,k
A1
identity, (4.6),
1 ik
kn
e,,,R',y^f,n)
one can
F 1'Y6H1 6V
derive from
(4.78)
(4.76)
6jRja,kF1 "v6Xvj
1F1'6'6jRj0,k (R
2
k
A1
_
kn
7
e,,,, R
'Yrn)
F 1
YdJ(H1 da
here, it follows, in particular, that the oneloop effective e 0, (4.5), does not depend on the gauge,
(4.79)
action
on mass
=
6F1 Since the effective action in
6HA" H1) AV

the Ward identities
,A1
Rom
HA')
R' J' ") 6x 'i

(Xt','A1 ik Xvk
+
shell.,
the functions Xt, and
(,A1 ik XIAHI"

Using
(i.e.,
background
fields in the
on
=
(480)
0.
onshell
the
mass
neighborhood
shell is well of the
defined,
mass
shell
it is
analytical
(4.5). Therefore,
OneLoop 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_ (R4A) _2_ OR
1
(4.81)
1
contributes to the divergences. Therefore, only ycoefficient, (4.75), in the divergent part of the effective action depends on the gauge parameters. The 0functions, (4.72)(4.74), do not depend on the gauge. So, from (4.79) we obtain the variation of the oneloop effective action with respect to the variation of the gauge
,jRj,,
div
1
k'F1
"136X,3i div
k

2
Rom here
one can
ejk,, ,kR 13 F1 13'rF1 "65(H1) 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
5H1 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
1div(,'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.
Higherderivative Quantum Gravity
UI (r.)
=
ej
Rjce,
k
'A1
ki(n, ao,flo,P)F1'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)=eRja,k R P F

8,y
1
I
(n)F1 `(r.)H1 YJ (a, #, P)
(4.85)
To calculate the
quantities U1, (4.84), and U2, (4.85), one has to find the F1 for arbitrary r., a, 8 and P and ghost propagators (r.) and H1 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 spacetime 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
F1 "'(r.)
IT A
1
(a, 0)
pV
' Al
1
2
=
ik
(_
=
4a
2
K
+Pj
1
VjAVv 
(_guv
(r., ao, flo)

E] +
9jAi,
=
E]
_
1_9
2f2
(gl'V Va VO
rI
3
2g112j(X, Y) Ei
VJ'VV
2
9
1126(X, Y)
(4.86)
,
,
(487)
EC01)Itv'O ci 2+ P2VILVvVaV)3 4g112j(X, Y)
+ go"O V/' VV)
(488)
where
E
(O)Av,a,8
E',,, Av, )31r.=ro 1 Pi
=
3w
1 + 4w =
9A(O'g3)v
(Oj

12i, 3
+ 1 + r.

3
)
,
g""gc',3
I
(489)
(4.90)
4 P2
Let
1) (

3w(w
+
3 + 1 +
W
r.
(4.61),
note that in the minimal gauge,
us
'A1
=
ik
Co1 O)Av,a,3
2 f2E
(0)
in
OneLoop Divergences
4.3
3

Arbitrary Gauge
)2
(4.91) 0
P2
=
pi
97
and, therefore,
91126(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]
3g112j(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]3g1126(X'Y) V
3)2(l


0)
Y=x (493)
the
Using
Green functions
of the coincidence limits of the
divergences
and their derivatives in the dimensional
regularization
div
1:1
2
2
91126(X, Y)
(n

4) (47r )2
V=X
Vt,Vv
[:]
div
3g112j(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(R4A)
,
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 (R4A),
Higherderivative 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 7coefficient reads Thus the offshell
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 quanycoefficient. 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 nonmetric part of the connection
T3,11 (4.34),
is
nonlocal,
the operator Z, (4.100), is an integrodifferential 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 nonlocal 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
+
Z1 mnTimn'i 10C
I
div
(4.102)
OneLoop 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' ,Al 10C
ik
B' Al 10C
ik
=
&
Using
the
N1 "H1N1 'OR k18
+
0(6)
N1 'AH1N1'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'
jDkR7
=
has to fix the
aA
'0
I
Tr
(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
(446). 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
nondegenerate 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 , 17Lh, 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 hL
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
__
r2

W+1
I
=
r2 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
Higherderivative 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
A1
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,
r1
4g112j(X, .) Y=X
div div
d 4X ettv
U4 where
2 f2
/IV
71
'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 j2 (R
g112 j2Y (R


4A)
4A)
.
(4.113)
Ultraviolet'Asymptotics
and
Group
4.4 Renormalization
101
Thus, the offshell divergences of the oneloop 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 7coefficient, (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 higherderivative quantum gravity is renormalizable off mass shell. Thus one can apply the renormalization group methods to study the highenergy 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 "nonessential" 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
onshell
Using the oneloop 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 0function, (1.47), and do not coupling
constants
the coefficients
play the role of generalized GellMannLow depend neither on the gauge condition nor on
the
parametrization of the
102
4.
Higherderivative Quantum Gravity
quantum field. The nonessential coupling constant k 2 (t) is, in fact, simply 4 field renormalization constant. Thus the ycoefficient, (475), (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 "zerocharge" 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 spin2 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
V2296401 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
V296401)
(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 "zerocharge" 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
V296401 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)
=
01(O)A(O)
O(t)
+
f drA(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 "zerocharge" 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
f2q(t)
+
0(f2)
(4.131)
.
t400
A2 (0) and
(4.131) depends on the initial for A (0) > A2 (0) where
C4 < 0
,
value
A(O), i.e.,
4.
104
Higherderivative Quantum Gravity 00
A2 (0)
!P(O)fdrA(,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 drA(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 lowspin 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
(4137)
twocomponent. 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 0function (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
"zerocharge" 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 higherderivative quantum gravity interacting with any lowspin 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 higherderivative 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 nonessential Einstein
constant k 2(t). The solution of the
equation (4.119)
cou
has the form
t
k2 (t)
=
k2(0) exp,
Of
dr&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.
Higherderivative 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)jr _
3
13
g65 (269 +
5 S
A 296401)
;ze
3.67
(4.142)
r
a
2
+ 2,,F296401) 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
f2r(t)
+ 0
(f 2(pr)
(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
t2.76 , 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 higherderivative 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 nontrivial metrics with large Euler characteristic, the violation of the condition (4.145) leads to the exponential growth of their weight and, therefore, to a foamlike structure of the spacetime 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
"nonphysical" (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]).
higherderivative quantum gravity in the region behavior in the conformal sector, "zerocharge" 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 "nonphysical" condition (4.146) to enough. First, the higherderivative 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 lowenergy limit of the higherderivative 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 higherderivative 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.
Higherderivative Quantum Gravity
cally free both in the tensor and conformal sectors. Besides, instead ontradictory "zerocharge" 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
higherderivative 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 nonrenormalizability [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 Smatrix
spacetime 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 nonvanishing 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 higherderivative 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 oneloop effective potential exactly. In the particular case of De Sitter background one can also check our result for the R'divergence of the oneloop 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 spacetime. Let the spacetime 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
27r 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 pseudoEuclidean 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), Sonshell
=
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 oneloop 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
hL/IV
0
a
=
and
h
7
61 A
el, A
+
=
1vj'a
2
satisfy
,
(4.158)
the differential constraints
4.
110
Higherderivative 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
AL
0
=
,UV
following JO
,
J61 Therefore, are
the
a are
+ V 1,
=
A
,
6a
the transverse traceless tensor field
physical gaugeinvariant components nonphysical 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
hL 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)ejAj
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 nonphysical fields E1 and a disappears and (4.164) takes the form
S2(g + h) I
onshell
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)
hL
(4.166) invariance, of the second varia
particular, the gauge shell, (4.150),
mass
The
An
(
(_34A
6S2(g
their
on
P2
+
h) Ionshell
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)
Ionshell
imposes the following restrictions (for A
f2
and
R
P2
+3nj
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)
Ionshell
=0.
(4.172)
4.
112
This
Higherderivative 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 oneloop 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
=
deL 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
2g1/2
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
(17 ) )
,
(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) oneloop 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.
Higherderivative 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 gaugeindependent and reparametrizationinvaVilkovisky's effective action, (4.21), on De Sitter background in the orthogonal gauge, (4.36), (4.38). In the oneloop 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) 12 (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 hL
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 DonderFockLandau 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
=
*
dhL deL 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
oneloop 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
Higherderivative 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)(4194)
On the
mass
r.
shell
(4.185), (4187). case one
has to
=
gauge
disappears
and
have
we
I
1
onshell
2)
=
2
log det. A2
1
(2A)
log det.A2
+
3
2
M
2 2
y2
4
+
+
V2)
A
V2
3
(4.195) pnshell 2
Ionshell 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 higherderivative 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
Higherderivative 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
IVIq
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 nonlocal, i.e., integrodifferential, operIn
ators.
Using the technique of the generalized (function and separating the dependence on the renormalization parameter p we get the oneloop 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
Higherderivative 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
24y X
where the coefficients
01, 03
and
04
are
6yA

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 singlevalued expression
Comparing
the
result for the coefficient
nonshell "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)(4215)
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
Tonshell
=
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(onshell 1)ren
+,
+
0(h2) (4.227)
where
ronshell 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.
Higherderivative Quantum Gravity
(4.171),
to the condition
it is nonphysical, 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
6y)
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 nonperturbative have nonperturbative 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
quantumgravitational origin [139]. 1 and, nonperturbative solution has the order Rk' the in inapplicable oneloop 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.
oneloop 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 nonlocal 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 nonlocal
coinciding points
the Green function at
background
up to
coinciding points
fields.
background fields
in De Witt coefficients
expression for the oneloop effective
background
are
nonlocal
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 twodimensional 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 SchwingerDe Witt series case
The Borel summation of the
SchwingerDe Witt expansion is explicit nonanalytic expression in the background oneloop effective action is obtained.
carried out and fields for the
an
Conclusion
126
oneloop divergences of the effective action in arbitrary as well as those of the Vilkovisky's effective action in higherderivative quantum gravity are calculated.
9. The offshell
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 higherderivative 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 nonflat 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 offshell
oneloop 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 parametrizationindependent 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