An Introduction to Quantum Gravity 1108.3269v1

8/13/2019 An Introduction to Quantum Gravity 1108.3269v1

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arXiv:1108

.3269v1

[hep-th]

16Aug2011

An introduction to quantum gravity

Giampiero Esposito,INFN, Sezione di Napoli,

Complesso Universitario di Monte S. Angelo,Via Cintia, Edificio 6, 80126 Napoli, Italy

August 17, 2011

Abstract

Quantum gravity was born as that branch of modern theoretical

physics that tries to unify its guiding principles, i.e., quantum me-

chanics and general relativity. Nowadays it is providing new insight

into the unification of all fundamental interactions, while giving rise to

new developments in mathematics. The various competing theories,

e.g. string theory and loop quantum gravity, have still to be checked

against observations. We review the classical and quantum founda-

tions necessary to study field-theory approaches to quantum gravity,the passage from old to new unification in quantum field theory, canon-

ical quantum gravity, the use of functional integrals, the properties

of gravitational instantons, the use of spectral zeta-functions in the

quantum theory of the universe, Hawking radiation, some theoretical

achievements and some key experimental issues.

1 Introduction

The aim of theoretical physics is to provide a clear conceptual framework for

the wide variety of natural phenomena, so that not only are we able to makeaccurate predictions to be checked against observations, but the underlyingmathematical structures of the world we live in can also become sufficientlywell understood by the scientific community. What are therefore the keyelements of a mathematical description of the physical world? Can we derive

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all basic equations of theoretical physics from a set of symmetry principles?

What do they tell us about the origin and evolution of the universe? Why isgravitation so peculiar with respect to all other fundamental interactions?The above questions have received careful consideration over the last

decades, and have led, in particular, to several approaches to a theory aimedat achieving a synthesis of quantum physics on the one hand, and generalrelativity on the other hand. This remains, possibly, the most importanttask of theoretical physics. In early work in the thirties, Rosenfeld [131,132]computed the gravitational self-energy of a photon in the lowest order ofperturbation theory, and obtained a quadratically divergent result. Withhindsight, one can say that Rosenfelds result implies merely a renormaliza-tion of charge rather than a non-vanishing photon mass [40]. A few years after

Rosenfelds papers [131,132], Bronstein realized that the limitation posed bygeneral relativity on the mass density radically distinguishes the theory fromquantum electrodynamics and would ultimately lead to the need to rejectRiemannian geometry and perhaps also to reject our ordinary concepts ofspace and time[20,135].

Indeed, since the merging of quantum theory and special relativity hasgiven rise to quantum field theory in Minkowski spacetime, while quantumfield theory and classical general relativity, taken without modifications, havegiven rise to an incomplete scheme such as quantum field theory in curvedspacetime[65], which however predicts substantially novel features like Hawk-

ing radiation [87, 88], here outlined in section 7, one is led to ask what wouldresult from the unification of quantum field theory and gravitation, despitethe lack of a quantum gravity phenomenology in earth-based laboratories.The resulting theory is expected to suffer from ultraviolet divergences [157],and the one-loop [94] and two-loop [74]calculations for pure gravity are out-standing pieces of work. As is well described in Ref. [157], if the couplingconstant of a field theory has dimension massd in h= c = 1 units, then theintegral for a Feynman diagram of order Nbehaves at large momenta like

pANddp, where A depends on the physical process considered but not onthe orderN. Thus, the harmful interactions are those having negative val-ues ofd, which is precisely the case for Newtons constant G, where d =

2,

since G= 6.67 1039GeV2 in h= c = 1 units. More precisely, since thescalar curvature contains second derivatives of the metric, the correspondingmomentum-space vertex functions behave like p2, and the propagator like

p2. Ind dimensions each loop integral contributes pd, so that withL loops,V vertices and P internal lines, the superficial degree D of divergence of a

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Feynman diagram is given by[53]

D= dL + 2V 2P. (1)Moreover, a topological relation holds:

L= 1 V +P, (2)which leads to [53]

D= (d 2)L + 2. (3)In other words, D increases with increasing loop order for d >2, so that itclearly leads to a non-renormalizable theory.

A quantum theory of gravity is expected, for example, to shed new light onsingularities in classical cosmology. More precisely, the singularity theoremsprove that the Einstein theory of general relativity leads to the occurrenceof spacetime singularities in a generic way [86]. At first sight one might betempted to conclude that a breakdown of all physical laws occurred in thepast, or that general relativity is severely incomplete, being unable to predictwhat came out of a singularity. It has been therefore pointed out that all thesepathological features result from the attempt of using the Einstein theory wellbeyond its limit of validity, i.e. at energy scales where the fundamental theoryis definitely more involved. General relativity might be therefore viewed as alow-energy limit of a richer theory, which achieves the synthesis of both the

basic principlesof modern physics and the fundamental interactionsinthe form currently known.

So far, no less than 16 major approaches to quantum gravity have beenproposed in the literature. Some of them make a direct or indirect use of theaction functional to develop a Lagrangian or Hamiltonian framework. Theyare as follows.

1. Canonical quantum gravity [16,17,43,44,32,99, 100,6,54,144].

2. Manifestly covariant quantization [116,33,94,74,7, 152,21,103].

3. Euclidean quantum gravity[68,90].

4. R-squared gravity[142].

5. Supergravity[64,148].

6. String and brane theory [162,98,10].

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7. Renormalization group and Weinbergs asymptotic safety [129,106].

8. Non-commutative geometry [26,75].

Among these 8 approaches, string theory is peculiar because it is not field-theoretic, spacetime points being replaced by extended structures such asstrings.

A second set of approaches relies instead upon different mathematicalstructures with a more substantial (but not complete) departure from con-ventional pictures, i.e.

9. Twistor theory[122, 123].

10. Asymptotic quantization [67,5].

11. Lattice formulation [114,22].

12. Loop space representation[133, 134, 136,145,154].

13. Quantum topology [101], motivated by Wheelers quantum geometrody-namics [159].

14. Simplicial quantum gravity [72,1, 109,2] and null-strut calculus[102].

15. Condensed-matter view: the universe in a helium droplet [155].

16. Affine quantum gravity [105].After such a concise list of a broad range of ideas, we hereafter focus

on the presentation of some very basic properties which underlie whatevertreatment of classical and quantum gravity, and are therefore of interest forthe general reader rather than (just) the specialist. He or she should revertto the above list only after having gone through the material in sections 27.

2 Classical and quantum foundations

Before any attempt to quantize gravity we should spell out how classicalgravity can be described in modern language. This is done in the subsectionbelow.

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2.1 Lorentzian spacetime and gravity

In modern physics, thanks to the work of Einstein [51], space and time areunified into the spacetime manifold (M, g), where the metricgis a real-valuedsymmetric bilinear map

g: Tp(M) Tp(M) Rof Lorentzian signature. The latter feature gives rise to the light-cone struc-ture of spacetime, with vectors being divided into timelike, null or spacelikedepending on whetherg(X, X) is negative, vanishing or positive, respectively.The classical laws of nature are written in tensor language, and gravity is thecurvature of spacetime. In the theory of general relativity, gravity couples to

the energy-momentum tensor of matter through the Einstein equations

R12

gR=8G

c4 T. (4)

The EinsteinHilbert action functional for gravity, giving rise to Eq. (4),is diffeomorphism-invariant, and hence general relativity belongs actually tothe general set of theories ruled by an infinite-dimensional [31] invariancegroup (or pseudo-group). With hindsight, following DeWitt [39], one cansay that general relativity was actually the first example of a non-Abeliangauge theory (about 38 years before YangMills theory [164]).

Note that the spacetime manifold is actually an equivalence class of pairs

(M, g), where two metrics are viewed as equivalent if one can be obtainedfrom the other through the action of the diffeomorphism group Diff(M). Themetric is an additional geometric structure that does not necessarily solveany field equation.

2.2 From Schrodinger to Feynman

Quantum mechanics deals instead, mainly, with a probabilistic descriptionof the world on atomic or sub-atomic scale. It tells us that, on such scales,the world can be described by a Hilbert space structure, or suitable gener-alizations. Even in the relatively simple case of the hydrogen atom, the ap-

propriate Hilbert space is infinite-dimensional, but finite-dimensional Hilbertspaces play a role as well. For example, the space of spin-states of a spin-sparticle is C2s+1 and is therefore finite-dimensional. Various pictures or for-mulations of quantum mechanics have been developed over the years, andtheir key elements can be summarized as follows:

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(i) In the Schrodinger picture, one deals with wave functions evolving in

time according to a first-order equation. More precisely, in an abstractHilbert spaceH, one studies the Schrodinger equation

ihd

dt = H, (5)

where the state vector belongs toH, while H is the Hamiltonianoperator. In wave mechanics, the emphasis is more immediately puton partial differential equations, with the wave function viewed as acomplex-valued map: (x, t) C obeying the equation

ih

t = h2

2m+V, (6)where is the Laplacian in Cartesian coordinates on R3 (with thissign convention, its symbol is positive-definite).

(ii) In the Heisenberg picture, what evolves in time are instead the opera-tors, according to the first-order equation

ihdA

dt = [A, H]. (7)

Heisenberg performed a quantum mechanical re-interpretation of kine-matic and mechanical relations [93] because he wanted to formulatequantum theory in terms of observables only.

(iii) In the Dirac quantization, from an assessment of the Heisenberg ap-proach and of Poisson brackets [41], one discovers that quantum me-chanics can be made to rely upon the basic commutation relationsinvolving position and momentum operators:

[qj ,qk] = [pj ,pk] = 0, (8)

[qj,pk] = ihjk. (9)

For generic operators depending on q,p variables, their formal Taylorseries, jointly with application of (8) and (9), should yield their com-mutator.

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(iv) Weyl quantization. The operators satisfying the canonical commuta-

tion relations (9) cannot be both bounded [57], whereas it would benice to have quantization rules not involving unbounded operators anddomain problems. For this purpose, one can consider the strongly con-tinuous one-parameter unitary groups having position and momentumas their infinitesimal generators. These read asV(t) eitq,U(s) eisp,and satisfy the Weyl form of canonical commutation relations, whichis given by

U(s)V(t) = eisthV(t)U(s). (10)

Here the emphasis was, for the first time, on group-theoretical meth-ods, with a substantial departure from the historical development, that

relied instead heavily on quantum commutators and their relation withclassical Poisson brackets.

(v) Feynman quantization(i.e., Lagrangian approach). The Weyl approachis very elegant and far-sighted, with several modern applications [57],but still has to do with a more rigorous way of doing canonical quanti-zation, which is not suitable for an inclusion of relativity. A spacetimeapproach to ordinary quantum mechanics was instead devised by Feyn-man [62](and partly Dirac himself [42]), who proposed to express theGreen kernel of the Schrodinger equation in the form

G[xf, tf; xi, ti] = all paths eiS/hd, (11)where d is a suitable (putative) measure on the set of all spacetimepaths (including continuous, piecewise continuous, or even discontinu-ous paths) matching the initial and final conditions. This point of viewhas enormous potentialities in the quantization of field theories, sinceit preserves manifest covariance and the full symmetry group, beingderived from a Lagrangian.

It should be stressed that quantum mechanics regards wave functionsonly as a technical tool to study bound states (corresponding to the discrete

spectrum of the Hamiltonian operator H), scattering states (correspondinginstead to the continuous spectrum ofH), and to evaluate probabilities (of

finding the values taken by the observables of the theory). Moreover, it ismeaningless to talk about an elementary phenomenon on atomic (or sub-atomic) scale unless it is registered [160], and quantum mechanics in the

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laboratory needs also an external observer and assumes the so-called reduc-

tion of the wave packet (see [57] and references therein). There exist indeeddifferent interpretations of quantum mechanics, e.g. Copenhagen [160], hid-den variables[15], many worlds[60,35].

2.3 Spacetime singularities

Now we revert to the geometric side. In Riemannian or pseudo-Riemanniangeometry, geodesics are curves whose tangent vector X moves by paralleltransport[85], so that eventually

dX

ds

+ XX = 0, (12)

where s is the affine parameter and are the connection coefficients. Ingeneral relativity, timelike geodesics correspond to the trajectories of freelymoving observers, while null geodesics describe the trajectories of zero-rest-mass particles (section 8.1 of Ref. [85]). Moreover, a spacetime (M, g) is saidto be singularity-free ifall timelike and null geodesics can be extended to ar-bitrary values of their affine parameter. At a spacetime singularity in generalrelativity, all laws of classical physics would break down, because one wouldwitness very pathological events such as the sudden disappearance of freelymoving observers, and one would be completely unable to predict what came

out of the singularity. In the sixties, Penrose [121]proved first an importanttheorem on the occurrence of singularities in gravitational collapse (e.g. for-mation of black holes). Subsequent work by Hawking [79, 80, 81, 82, 83],Geroch [66], Ellis and Hawking [84, 52], Hawking and Penrose [86] provedthat spacetime singularities are generic properties of general relativity, pro-vided that physically realistic energy conditions hold. Very little analytic useof the Einstein equations is made, whereas the key role emerges of topologicaland global methods in general relativity.

On the side of singularity theory in classical cosmology, explicit mentionshould be made of the work in Ref. [14], since it has led to significant progressby Damour et al. [27], despite having failed to prove singularity avoidance

in classical cosmology. As pointed out in Ref. [27], the work by Belinsky etal. is remarkable because it gives a description of the generic asymptotic be-haviour of the gravitational field in four-dimensional spacetime in the vicinityof a spacelike singularity. Interestingly, near the singularity the spatial pointsessentially decouple, i.e. the evolution of the spatial metric at each spatial

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point is asymptotically governed by a set of second-order, non-linear ordinary

differential equations in the time variable [14]. Moreover, the use of quali-tative Hamiltonian methods leads naturally to a billiard description of theasymptotic evolution, where the logarithms of spatial scale factors define ageodesic motion in a region of the Lobachevskii plane, interrupted by geomet-ric reflections against the walls bounding this region. Chaos follows becausethe Bianchi IX billiard has finite volume [27]. A self-contained derivationof the billiard picture for inhomogeneous solutions in D dimensions, withdilaton andp-form gauge fields, has been obtained in Ref. [27].

2.4 Unification of all fundamental interactions

The fully established unifications of modern physics are as follows.(i) Maxwell: electricity and magnetism are unified into electromagnetism.

All related phenomena can be described by an antisymmetric rank-twotensor field, and derived from a one-form, called the potential.

(ii) Einstein: space and time are unified into the spacetime manifold. More-over, inertial and gravitational mass, conceptually different, are actu-ally unified as well.

(iii) Standard model of particle physics: electromagnetic, weak and strongforces are unified by a non-Abelian gauge theory, normally consideredin Minkowski spacetime (this being the base space in fibre-bundle lan-

guage).

The physics community is now familiar with a picture relying upon fourfundamental interactions: electromagnetic, weak, strong and gravitational.The large-scale structure of the universe, however, is ruled by gravity only.All unifications beyond Maxwell involve non-Abelian gauge groups (eitherYangMills or Diffeomorphism group). At least three extreme views havebeen developed along the years, i.e.,

(i) Gravity arose first, temporally, in the very early Universe, then all otherfundamental interactions.

(ii) Gravity might result from Quantum Field Theory (this was the Sakharovidea [139]).

(iii) The vacuum of particle physics is regarded as a cold quantum liquidin equilibrium. Protons, gravitons and gluons are viewed as collectiveexcitations of this liquid [155].

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3 Canonical quantum gravity

Although Hamiltonian methods differ substantially from the Lagrangian ap-proach used in the construction of the functional integral (see the follow-ing sections), they remain nevertheless of great importance both in cos-mology and in light of modern developments in canonical quantum gravity[6,136,144], which is here presented within the original framework of quan-tum geometrodynamics. For this purpose, it may be useful to describe themain ideas of the Arnowitt-Deser-Misner (hereafter referred to as ADM) for-malism. This is a canonical formalism for general relativity that enables oneto re-write Einsteins field equations in first-order form and explicitly solvedwith respect to a time variable. For this purpose, one assumes that four-

dimensional spacetime (M, g) can be foliated by a family of t = constantspacelike surfaces St, giving rise to a 3 + 1 decomposition of the original4-geometry. The basic geometric data of this decomposition are as follows[53].

(1) The induced 3-metrichof the three-dimensional spacelike surfaces St.This yields the intrinsic geometry of the three-space. his also called the firstfundamental form ofSt, and is positive-definite with our conventions.

(2) The way each St is imbedded in (M, g). This is known once we areable to compute the spatial part of the covariant derivative of the normal nto St. On denoting by the four-connection of (M, g), one is thus led todefine the tensor

Kij jni. (13)Note that Kijis symmetric if and only if is symmetric. In general relativity,an equivalent definition ofKij isKij 12 (Lnh)ij, whereLn denotes the Liederivative along the normal to St. The tensor Kis called extrinsic-curvaturetensor, or second fundamental form ofSt.

(3) How the coordinates are propagated off the surface St. For this pur-pose one defines the vector (N, N1, N2, N3)dt connecting the point (t, xi)with the point (t+dt, xi). Thus, given the surface x0 = t and the surfacex0 =t + dt,N dt dspecifies a displacement normal to the surface x0 =t.Moreover,Nidt yields the displacement from the point (t, xi) to the foot of

the normal to x0 =t through (t+dt, xi). In other words, the Ni arise sincethe xi = constant lines do not coincide in general with the normals to thet= constant surfaces. According to a well-established terminology, N is thelapse function, and the Ni are the shift functions. They are the tool needed

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On using the ADM variables described so far, the form of the action

integral I for pure gravity that is stationary under variations of the metricvanishing on the boundary is (in c = 1 units)

I 116G

M

(4)Rg d4x + 1

8G

M

Kii

h d3x

= 1

16G

M

(3)R+KijK

ij

Kii2

N

h d3x dt. (24)

The boundary term appearing in (24) is necessary since (4)Rcontains secondderivatives of the metric, and integration by parts in the EinsteinHilbertpart

IH 1

16G M (4)R g d4xof the action also leads to a boundary term equal to 1

8G

MK

ii

h d3x. On

denoting byG the Einstein tensor G (4)R 12 g (4)R, and defining

1

2g

g

+

g

g

, (25)

one then finds [165]

(16G)IH= M

g G gd4x+M

g

gg

d3x

,

(26)which clearly shows that IHis stationary if the Einstein equations hold, andthe normal derivatives of the variations of the metric vanish on the boundaryM. In other words, IH is not stationary under arbitraryvariations of themetric, and stationarity is only achieved after adding to IH the boundaryterm appearing in (24), if g is set to zero on M. Other useful formsof the boundary term can be found in [68, 165]. Note also that, strictly, inwriting down (24) one should also take into account a term arising from IH[32]:

It 18G

dt

M

d3x i

h

Kll Ni hij N|j

. (27)

However, we have not explicitly included It since it does not modify theresults derived or described hereafter.

We are now ready to apply Diracs technique to the Hamiltonian quanti-zation of general relativity. This requires that all classical constraints whichare first-class are turned into operators that annihilate the wave functional

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[53]. Hereafter, we assume that this step has already been performed. As

we know, consistency of the quantum constraints is proved if one can showthat their commutators lead to no new constraints [53]. For this purpose, itmay be useful to recall the equal-time commutation relations of the canonicalvariables, i.e.

N(x), (x)

= i(x, x), (28)Nj(x),

k(x)

= ik

j , (29)hjk ,

lm

= i lm

jk . (30)

Note that, following [32], primes have been used, either on indices or on thevariables themselves, to distinguish different points of three-space. In other

words, one definesj

i ji (x, x), (31) k

l

ij klij (x, x), (32)

klij 1

2

ki

lj+

likj

. (33)

The reader can check that, since [32]

H

h

KijKij K2 (3)R

, (34)

Hi

2ij

,jhil2hjl,k hjk,l

jk , (35)

one has(x), i(x)

=

(x),Hi(x)

=(x),H(x)

=i(x),Hj(x)

=

i(x),H(x)

= 0. (36)

It now remains to compute the three commutatorsHi,Hj

,Hi,H

,H,H

.

The first two commutators are obtained by using Eq. (35) and definingHi hijHj. Interestingly,Hi is homogeneous bilinear in the hij and ij ,with the momenta always to the right. As we said before, following Dirac,

the operator version of constraints should annihilate the wave function sincethe classical constraints are first-class (i.e. their Poisson brackets are linearcombinations of the constraints themselves). This condition reads as

StH d3x = 0 , (37)

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StHii d3x = 0 i. (38)

In the applications to cosmology, Eq. (37) is known as the WheelerDeWittequation, and the functionalis then called thewave function of the universe[78].

We begin by computing [32]hjk , i

StHkk d3x

= hjk,l l hlk l ,j hjl l ,k, (39)

jk , iStHkk d3x

=

jkl

,l

+lk j,l+jl k,l. (40)

This calculation shows that theHi are generators of three-dimensional co-ordinate transformations x

i

= xi

+i

. Thus, by using the definition ofstructure constants of the general coordinate-transformation group [32], i.e.

ck

ij k

i,l l

j k

j,l l

i , (41)

the results (39)(40) may be used to show thatHj(x),Hk(x)

= i

StHl cljk d3x, (42)

Hj(x),H(x)

= iH ,j(x, x). (43)Note that the only term ofH which might lead to difficulties is the onequadratic in the momenta. However, all factors appearing in this termhave homogeneous linear transformation laws under the three-dimensionalcoordinate-transformation group. They thus remain undisturbed in positionwhen commuted withHj [32].

Last, we have to study the commutatorH(x),H(x)

. The following

remarks are in order:(i) Terms quadratic in momenta contain no derivatives ofhij or

ij withrespect to three-space coordinates. Hence they commute;

(ii) The terms

h(x)

(3)R(x)

and

h(x)

(3)R(x)

contain no momenta,so that they also commute;

(iii) The only commutators we are left with are the cross-commutators,

and they can be evaluated by using the variational formula [32]

h (3)R

=

h hijhkl

hik,jl hij,kl

h

(3)Rij 12

hij

(3)R

hij , (44)

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which leads to

StH 1 d3x,

StH 2d3x= i

StHl1 2,l 1,l 2d3x. (45)

The commutators (42)(43) and (45) clearly show that the constraint equa-tions of canonical quantum gravity are first-class. The Wheeler-DeWitt equa-tion (37) is an equation on the superspace (here is a Riemannian 3-manifolddiffeomorphic to St)

S() Riem()/Diff().

In this quotient space, two Riemannian metrics on are identified if they

are related through the action of the diffeomorphism group Diff().Two very useful classical formulae frequently used in Lorentzian canonical

gravity are

H (16G)Gijklpijpkl

h

16G

(3)R

, (46)

H (16G)1

GijmlKijKml

h

(3)R

, (47)

where the rank-4 tensor density is the DeWitt supermetric on superspace,with covariant and contravariant forms

Gijkl

1

2hhikhjl +hilhjk hijhkl, (48)Gijkl

h

2

hikhjl +hilhjk 2hijhkl

, (49)

and pij is here defined ash

16G

Kij hijK

. Note that the factor2

multiplying hijhkl in (49) is needed so as to obtain the identity

GijmnGmnkl =

1

2

ki lj+

likj

. (50)

Equation (46) clearly shows that

Hcontains a part quadratic in the momenta

and a part proportional to (3)R (cf. (34)). On quantization, it is then hardto give a well-defined meaning to the second functional derivative

2

hijhkl,

whereas the occurrence of (3)R makes it even more difficult to solve exactlythe Wheeler-DeWitt equation.

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It should be stressed that wave functions built from the functional integral

(see the following sections) which generalizes the path integral of ordinaryquantum mechanics (see (11)) do not solve the WheelerDeWitt equation(37) unless some suitable assumptions are made [77], and counterexampleshave been built, i.e. a functional integral for the wave function of the universewhich does not solve the WheelerDeWitt equation[37].

4 From old to new unification

Here we outline how the space-of-histories formulation provides a commonground for describing the old and new unifications of fundamental theories.

4.1 Old unification

Quantum field theory begins once an action functional S is given, since thefirst and most fundamental assumption of quantum theory is that every iso-

lated dynamical system can be described by a characteristic action functional

[31]. The Feynman approach makes it necessary to consider an infinite-dimensional manifold such as the space of all field histories i. On thisspace there exist (in the case of gauge theories) vector fields

Q = Qi

i (51)

that leave the action invariant, i.e. [39]

QS= 0. (52)

The Lie brackets of these vector fields lead to a classification of all gaugetheories known so far.

4.2 Type-I gauge theories

The peculiar property of type-I gauge theories is that these Lie brackets areequal to linear combinations of the vector fields themselves, with structureconstants, i.e. [38]

[Q, Q] =C Q, (53)

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where C

i = 0. The Maxwell, YangMills, Einstein theories are all ex-

ample of type-I theories (this is the unifying feature). All of them, beinggauge theories, need supplementary conditions, since the second functionalderivative ofS is not an invertible operator. After imposing such conditions,the theories are ruled by an invertible differential operator of DAlemberttype (or Laplace type, if one deals instead with Euclidean field theory), or anon-minimal operator at the very worst (for arbitrary choices of gauge pa-rameters). For example, when Maxwell theory is quantized via functionalintegrals in the Lorenz[110]gauge, one deals with a gauge-fixing functional

(A) = A, (54)

and the second-order differential operator acting on the potential in thegauge-fixed action functional reads as

P = +R +

1 1

, (55)

where is an arbitrary gauge parameter. The Feynman choice= 1 leadsto the minimal operator

P = +R ,which is the standard wave operator on vectors in curved spacetime. Such

operators play a leading role in the one-loop expansion of the Euclideaneffective action, i.e. the quadratic order in hin the asymptotic expansion ofthe functional ruling the quantum theory with positive-definite metrics.

The closure property expressed by Eq. (53) implies that the gauge groupdecomposes the space of histories into sub-spaces to which the Q aretangent. These sub-spaces are known as orbits, and may be viewed as aprincipal fibre bundle of which the orbits are the fibres. The space of orbitsis, strictly, the quotient space /G, whereG is the proper gauge group, i.e.the set of transformations of into itself obtained by exponentiating theinfinitesimal gauge transformation

i =Qi, (56)

and taking products of the resulting exponential maps. Suppose one performsthe transformation[39]

i IA, K (57)

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and it therefore reduces to (53) only on the mass-shell, i.e. for those field

configurations satisfying the EulerLagrange equations. An example is givenby theories with gravitons and gravitinos such as BoseFermi supermulti-plets of both simple and extended supergravity in any number of spacetimedimensions, without auxiliary fields[148].

4.5 From general relativity to supergravity

It should be stressed that general relativity is naturally related to supersym-metry, since the requirement of gauge-invariant RaritaSchwinger equations[128] in curved spacetime implies Ricci-flatness in four dimensions [29], whichis then equivalent to vacuum Einstein equations. Of course, despite such a

relation does exist, general relativity can be (and is) formulated without anyuse of supersymmetry.

The Dirac operator [56] is more fundamental in this framework, sincethe m-dimensional spacetime metric is entirely re-constructed from the -matrices, in that

g= 2[m/2]1tr(+). (62)

In four-dimensional spacetime, one can use the tetrad formalism, with Latinindices (a, b) corresponding to tensors in flat space (the tangent frames, thefreely falling lifts) while Greek indices (, ) correspond to coordinates incurved space. The contravariant form of the spacetime metric g is then given

by g=abeaeb , (63)

whereab is the Minkowski metric and ea are the tetrad vectors. The curved-space-matrices are then obtained from the flat-space-matricesa andfrom the tetrad according to

=aea . (64)

In Ref. [64], the authors assumed that the action functional describing theinteraction of tetrad fields and RaritaSchwinger fields in curved spacetime,subject to the Majorana constraint (x) =C(x)

T, reads as

I= d4x14

2gR 12

(x)5D(x), (65)where the covariant derivative of RaritaSchwinger fields is defined by

D(x) (x) +1

2ab

ab. (66)

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With a standard notation, are the Christoffel symbols built from the

curved spacetime metric g

, ab is the spin-connection (the gauge fieldassociated to the generators of the Lorentz algebra)

ab =1

2

ea(eb eb) +eaeb (ec)ec

(a b), (67)

while ab is proportional to the commutator of -matrices in Minkowskispacetime, i.e.

ab 14

[a, b]. (68)

To investigate the possible supersymmetry possessed by the above actionfunctional, the authors of Ref. [64] considered the transformation laws

(x) =1D(x), (69)

ea(x) = i(x)a(x), (70)

g(x) = i(x)(x) +(x)

, (71)

where the supersymmetry parameter is taken to be an arbitrary Majoranaspinor field(x) of dimension square root of length. The assumption of localsupersymmetry was non-trivial, and was made necessary by the coordinate-invariant Lagrangian (i.e. at that stage one had to avoid, for consistency, thecoordinate-dependent notion of constant, space-time-independent spinor).After a lengthy calculation the authors of Ref. [64] managed to prove fullgauge invariance of the supergravity action. With geometrical hindsight, onecan prove it in a quicker and more elegant way by looking at a formulationof Supergravity as a YangMills Theory [149].

4.6 New unification

In modern high energy physics, the emphasis is no longer on fields (sections ofvector bundles in classical field theory [156], operator-valued distributions inquantum field theory[161]), but rather on extended objects such as strings

[28]. In string theory, particles are not described as points, but insteadas strings, i.e., one-dimensional extended objects. While a point particlesweeps out a one-dimensional worldline, the string sweeps out a worldsheet,i.e., a two-dimensional real surface. For a free string, the topology of theworldsheet is a cylinder in the case of a closed string, or a sheet for an

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open string. It is assumed that different elementary particles correspond to

different vibration modes of the string, in much the same way as differentminimal notes correspond to different vibrational modes of musical stringinstruments[28]. The five different string theories [4] are different aspects ofa more fundamental unified theory, called M-theory [13].

In the latest developments, one deals with branes, which are classicalsolutions of the equations of motion of the low-energy string effective action,that correspond to new non-perturbative states of string theory, break half ofthe supersymmetry, and are required by duality arguments in theories withopen strings. They have the peculiar property that open strings have theirend-points attached to them[45,46]. Branes have made it possible not onlyto arrive at the formulation ofMtheory, but also to study perturbative and

non-perturbative properties of the gauge theories living on the world-volume[47]. The so-called Dirichlet branes [124], or Dp branes, admit indeed twodistinct descriptions. On the one hand, they are classical solutions of thelow-energy string effective action (as we said before) and may be thereforedescribed in terms of closed strings. On the other hand, their dynamics isdetermined by the degrees of freedom of the open strings with endpoints at-tached to their world-volume, satisfying Dirichlet boundary conditions alongthe directions transverse to the branes. They may be thus described in termsof open strings as well. Such a twofold description of Dp branes laid the foun-dations of the Maldacena conjecture [112] providing the equivalence between

a closed string theory, as the IIB theory on five-dimensional anti-de Sitterspace times the 5-sphere, and N= 4 super YangMills with degrees of free-dom corresponding to the massless excitations of the open strings havingtheir endpoints attached to a D3 brane.

For the impact of braneworld picture on phenomenology and unification,we refer the reader to the seminal work in Refs. [126, 127], while for therole of extra dimensions in cosmology we should mention also the work inRefs. [137,138]. With the language of pseudo-Riemannian geometry, branesare timelike surfaces embedded into bulk spacetime [11, 10]. According tothis picture, gravity lives on the bulk, while standard-model gauge fields areconfined on the brane[12]. For branes, the normal vector Nis spacelike with

respect to the bulk metric GAB, i.e.,

GABNANB =NCN

C >0. (72)

For a wide class of brane models, the action functional Spertaining to thecombined effect of bulk and brane geometry can be taken to split into the

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sum[10] (g(x) being the brane metric)

S= S4[g(x)] +S5[GAB(X)], (73)

while the effective action [38] is formally given by

ei =

DGAB(X) eiS gauge fixing term. (74)

In the functional integral, the gauge-fixed action reads as (here there is sum-mation as well as integration over repeated indices [31,38,10])

Sg.f.= S4+S5+1

2

FAABFB +

1

2

, (75)

where FA and are bulk and brane gauge-fixing functionals, respectively,while AB and are non-singular matrices of gauge parameters, similarlyto the end of section 4.2. The gauge-invariance properties of bulk and braneaction functionals can be expressed by saying that there exist vector fieldson the space of histories such that (cf. Eq. (52))

RBS5 = 0, RS4 = 0, (76)

whose Lie brackets obey a relation formally analogous to Eq. (53) for ordi-nary type-I theories, i.e.

[RB, RD] =CABD RA, (77)

[R, R] =CR. (78)

Equations (77) and (78) refer to the sharply different Lie algebras of dif-feomorphisms on the bulk and the brane, respectively. The bulk and braneghost operators are therefore

QAB RBFA =FA,a RaB, (79)

J

R

=,i

Ri

, (80)

respectively, where the commas denote functional differentiation with respectto the field variables. The full bulk integration means integrating first withrespect to all bulk metrics GAB inducing on the boundary M the givenbrane metric g(x), and then integrating with respect to all brane metrics.

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Thus, one first evaluates the cosmological wave function [10] of the bulk

spacetime (which generalizes the wave function of the universe encounteredin canonical quantum gravity), i.e.

Bulk =GAB[M]=g

(GAB, SC, TD)ei

S5, (81)where is taken to be a suitable measure, the SC, T

D are ghost fields, and

S5 S5[GAB] +12

FAABFB +SAQ

ABTB. (82)

Eventually, the effective action results from

ei =(g, , )eiS4Bulk, (83)

where is another putative measure, and are brane ghost fields, andS4 S4+1

2

+J. (84)

We would like to stress here that infinite-dimensional manifolds are thenatural arena for studying the quantization of the gravitational field, evenprior to considering a space-of-histories formulation. There are, indeed, at

least three sources of infinite-dimensionality in quantum gravity:(i) The infinite-dimensional Lie group (or pseudo-group) of spacetime dif-

feomorphisms, which is the invariance group of general relativity in thefirst place [31],[143].

(ii) The infinite-dimensional space of histories in a functional-integral quan-tization [38,39].

(iii) The infinite-dimensional Geroch space of asymptotically simple space-times [67].

5 Functional integrals and background fields

We now study in greater detail some aspects of the use of functional integralsin quantum gravity, after the previous (formal) applications to a space ofhistories formulation.

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5.1 The one-loop approximation

In the one-loop approximation (also called stationary phase or JWKB meth-od) one first expands boththe metric g and the fields coupled to it abouta metric g0 and a field 0 which are solutions of the classical field equations:

g= g0+g, (85)

= 0+. (86)

One then assumes that the fluctuations g and are so small that the dom-inant contribution to the functional integral for the in-out amplitude comesfrom the quadratic order in the Taylor-series expansion of the action about

the background fields g0 and0 [90]:

IE[g, ] = IE[g0, 0] +I2[g, ] + higher order terms, (87)

so that the logarithm of the quantum-gravity amplitudeAcan be expressedas

logA IE[g0, 0] + log D[g, ]eI2[g,]. (88)

It should be stressed thatbackground fields need not be a solution of any fieldequation [36], but this possibility will not be exploited in our presentation.For our purposes we are interested in the second term appearing on the right-hand side of (88). An useful factorization is obtained if0 can be set to zero.

One then finds that I2[g, ] =I2[g] +I2[], which implies[91]

logA IE[g0] + log D[]eI2[] + log D[g]eI2[g]. (89)

The one-loop term for matter fields with various spins (and boundary con-ditions) is extensively studied in the literature. We here recall some basicresults, following again Ref. [91].

A familiar form ofI2[] is

I2[] =1

2 Bg0 d4x, (90)

where the elliptic differential operatorBdepends on the background metricg0. Note thatB is a second-order operator for bosonic fields, whereas itis first-order for fermionic fields. In light of (90) it is clear that we are

interested in the eigenvalues

n

ofB, with corresponding eigenfunctions

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When fermionic fields appear in the functional integral for the in-out

amplitude, one deals with a first-order elliptic operator, the Dirac opera-tor, acting on independent spinor fields and. These are anticommutingGrassmann variables obeying the Berezin integration rules

dw= 0,

w dw= 1. (95)

The formulae (95) are all what we need, since powers ofw greater than orequal to 2 vanish in light of the anticommuting property. The reader canthen check that the one-loop amplitude for fermionic fields is

A

(1) = det

1

22B

. (96)

The main difference with respect to bosonic fields is the direct proportionalityto the determinant. The following comments can be useful in understandingthe meaning of (96).

Let us denote again by the curved-space -matrices, and by i theeigenvalues of the Dirac operator D, and suppose that no zero-modes ex-ist. More precisely, the eigenvalues ofDoccur in equal and opposite pairs:1,2, ..., whereas the eigenvalues of the Laplace operator on spinors oc-cur as (1)

2 twice, (2)2 twice, and so on. For Dirac fermions (D) one thus

finds

detDD

=

i=1| i|

i=1| i|

=

i=1| i|2, (97)

whereas in the case of Majorana spinors (M), for which the number of degreesof freedom is halved, one finds

detM

D

=i=1

| i|=

detD

D

. (98)

5.2 Zeta-function regularization of functional integrals

The formal expression (94) for the one-loop quantum amplitude clearly di-verges since the eigenvalues n increase without bound, and a regularizationis thus necessary. For this purpose, the following technique has been de-scribed and applied by many authors [48,89,91].

Bearing in mind that Riemanns zeta-function R(s) is defined as

R(s) n=1

ns, (99)

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one first defines a generalized (also called spectral) zeta-function (s) ob-

tained from the (positive) eigenvalues of the second-order, self-adjoint oper-atorB. Such a(s) can be defined as (cf. [141])

(s) n=n0

m=m0

dm(n)sn,m. (100)

This means that all the eigenvalues are completely characterized by two in-teger labels n and m, while their degeneracy dm only depends on n. Notethat formal differentiation of (100) at the origin yields

det

B

= e(0). (101)

This result can be given a sensible meaning since, in four dimensions, (s)converges for Re(s) > 2, and one can perform its analytic extension to ameromorphic function of s which only has poles at s = 1

2, 1, 3

2, 2. Since

det

B

= (0)detB

, one finds the useful formula

logA= 1

2(0) +

1

2log

22

(0). (102)

As we said following (90), it may happen quite often that the eigenvaluesappearing in (100) are unknown, since the eigenvalue condition, i.e. theequation leading to the eigenvalues by virtue of the boundary conditions, isa complicated equation which cannot be solved exactly for the eigenvalues.However, since the scaling properties of the one-loop amplitude are still givenby(0) (and (0)) as shown in (102), efforts have been made to compute(0) also in this case. The various steps of this program are as follows [89].

(1) One first studies the heat equation for the operatorB, i.e.

F(x,y, ) + BF(x,y, ) = 0, (103)

where the Greens function F satisfies the initial condition F(x,y, 0) =(x, y).

(2) Assuming completeness of the set n of eigenfunctions ofB, thefield can be expanded as

=n=ni

ann.

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(3) The Greens functionF(x,y, ) is then given by

F(x,y, ) =n=n0

m=m0

en,mn,m(x) n,m(y). (104)

(4) The corresponding integrated heat kernel is then

G() =M

TrF(x,x,)

g d4x=n=n0

m=m0

en,m. (105)

(5) In light of (100) and (105), the generalized zeta-function can be alsoobtained as an integral transform (also called inverse Mellin transform) ofthe integrated heat kernel, i.e. [89,71]

(s) = 1(s)

0

s1G()d. (106)

(6) The hard part of the analysis is now to prove that G() has anasymptotic expansion as 0+ [76]. This property has been proved forall boundary conditions such that the Laplace operator is self-adjoint andthe boundary-value problem is strongly elliptic [71, 8]. The correspondingasymptotic expansion ofG() can be written as

G() A02 +A 12

32 +A1

1 +A 32

12 +A2+ O

, (107)

which implies

(0) =A2. (108)

The result (108) is proved by splitting the integral in (106) into an integralfrom 0 to 1 and an integral from 1 to . The asymptotic expansion of1

0 s1G()d is then obtained by using (107).In other words, for a given second-order self-adjoint elliptic operator,

we study the corresponding heat equation, and the integrated heat kernelG(). The (0) value is then given by the constant term appearing in theasymptotic expansion of G() as 0+. The (0) value also yields theone-loop divergences of the theory for bosonic and fermionic fields[55].

5.3 Gravitational instantons

This section is devoted to the study of the background gravitational fields.These gravitational instantons are complete four-geometries solving the Ein-stein equationsR(X, Y)g(X, Y) = 0 when the four-metricg has signature

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+4 (i.e. it is positive-definite, and thus called Riemannian). They are of in-

terest because they occur in the tree-level approximation of the partitionfunction, and in light of their role in studying tunnelling phenomena. More-over, they can be interpreted as the stationary phase metrics in the pathintegrals for the partition functions, Z, of the thermal canonical ensembleand the volume canonical ensemble. In these cases the action of the instan-ton gives the dominant contribution to log Z. Following[125], essentiallythree cases can be studied.

5.3.1 Asymptotically locally Euclidean instantons

Even though it might seem natural to define first the asymptotically Eu-

clidean instantons, it turns out that there is not much choice in this case,since the only asymptotically Euclidean instanton is flat space. It is in factwell-known that the action of an asymptotically Euclidean metric with van-ishing scalar curvature is 0, and it vanishes if and only if the metric isflat. Suppose now that such a metric is a solution of the Einstein equationsR(X, Y) = 0. Its action should be thus stationary also under constant con-formal rescalings g k2g of the metric. However, the whole action rescalesthen asIE k2IE, so that it can only be stationary and finite ifIE= 0. Byvirtue of the theorem previously mentioned, the metric g must then be flat[70,108].

In the asymptotically locally Euclidean case, however, the boundary at

infinity has topology S3/ rather than S3, where is a discrete subgroupof the group SO(4). Many examples can then be found. The simplestwas discovered by Eguchi and Hanson [49, 50], and corresponds to = Z2and M = RP3. This instanton is conveniently described using three left-invariant 1-forms

i

on the 3-sphere, satisfying the SU(2) algebra di =

12 jki j k, and parametrized by Euler angles as follows:1= (cos )d+ (sin )(sin )d, (109)

2= (sin )d+ (cos )(sin )d, (110)3= d + (cos )d, (111)

where [0, ], [0, 2]. The metric of the EguchiHanson instantonmay be thus written in the Bianchi-IX form[125]

g1 =

1 a

4

r4

1dr dr+ r

2

4

(1)

2 + (2)2 +

1 a

4

r4

(3)

2

, (112)

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where r [a,[. The singularity ofg1 at r =a is only a coordinate singu-larity. We may get rid of it by defining 4

2

a2 1 a4

r4 , so that, as r a, themetric g1 is approximated by the metric

g2= d d+2

d+ (cos )d2

+a2

4

d d+ (sin )2d d

. (113)

Regularity ofg2 at = 0 is then guaranteed provided that one identifies with period 2. This implies in turn that the local surfaces r = constant havetopology RP3 rather than S3, as we claimed. Note that at r = a = 0the metric becomes that of a 2-sphere of radius a2 . Following Ref. [69], wesay that r = a is a bolt, where the action of the Killing vector has a

two-dimensional fixed-point set[125].A whole family of multi-instanton solutions is obtained by taking thegroup = Zk. They all have a self-dual Riemann-curvature tensor, andtheir metric takes the form

g = V1

d+ dx2

+V dx dx. (114)

Following [125], V = V(x) and = (x) on an auxiliary flat 3-space withmetric dx dx. This metricg solves the Einstein vacuum equations providedthat gradV= curl, which impliesV= 0. If one takes

V =

ni=1

1| x xi| , (115)

one obtains the desired asymptotically locally Euclidean multi-instantons. Inparticular, ifn= 1 in (115),g describes flat space, whereasn= 2 leads to theEguchiHanson instanton. Ifn >2, there are (3n 6) arbitrary parameters,related to the freedom to choose the positions xi of the singularities in V.These singularities correspond actually to coordinate singularities in (113),and can be removed by using suitable coordinate transformations [125].

5.3.2 Asymptotically flat instantons

This name is chosen since the underlying idea is to deal with metrics in thefunctional integral which tend to the flat metric in three directions but areperiodic in the Euclidean-time dimension. The basic example is provided

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by the Riemannian version g(1)R (also called Euclidean) of the Schwarzschild

solution, i.e.

g(1)R =

1 2M

r

d d+

1 2M

r

1dr dr+r22, (116)

where 2 = d d+ (sin )2d d is the metric on a unit 2-sphere. Itis indeed well-known that, in the Lorentzian case, the metric gL is moreconveniently written by using KruskalSzekeres coordinates

gL= 32M3r1e

r2M

dz dz+dy dy

+r22, (117)

where zand y obey the relations

z2 +y2 = r

2M 1

e r2M, (118)

(y+z)

(y z)= e t2M . (119)

In the Lorentzian case, the coordinate singularity at r = 2M can be thusavoided, whereas the curvature singularity atr = 0 remains and is describedby the surface z2 y2 = 1. However, if we set = iz, the analytic continu-ation to the section of the complexified space-time where is real yields thepositive-definite (i.e. Riemannian) metric

g(2)R = 32M

3r1e r2M

d d+dy dy

+r22, (120)

where2 +y2 =

r2M

1

e r2M . (121)

It is now clear that also the curvature singularity at r= 0 has disappeared,since the left-hand side of (121) is 0, whereas the right-hand side of (121)would be equal to1 at r = 0. Note also that, by setting z =i andt= i in (119), and writing 2 +y2 as (y+ i)(y i) in (121), one finds

y+ i= e i4M r2M

1 e r4M , (122)

y= cos

4M

r2M

1 e r4M, (123)

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which imply that the Euclidean time is periodic with period 8M. This

periodicity on the Euclidean section leads to the interpretation of the Rie-mannian Schwarzschild solution as describing a black hole in thermal equi-librium with gravitons at a temperature (8M)1 [125]. Moreover, the factthat any matter-field Greens function on this Schwarzschild background isalso periodic in imaginary time leads to some of the thermal-emission prop-erties of black holes. This is one of the greatest conceptual revolutions inmodern gravitational physics.

Interestingly, a new asymptotically flat gravitational instanton has beenfound by Chen and Teo in Ref. [24]. It has an U(1) U(1) isometry groupand some novel global features with respect to the other two asymptoticallyflat instantons, i.e. Euclidean Schwarzschild and Euclidean Kerr.

There is also a local version of the asymptotically flat boundary conditionin which Mhas the topology of a non-trivial S1-bundle over S2, i.e. S3/,where is a discrete subgroup of SO(4). However, unlike the asymptot-ically Euclidean boundary condition, the S3 is distorted and expands withincreasing radius in only two directions rather than three [125]. The simplestexample of an asymptotically locally flat instanton is the self-dual (i.e. withself-dual curvature 2-form) Taub-NUT solution, which can be regarded as aspecial case of the two-parameter Taub-NUT metrics

g=(r+M)

(r

M)

dr dr + 4M2 (r M)(r+M)

(3)2+

r2M2

(1)

2 + (2)2

, (124)

where the

i

have been defined in (109)(111). The main properties of

the metric (124) are(I)r [M,[, andr = Mis a removable coordinate singularity provided

that is identified modulo 4;(II) the r = constant surfaces have S3 topology;(III) r = M is a point at which the isometry generated by the Killing

vector

has a zero-dimensional fixed-point set.

In other words, r= Mis a nut, using the terminology in Ref. [69].There is also a family of asymptotically locally flat multi-Taub-NUT in-

stantons. Their metric takes the form (114), but one should bear in mindthat the formula (115) is replaced by

V = 1 +ni=1

2M

| x xi|. (125)

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(3) The Einstein metric on the product manifold S2 S2 is obtained asthe direct sum of the metrics on two 2-spheres, i.e.

g= 1

2i=1

di di+ (sin i)2di di

. (128)

The metric (128) is invariant under the SO(3) SO(3) isometry group ofS2 S2, but is not of Bianchi-IX type as (126)-(127). This can be achievedby a coordinate transformation leading to [125]

gII I=d d+ (cos )2(1)2 + (sin )2(2)2 + (3)2, (129)

where = 2 and 0, 2 . Regularity at = 0, 2 is obtained providedthat is identified modulo 2 (cf. (126)). Remarkably, this is a regularBianchi-IX Einstein metric in which the coefficients of1, 2 and 3 are alldifferent.

(4) The nontrivialS2-bundle overS2 has a metric which, by setting = 3,may be cast in the form[118,125]

gIV= (1 +2)f1(x)dx dx +f2(x)

(1)

2 + (2)2

+f3(x)(3)2, (130)

where x [0, 1], is the positive root of

w4

+ 4w3

6w2

+ 12w 3 = 0, (131)and the functionsf1, f2, f3 are defined by

f1(x) (1 2x2)

(3 2 2(1 +2)x2)(1 x2) , (132)

f2(x) (1 2x2)

(3 + 62 4) , (133)

f3(x) (3 2 2(1 +2)x2)(1 x2)(3

2)(1

2x2) . (134)

The isometry group corresponding to (130) may be shown to be U(2).(5) Another compact instanton of fundamental importance is the K3

surface, whose explicit metric has not yet been found. K3 is defined as thecompact complex surface whose first Betti number and first Chern class are

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vanishing. A physical picture of the K3 gravitational instanton has been

obtained by Page[119].Two topological invariants exist which may be used to characterize the var-ious gravitational instantons studied so far. These invariants are the Eulernumber and the Hirzebruch signature. The Euler number can be definedas an alternating sum of Betti numbers, i.e.

B0 B1+B2 B3+B4. (135)The Hirzebruch signature can be defined as

B+2B2, (136)whereB+2 is the number of self-dual harmonic 2-forms, andB

2 is the number

of anti-self-dual harmonic 2-forms [in terms of the Hodge-star operator Fab12

abcdFcd, self-duality of a 2-form F is expressed as F = F, and anti-self-

duality as F =F]. In the case of compact four-dimensional manifoldswithout boundary, and can be expressed as integrals of the curvature[91]

= 1

1282

M

R R

g d4x, (137)

= 1

962

M

R R

g d4x. (138)For the instantons previously listed one finds [125]

EguchiHanson: = 2, = 1.

Asymptotically locally Euclidean multi-instantons: = n, =n 1.Schwarzschild: = 2, = 0.

Taub-NUT:= 1, = 0.

Asymptotically locally flat multi-Taub-NUT instantons: = n,=n 1.S4: = 2, = 0.

CP2: = 3, = 1.

S2 S2: = 4,= 0.S2-bundle over S2: = 4, = 0.

K3: = 24, = 16.

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6 Spectral zeta-functions in one-loop quan-

tum cosmology

In the late nineties a systematic investigation of boundary conditions inquantum field theory and quantum gravity has been performed (see Refs.[111,150,55,117,8] and the many references therein). It is now clear thatthe set of fully gauge-invariant boundary conditions in quantum field theory,providing a unified scheme for Maxwell, YangMills and General Relativityis as follows:

AM

= 0, (139)

(A)M = 0, (140)[]M= 0, (141)where is a projection operator,A is the Maxwell potential, or the YangMills potential, or the metric (more precisely, their perturbation about abackground value which can be set to zero for Maxwell or YangMills), isthe gauge-fixing functional, denotes the set of ghost fields for these bosonictheories. Equations (139) and (140) are both preserved under infinitesimalgauge transformations provided that the ghost obeys homogeneous Dirichletconditions as in Eq. (141). For gravity, it may be convenient to choose soas to have an operator Pof Laplace type in the Euclidean theory.

6.1 Eigenvalue condition for scalar modes

In a quantum theory of the early universe via functional integrals, the semi-classical analysis remains a valuable tool, but the tree-level approximationmight be an oversimplification. Thus, it seems appropriate to consider atleast the one-loop approximation. On the portion of flat Euclidean 4-spacebounded by a 3-sphere, called Euclidean 4-ball and relevant for one-loopquantum cosmology [78, 92, 55] when a portion of 4-sphere bounded by a3-sphere is studied in the limit of small 3-geometry [140], the metric pertur-bationshcan be expanded in terms of scalar, transverse vector, transverse-

traceless tensor harmonics on the 3-sphere S

3

of radiusa. For vector, tensorand ghost modes, boundary conditions reduce to Dirichlet or Robin. Forscalar modes, one finds eventually the eigenvalues E=X2 from the rootsXof [58,59]

Jn(x)n

xJn(x) = 0, (142)

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Jn(x) + x

2 n

xJn(x) = 0, (143)

where Jn are the Bessel functions of first kind. Note that both x andxsolve the same equation.

6.2 Four spectral zeta-functions for scalar modes

By virtue of the Cauchy theorem and of suitable rotations of integrationcontours in the complex plane[19], the eigenvalue conditions (142) and (143)give rise to the following four spectral zeta-functions [58,59]:

A,B(s)

sin(s)

n=3 n(2s2)

0

dzz

log FA,B(zn)

z2s , (144)

where, denoting by In the modified Bessel functions of first kind (here +n, n + 2),

FA (zn) z

znIn(zn) nIn(zn)

, (145)

FB (zn) z

znIn(zn) +

z2n2

2 n

In(zn)

. (146)

Regularity at the origin is easily proved in the elliptic sectors, correspondingtoA (s) and

B (s) [58,59].

6.3 Regularity at the origin of+B

With the notation in Refs. [58,59], if one defines the variable (1+z2) 12 ,one can write the uniform asymptotic expansion ofF+B in the form [58,59]

F+B en()

h(n)

(1 2)

1 + j=1

rj,+()

nj

. (147)On splitting the integral

1

0 d =

0 d+1d with small, one gets an

asymptotic expansion of the left-hand side of Eq. (144) by writing, in thefirst interval on the right-hand side,

log

1 + j=1

rj,+()

nj

j=1

Rj,+()

nj , (148)

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6.4 Interpretation of the result

Since all other(0) values for pure gravity obtained in the literature are neg-ative, the analysis here briefly outlined shows that only fully diffeomorphism-invariant boundary conditions lead to a positive(0) value for pure gravity onthe 4-ball, and henceonly fully diffeomorphism-invariant boundary conditionslead to a vanishing cosmological wave function for vanishing 3-geometries at

one-loop level, at least on the Euclidean 4-ball. If the probabilistic interpre-tation is tenable for the whole universe, this means that the universe hasvanishing probability of reaching the initial singularity at a = 0, which istherefore avoided by virtue of quantum effects [58, 59], since the one-loopwave function is proportional to a(0) [140].

Interestingly, quantum cosmology can have observational consequences aswell. For example, the work in Ref. [104]has derived the primordial powerspectrum of density fluctuations in the framework of quantum cosmology,by performing a BornOppenheimer approximation of the WheelerDeWittequation for an inflationary universe with a scalar field. In this way onefirst recovers the scale-invariant power spectrum that is found as an ap-proximation in the simplest inflationary models. One then obtains quantumgravitational corrections to this spectrum, discussing whether they lead tomeasurable signatures in the Cosmic Microwave Background anisotropy spec-trum [104].

7 Hawkings radiation

Hawkings theoretical discovery of particle creation by black holes[87,88]hasled, along the years, to many important developments in quantum field theoryin curved spacetime, quantum gravity and string theory. Thus, we devotethis section to a brief review of such an effect, relying upon the DAMTPlecture notes by Townsend [147]. For this purpose, we consider a masslessscalar field in a Schwarzschild black hole spacetime. The positive-frequencyoutgoing modes of are known to behave, near future null infinityF+, as

eiu

. (155)

According to a geometric optics approximation, a particles worldline is anull rayof constant phaseu, and we trace this ray backwards in time fromF+. The later it reachesF+, the closer it must approach the future event

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In this integral, let us choose the branch cut in the complex v-plane to lie

along the real axis. For positive let us rotate contour to the positiveimaginary axis and then set v = ix to get

() = i 0

ex+ i

log

xei/2

dx

= e2

0e

x+ i

log(x)dx. (162)

Since is positive the integral converges. When is negative one can rotatethe contour to the negative imaginary axis and then set v = ix to get

() = i 0 ex+i

logxei/2dx= e

2

0

ex+ i

log(x)dx. (163)

From the two previous formulae one gets

() = e() if> 0. (164)Thus, a mode of positive frequency onF+ matches, at late times, ontopositive and negative modes onF. For positive one can identify

A

=(), (165)B =() = e(), (166)

as the Bogoliubov coefficients. These formulae imply that

Bij = e Aij . (167)

On the other hand, the matrices A and B should satisfy the Bogoliubovrelations, from which

ij = (AA BB)ij

= l

(AilAjl BilBjl)

=e(i+j)

1l

BilBjl , (168)

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where we have inserted the formula relating Bij to Aij . Now one can take

i= j to get(BB)ii=

1

e2i 1

. (169)

Eventually, one needs the inverse Bogoliubov coefficients corresponding toa positive-frequency mode onF matching onto positive- and negative-frequency modes onF+. Since the inverse B coefficient is found to be

B= BT, (170)the late-time particle flux throughF+, given a vacuum onF, turns out tobe

NiF+ =(B)Bii =BBTii =BBTii. (171)From reality of (BBT)ii, the previous formulae lead to

NiF+ = 1e2i 1

. (172)

Remarkably, this is the Planck distribution for black body radiation from aSchwarzschild black hole at the Hawking temperature

TH=h

2. (173)

8 Achievements and open problems

At this stage, the general reader might well be wondering what has beengained by working on the quantum gravity problem over so many decades.Indeed, at the theoretical level, at least the following achievements can bebrought to his (or her) attention:

(i) The ghost fields[63] necessary for the functional-integral quantization ofgravity and YangMills theories[33,61] have been discovered, jointly with adeep perspective on the space of histories formulation.

(ii) The VilkoviskyDeWitt gauge-invariant effective action[151,38] has beenobtained and thoroughly studied.

(iii) We know that black holes emit a thermal spectrum and have temperatureand entropy by virtue of semiclassical quantum effects [87, 88, 40]. A full

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least four items can be brought to the attention of the reader within the

experimental framework.(i) Colella et al. [25] have used a neutron interferometer to observe thequantum-mechanical phase shift of neutrons caused by their interaction withthe Earths gravitational field.

(ii) Page and Geilker [120] have considered an experiment that gave resultsinconsistent with the simplest alternative to quantum gravity, i.e. the semi-classical Einstein equation. This evidence supports, but does not prove, thehypothesis that a consistent theory of gravity coupled to quantized mattershould also have the gravitational field quantized [30].

(iii) Balbinot et al. have shown [9] that, in a black hole-like configurationrealized in a BoseEinstein condensate, a particle creation of the Hawkingtype does indeed take place and can be unambiguously identified via a char-acteristic pattern in the density-density correlations. This has opened theconcrete possibility of the experimental verification of this effect.

(iv) Mercati et al. [115], for the study of the Planck-scale modifications [3]of the energy-momentum dispersion relations, have considered the possiblerole of experiments involving nonrelativistic particles and particularly atoms.They have extended a recent result, establishing that measurements of atom-recoil frequency can provide insight that is valuable for some theoretical

models.We are already facing unprecedented challenges, where the achievements

of spacetime physics and quantum field theory are called into question. Theyears to come will hopefully tell us whether the many new mathematical con-cepts considered in theoretical physics lead really to a better understandingof the physical universe and its underlying structures.

Acknowledgments The author is grateful to the Dipartimento di Sci-enze Fisiche of Federico II University, Naples, for hospitality and support.He dedicates to Maria Gabriella his work.

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