Quantum Fields in Anti-de Sitter Spae
and the Maldaena Conjeture
Nelson R. F. Braga
InstitutodeFsia
UniversidadeFederaldoRiodeJaneiro,RJ,21945-970{Brazil
Reeivedon7February,2002
WereviewtherelationbetweentheMaldaenaonjeture,alsoknownasAdS=CFTorrespondene,
andtheso-alledholographipriniplethatseemstobeanessentialingredientforaquantumgravity
theory. Wealsoillustratetheideaofholographybyshowingthattheurvatureoftheanti-deSitter
spaeredues thenumberof degrees offreedom makingit possible to nda mapping between a
quantumtheorydenedonthebulkandanotherdenedontheorrespondingboundary.
I Introdution
Theinterestoftheoretialphysiistsinstudying
quan-tum elds in antide Sitter(AdS) spae isnot new[1℄.
Inpartiular thequestionof thequantizationof elds
in this spaeirunventing the problem of thelak of
a Cauhy surfae was addressed in[2, 3℄. There was,
however,aremarkableinreaseintheattentiondevoted
to thissubjetsinetheappearaneof theMaldaena
onjeture[4℄ (see[5,6℄forreviews)ontheequivalene
(orduality)ofthelargeN limitofSU(N)
superonfor-maleldtheoriesinndimensionsandsupergravityand
stringtheoryinantideSitterspaetimeinn+1
dimen-sions also alled AdS/CFTorrespondene. This
or-respondene was elaboratedby Gubser,Klebanovand
Polyakov[7℄andbyWitten[8℄interpretingthe
bound-aryvaluesofbulkeldsas souresofboundarytheory
orrelators.
The large sienti impat aused by Maldaena
work anbeassoiatedwith(atleast)tworeasons:
(i) It represents an expliit example of a
holo-graphi mappingbetween twoquantum theories
that livein dierent dimensions. That means: a
realization oftheHolographi Priniple.
(ii) It also represents an important step in the
searhforastringtheorydesriptionofQCDthat
wouldleadtoaformulationforQCDatlow
ener-gies (where perturbativealulationsannot be
usedbeauseofthestrongoupling)
Asa remark letus mention that the largeN limit
ofSU(N)gaugetheorieswasstudiedby'tHooftalong
time ago. He onsidered anexpansionin the
parame-ter1=NwereQCDgetsasimplerform(planardigrams
II Holographi priniple
The original motivation for the holographi priniple
wasthestudy ofblakholeentropy. Let usstartwith
thefollowingquestion: Whathappenswiththeentropy
of the Universe when some portion of matter (say a
roket or anything else) is absorbed by a blak hole?
Classialy(wemeanwithoutquantummehanis)blak
holesanonlyabsorbpartiles. So,oneouldnot
asso-iateanytemperaturewiththem. Thusitwouldmake
no sense to think that the blak hole itself has some
entropyand theentropyoftheUniverse would notbe
onservedwhenwethrowsomematterinsideit.
Quantum eets howeverhange thispiture. The
thermal radiation found outby Hawkinglead himself
and Bekenstein[10, 11℄ to develop a thermodynamial
modelforblakholes. Basedontheanalysisofthe
dy-namisofblakholestheyfoundageneralisedformof
the seondlawof thermodynamiswhere ablak hole
hasanentropyproportionaltoareaofitshorizon
S
BH =
A
4~G
(1)
The generalized form of the seond law of
thermody-namisreads:
S
BH
+S
RestofUniv: 0
holes. This is still an open problem beause it would
depend essentiallyon theformulationofasatisfatory
quantum theory for gravity. There are howeversome
important results in this diretion for example
alu-lating blak hole entropy using string theory as refs.
[12, 13℄.
Aninterestingonsequeneofeq. (1)isthatit
on-trasts with our standard idea(without taking gravity
into aount) that the entropy is an extensive
quan-tity. That means: entropy should be proportional to
the volume notto the area. This resultlead 't Hooft
andSusskindto formulatetheHolographiPriniple:
\Physisofaquantumsystemwithgravityinsome
volume V an be desribed in terms of degrees of
freedom thatan beontainedinitsboundary".
The general idea is that somesystems tend naturally
to blakholesand fortheothersweanndproesses
thattransformthemintoblakholesandinreasetheir
entropy. So: the maximum entropy is limited by the
boundaryarea(in Plankunits). That means:
Quan-tum mehanis plusgravityin 3 spatial dimensionsis
equivalentto an imagethat anbe mappedin a
bidi-mensional projetion. The area of the boundary in
unitsofPlankarearepresentsthemaximumnumberof
degreesoffreedomintheinteriorvolume. Itispresently
believedthata(andidateto)quantumgravitytheory
should satisfythispriniple.
III Implementation of
Malda-ena onjeture
Soonafter Maldaena's seminal artile, Gubser ,
Kle-banovand Polyakov[7℄ and Witten[8 ℄ haveelaborated
thegeneralonjeture showinghowto alulate
Phys-ial quantities of a onformal theory in the
bound-ary of an anti-de Sitter (AdS) spae in terms of a
bulk theory. In order to see how this orrespondene
holds, let us rst remind that an anti-de Sitter spae
of n +1 dimensions (AdS
n+1
) is a spae of
on-stant negative urvature that an be taken as a
Hy-perboloidinalargern+2dimensional atspae with
oordinates (X 0
;X 1
;:::;X n
;X n+1
) and metri
ab =
diag(+; ; ;:::; ;+):
(X
0 )
2 +(X
n+1 )
2 P
n
i=1 (X
i )
2
= 2
=onstant;
We an introdue oordinate systems inside AdS like
the global oordinates ;;
i
, more used before the
disoveryoftheAdS=CFT orrespondene,denedby:
X
0
= seos
X
i
= tan
i
X
n+1
= sesin
where P
n
i=1
2
i
=1;0<=2; 0 <2
Notethat thetime variable is ompat, sowe must
\unwrap" itby atuallyonsidering theAdS overing
spae(thatmeansaninnitesetofopiesofAdSspaes
inthe diretion.
For the AdS/CFT orrespondene the so alled
Poinare oordinates (z;x i
; t) ( with z 0) are
moreusefull. Theyaredened bytherelations
X
0 =
1
2z
z 2
+ 2
+~x 2
t 2
X
i =
x i
z
X
n =
1
2z
z 2
2
+ ~x 2
t 2
X
n+1 =
t
z
andtheorrespondingmeasure reads
ds 2
=
2
z 2
(dz) 2
+(d~x ) 2
(dt) 2
TheAdSboundary,wheretheConformalFieldTheory
isdenedorrespondstotheregionz = 0 plusapoint
atinnity: z=1.
Witten[8 ℄ has interpreted the AdS=CFT
orrespon-dene in terms of a holography mapping: boundary
values of elds that have dynamis dened inside the
AdS spae at as soures of orrelation funtions of
the boundary onformal eld theory (CFT). The
sim-plest illustrative example is that of a salar massless
eld (massless) in AdS (more details an be found in
[7, 8, 16, 17℄. Taking asalar eld in the bulk, with
boundaryvalue
(z;x i
;t) !
0 (x
i
;t) as z ! 0.
andonformaloperatorsO(x i
;t)ontheboundary,the
generatoroforrelation funtions for this CFT
opera-tors(now: x i
;txforsimpliity)
Z[
0
℄ = hexp Z
d n
x
0
(x)O(x)i
suhthat
Æ
Æ (x ) Æ
Æ (x ) Z[
0
℄ = hO(x
1 )O(x
where the elds
0
at as soures will be alulated
from the on shell ation I() for thebulk salar eld
Z[
0
℄ () expf I()g
Consideringthe ation for a salareld in urved
spaetime
L = 1
2 Z
d n+1
x p
g
;
withequationofmotion
r
r
= 1
p
g
p
g
= 0:
weanrelatetheeldinAdSbulkintermsofboundary
values:
(z;x) = Z
d n
x 0
(z) n
((z) 2
+(x x 0
) 2
) n
0 (x
0
):
andwritetheonshellationas
I[℄ = n
2 Z
d n
xd n
x 0
0 (x
0
)
0 (x)
(x x 0
) 2n
Sowend,forexample,thetwopointorrelation
fun-tionfortheoperatorsO(x)
hO(x)O(y)i 1
(x y) 2n
asexpetedfromonformalinvariane.
LetusremarkthattheoperatorsOoftheCFTarenot
in generalsalarelds but ratheromposite operators
whose onformaldimension depends onthedimension
ofthespae.
IV Counting degrees of freedom
in AdS spae
Considering the geometry of AdS spae in Poinare
oordinatesweseethatahypersurfaeofarea A
or-responding to x 1
:::x n 1
at somesmall z =Æ will
generateanite volumeV ifwemoveitalong z till
z!1
A =
Z
d n 1
x
(z=) n 1
=
Æ
n 1
x 1
:::x n 1
V =
Z
d n 1
xdz
(z=) n
= A
n 1 :
A V
-Æ
z
Figure1.
If we assoiate (in a regularised way) degreesof
free-domwithvolumeellsofthespaeweseethat wean
map theV ellsin areaellsof A. This redution
ofdegreesoffreedomassoiatedwiththefatthat the
volumeis proportionalto theareashould be reeted
in aquantum theoryin AdS . Howanitbeso? The
hint omes from the analisys of the Cauhy problem
in the AdS spae onsidered in [2, 3℄: massless
parti-les anenter or left the AdS from spatial innity in
nite times. So aonsistentquantization(that means
awelldened Cauhy problem)requires a
ompati-ationofthespae. Onehastoaddasurfaeatinnity
where boundaryonditionshaveto beimposed. (Like
the overof a box: nothingenters orlefts thespae).
These referenes onsidered global oordinates where
theboundaryisthehypersurfae ==2.
Now onsidering the Cauhy problem in Poinare
Coordinates,partileswithoutmassmayomefromor
goto z ! 1 in anitetime. Buthowanwe
om-patify the z oordinate that has innite range? The
answer [18, 19, 20℄ is that oneneeds to introdue
an-otheroordinatehart(z 0
;x i
;t) relatedtotheoriginal
oneby
1
z 0
= 1
Æ 1
z
(2)
werewetaketherangeoftheoordinatesz asÆz
R in order to avoid the singularity at the originand
to stoptherst hartat somearbitrarilylargeR . For
z 0
wetake and Æ z 0
R 0
where R 0
= ÆR =(R Æ)
in order to map the rest of the spae. The two sets
mustbeompatbeauseweneedtoimposeboundary
onditionsthatmaththem togheter.
Now the \point at innity" in the z oordinate is
mappedatz 0
=Æ
Theimportantonsequeneofthisrepresentationofthe
ompatAdSspaeintermsoftwoompatoordinate
(z;~x;t) == 1
X
p=1 Z
d n 1
k
(2) n 1
z n=2
J
(u
p z)
R w
p (
~
k)J
+1 (u
p R )
fa
p (
~
k)e iwp(
~
k )t+i ~
k~x
+ ::g (3)
d
Theimportantonsequeneofthisdisretizationofthe
eld spetrum is that it makes possible to nd a one
to onemapping betweenbulk and boundary theories.
This wouldnotbepossibleifthespetrumwere
onti-nous. Weanunderstand this by ananalogywith the
fat that itispossibleto ndaonetoonemappingof
anenumerableset oflinesinto onesingleline butitis
not possible to map (one to one) a plane and a line.
This is explainedin [21℄ werean expliitmapping
be-tweensalareldsinAdSspaeandsalareldsonthe
boundaryispresented.
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