Quantum Cosmology: How to Interpret
and Obtain Results
Nelson Pinto-Neto
CentroBrasileirode PesquisasFsias,
Rua Dr.XavierSigaud150,Ura
22290-180,Riode Janeiro, RJ,Brazil
E-mails: nelsonpnlafex.bpf.br
Reeived7January,2000
WearguethattheCopenhageninterpretationofquantummehanisannotbeappliedtoquantum
osmology. Amongthealternativeinterpretations, wehoose toapplytheBohm-deBroglie
inter-pretation ofquantummehanistoanonialquantumosmology. Forminisuperspaemodels,we
showthatthereisnoproblemoftimeinthisinterpretation,andthatquantumeetsanavoidthe
initialsingularity,reateinationandisotropizetheUniverse. Forthegeneralase,itisshownthat,
irrespetiveofanyregularizationorhoieoffatororderingoftheWheeler-DeWittequation,the
uniquerelevantquantumeetwhihdoesnotbreakspaetimeisthehangeofitssignaturefrom
lorentzian to eulidean. The otherquantumeets are eithertrivial or break the four-geometry
ofspaetime. A Bohm-deBroglie piture ofa quantumgeometrodynamis isonstruted, whih
allows theinvestigationoftheselatterstrutures.
I Introdution
Almostallphysiistsbelievethatquantummehanisis
auniversal andfundamental theory, appliableto any
physialsystem,fromwhihlassialphysisanbe
re-overed. The Universe is, of ourse, a valid physial
system: thereis atheory,Standard Cosmology,whih
isable todesribeitin physial terms,and make
pre-ditions whih an be onrmed or refuted by
obser-vations. In fat, the observations until now onrm
the standard osmologial senario. Hene,
suppos-ing the universality of quantum mehanis, the
Uni-verseitselfmustbedesribedbyquantumtheory,from
whihweouldreoverStandardCosmology. However,
the Copenhagen interpretation of quantum mehanis
[1, 2, 3℄ 1
, whih is the one taught in undergraduate
oursesand employedbythe majority ofphysiists in
allareas(speiallythevonNeumann'sapproah),
an-notbeusedin aQuantumTheoryofCosmology. This
isbeauseitimposestheexisteneofalassialdomain.
InvonNeumann'sview,forinstane,theneessityofa
lassialdomainomesfromthewayitsolvesthe
mea-surementproblem(see Ref. [4℄foragooddisussion).
In an impulsive measurementof someobservable,the
wavefuntionoftheobservedsystemplusmarosopi
apparatus splits into many branhes whih almost do
notoverlap(in orderto beagoodmeasurement),eah
one ontainingtheobservedsystemin aneigenstateof
themeasuredobservable,and thepointer ofthe
appa-ratus pointing to the respetive eigenvalue. However,
in theend ofthemeasurement,weobserveonlyoneof
theseeigenvalues,andthemeasurementisrobustinthe
sense thatifwerepeat itimmediatelyafter,weobtain
the same result. So it seems that the wave funtion
ollapses, the other branhes disappear. The
Copen-hagen interpretationassumesthat thisollapseisreal.
However, a real ollapse annot be desribed by the
unitaryShrodingerevolution. Hene,theCopenhagen
interpretationmustassumethatthereisafundamental
proessinameasurementwhihmustouroutsidethe
quantumworld,inalassialdomain. Of ourse,ifwe
wanttoquantizethewhole Universe,there isnoplae
for alassialdomain outsideit, and theCopenhagen
interpretation annot be applied. Hene, if someone
insists with the Copenhagen interpretation, she orhe
must assumethat quantum theory isnotuniversal, or
atleasttrytoimproveitbymeansoffurtheronepts.
One possibility is by invoking the phenomenon of
de-oherene [5℄. Infat, the interation of the observed
quantum system with its environment yields an
ee-tivediagonalizationoftheredueddensitymatrix,
ob-tainedbytraingouttheirrelevantdegreesoffreedom.
Deoherene anexplainwhythesplittingofthewave
1
Althoughthesethreeauthorshavedierentviewsfromquantumtheory,the rstemphasizingtheindivisibilityofquantum
funtion is given in terms of the pointer basis states,
and why wedo not see superpositions of marosopi
objets. In this way, lassial properties emerge from
quantumtheorywithouttheneedofbeingassumed. In
the framework ofquantum gravity, itanalso explain
howa lassialbakground geometry an emerge in a
quantum universe [6℄. In fat, it is the rst quantity
to beomelassial. However,deohereneisnotyeta
ompleteanswertothemeasurementproblem[7,8℄. It
doesnot explain theapparentollapse after the
mea-surementisompleted,orwhyallbut oneofthe
diag-onalelementsofthedensitymatrix beome nullwhen
the measurementis nished. The theory is unable to
giveanaountof theexisteneoffats,their
unique-nessasopposedtothemultipliityofpossible
phenom-ena. Further developments are still in progress, like
the onsistenthistoriesapproah[9℄,whih ishowever
inomplete until now. The important role played by
theobserversin thesedesriptionsisnotyet explained
[10℄, and stillremains theproblemon howto desribe
aquantum universewhen thebakgroundgeometryis
notyetlassial.
Nevertheless, there are some alternative solutions
tothisquantumosmologialdilemmawhih, together
with deoherene,an solvethemeasurementproblem
maintaining the universality of quantum theory. One
an say that the Shrodinger evolutionis an
approxi-mationofamorefundamentalnon-lineartheorywhih
an aomplish the ollapse [11, 12℄, or that the
ol-lapse is eetive but not real, in the sense that the
otherbranhesdisappearfromtheobserverbutdonot
disappear from existene. In this seond ategory we
an ite the Many-WorldsInterpretation [13℄ and the
Bohm-deBroglieInterpretation[14,15℄. Intheformer,
allthepossibilitiesinthesplittingareatuallyrealized.
In eah branh there is an observer with the
knowl-edge of the orresponding eigenvalue of this branh,
but she orhe is notaware of theother observersand
the otherpossibilitiesbeausethebranhesdonot
in-terfere. In thelatter, apoint-partile in onguration
spaedesribingtheobservedsystemand apparatusis
supposed to exist,independently onanyobservations.
Inthesplitting,thispointpartilewillenterintooneof
thebranhes(whihonedependsontheinitialposition
of thepointpartile before themeasurement,whihis
unknown), and the other branhes will be empty. It
anbeshown[15℄thattheemptywavesanneither
in-terat withother partiles,norwiththepointpartile
ontainingthe apparatus. Hene, no observer anbe
aware of the other branhes whih are empty. Again
we havean eetive but not real ollapse (the empty
wavesontinueto exist), but now with no
multiplia-tionofobservers. Ofoursetheseinterpretationsanbe
used in quantum osmology. Shrodinger evolutionis
alwaysvalid,andthereisnoneedofalassialdomain
outsidetheobservedsystem.
tionoftheBohm-deBroglieinterpretationtoquantum
osmology [16, 17, 18, 19, 20℄. In this approah, the
fundamental objetof quantum gravity, the geometry
of 3-dimensional spaelike hypersurfaes, is supposed
toexist independentlyonanyobservationor
measure-ment, as well as its anonial momentum, the
extrin-siurvatureofthespaelikehypersurfaes. Its
evolu-tion,labeledbysometimeparameter,isditatedbya
quantum evolution that is dierent from the lassial
onedue to thepreseneof aquantum potentialwhih
appears naturallyfrom theWheeler-DeWitt equation.
This interpretation hasbeenapplied to many
minisu-perspae models [16, 19, 21, 22, 23, 24℄, obtained by
theimposition of homogeneity of the spaelike
hyper-surfaes. The lassial limit, the singularityproblem,
theosmologialonstantproblem, andthe timeissue
havebeendisussed. Forinstane,insomeofthese
pa-persitwasshownthatinmodelsinvolvingsalarelds
orradiation,whiharenierepresentativesofthe
mat-terontentoftheearlyuniverse,thesingularityanbe
learly avoided by quantum eets. In the Bohm-de
Broglieinterpretationdesription,thequantum
poten-tial beomes important near the singularity, yielding
a repulsive quantum fore ounterating the
gravita-tionaleld,avoidingthesingularityandyielding
ina-tion. Thelassiallimit(given by thelimitwhere the
quantum potential beomes negligiblewith respet to
thelassialenergy)for largesale fatorsare usually
attainable,butforsomesalareldmodelsitdepends
onthequantum stateandinitial onditions. Infat it
is possible to have small lassial universes and large
quantumones[24℄. Aboutthetimeissue,itwasshown
that foranyhoieof the lapsefuntion the quantum
evolution of the homogeneous hypersurfaes yield the
samefour-geometry[19℄. Whatremainedtobestudied
isifthisfatremainsvalidinthefulltheory,wherewe
arenot restritedto homogeneous spaelike
hypersur-faes. The question is, given an initial hypersurfae
with onsistent initial onditions, does the evolution
of the initial three-geometry driven by the quantum
bohmian dynamis yields the same four-geometry for
anyhoieofthelapseandshiftfuntions,andifitdoes,
whatkindofspaetime strutureisformed? Weknow
thatthisistrueifthethree-geometryisevolvedbythe
dynamisoflassialGeneralRelativity(GR),yielding
anon degenerate four geometry, but it anbefalse if
the evolving dynamis is the quantum bohmian one.
Theidea was to put the quantum bohmian dynamis
in hamiltonian form, and then usestrong results
pre-sentedintheliteratureexhibitingthemostgeneralform
thatahamiltonianshouldhaveinorderto formanon
degeneratefour-geometryfrom theevolution of
three-geometries [25℄. Our onlusion is that, in general,
thequantumbohmianevolutionofthethree-geometries
does not yield any non degenerate four-geometry at
all [20℄. Only for very speial quantum states a
obtained,anditmustbeeulidean. Inthegeneralase,
thequantumbohmianevolutionisonsistent(still
inde-pendentonthehoieof thelapseandshiftfuntions)
but yieldingadegeneratefour-geometry,where speial
vetorelds,thenulleigenvetorsofthefourgeometry,
are present 2
. Wearrivedat these onlusions without
assuminganyregularizationandfatororderingofthe
Wheeler-DeWittequation. Asweknow, the
Wheeler-DeWitt equation involvesthe appliation ofthe
prod-utofloaloperatorsonstatesatthesamespaepoint,
whihisilldened[27℄. Heneweneedtoregularizeit
inordertosolvethefatororderingproblem,andhave
atheoryfreeofanomalies(forsomeproposals,seeRefs
[28, 29,30℄). Ouronlusions are ompletely
indepen-dentonthese issues.
This paperisorganizedasfollows: inthenext
se-tion we review the Bohm-de Broglie interpretation of
quantum mehanis for non-relativisti partiles and
quantumeldtheoryinatspaetime. InsetionIIIwe
applytheBohm-deBroglieinterpretationto anonial
quantumgravityintheminisuperspaease. Weshow
thatthere isnoproblemoftimein thisinterpretation,
and that quantum eets an avoid the initial
singu-larity,reate ination, andisotropizetheUniverse. In
setion IV we treat the general ase. We prove that
the bohmian evolution of the 3-geometries,
irrespe-tive of any regularization and fator ordering of the
Wheeler-DeWittequation,anbeobtainedfroma
spe-ihamiltonian, whih isof oursedierent fromthe
lassialone. Wethenusethishamiltoniantoobtaina
pitureofthesenewquantumstrutures. Weendwith
onlusionsandmanyperspetivesforfuturework.
II The Bohm-de Broglie
Inter-pretation
Inthis setionwewill review theBohm-deBroglie
in-terpretationofquantummehanis. Wewillrstshow
how this interpretation works in the ase of a single
partiledesribedbyaShrodingerequation,andthen
wewillobtain,byanalogy,theausalinterpretationof
quantumeldtheoryin atspaetime.
Let us begin with the Bohm-de Broglie
interpre-tation of the Shrodinger equation desribing a single
partile. In the oordinate representation, for a
non-relativisti partile with Hamiltonian H = p 2
=2m+
V(x);theShrodingerequationis
ih
(x;t)
t =
h 2
2m r
2
+V(x)
(x;t): (1)
Weantransformthisdierentialequationovera
om-plex eld into apair of oupled dierential equations
overreal elds, bywriting =Aexp(iS=h), where A
and S are real funtions,and substituting it into (1).
Weobtainthefollowingequations.
S
t +
(rS) 2
2m +V
h 2
2m r
2
A
A
=0; (2)
A 2
t +r
A 2
rS
m
=0: (3)
Theusual probabilistiinterpretation, i.e. the
Copen-hagen interpretation, understands equation (3) as a
ontinuity equation for the probability density A 2
for
ndingthepartileatpositionxand timet. All
phys-ial information about thesystem is ontained in A 2
,
andthetotalphaseSofthewavefuntionisompletely
irrelevant. Inthisinterpretation,nothingissaidabout
Sanditsevolutionequation(2). Nowsupposethatthe
term h 2
2m r
2
A
A
is notpresentin Eqs. (2) and(3). Then
weould interpret them asequations for an ensemble
of lassial partiles under the inuene of a lassial
potentialV throughtheHamilton-Jaobiequation(2),
whoseprobabilitydensitydistributioninspaeA 2
(x;t)
satisestheontinuityequation(3),whererS(x;t)=m
istheveloityeldv(x;t)oftheensembleofpartiles.
Whenthe term h 2
2m r
2
A
A
ispresent,wean still
under-stand Eq. (2) as a Hamilton-Jaobi equation for an
ensemble of partiles. However, their trajetoriesare
no morethe lassialones,due to the presene of the
quantumpotentialtermin Eq. (2).
The Bohm-de Broglie interpretation of quantum
mehanis is based on the two equations (2) and (3)
in thewayoutlinedabove,notonlyonthe lastone as
itis theCopenhageninterpretation. Thestartingidea
isthatthepositionxandmomentumparealwayswell
dened,withthepartile'spathbeingguidedbyanew
eld,thequantumeld. Theeld obeysShrodinger
equation(1),whihanbewrittenasthetworeal
equa-tions (2) and (3). Equation (2) is interpreted as a
Hamilton-Jaobi type equation for the quantum
par-tile subjeted to an external potential, whih is the
lassialpotentialplusthenewquantum potential
Q h 2
2m r
2
A
A
: (4)
Onetheeld ,whoseeetonthepartiletrajetory
isthroughthequantumpotential(4),isobtainedfrom
Shrodinger equation, wean also obtain the partile
trajetory,x(t);byintegratingthedierentialequation
p = mx_ = rS(x;t), whih is alled the guidane
re-lation (a dot meanstime derivative). Of ourse, from
this dierential equation, the non-lassial trajetory
x(t) an only be known if the initial position of the
partile isgiven. However,wedonotknowtheinitial
2
For instane, the fourgeometry of Newtonian spaetime isdegenerate [26℄, and itssingle null eigenvetor isthe normalof the
position of the partile beause we do not know how
to measure it without disturbanes (it is the hidden
variable of the theory). To agree with quantum
me-hanialexperiments,wehaveto postulatethat, fora
statistial ensemble of partiles in the samequantum
eld , the probability density distribution of initial
positions x
0
is P(x
0 ;t
0 ) = A
2
(x
0 ;t = t
0
). Equation
(3) guaranteesthatP(x;t)=A 2
(x;t) foralltimes. In
this way,the statistialpreditionsofquantum theory
in theBohm-deBroglieinterpretationarethesameas
in theCopenhageninterpretation 3
.
It is interesting to note that the quantum
poten-tial depends only on the form of , not on its
abso-lutevalue,asanbeseenfrom equation(4). Thisfat
brings homethenon-loal andontextual haraterof
thequantumpotential 4
. Thisisaneessaryfeature
be-ause Bell'sinequalities togetherwith Aspet's
exper-imentsshowthat, in general, a quantum theorymust
be either non-loal or non-ontologial. As the
Bohm-deBroglieinterpretationisontologial,thanitmustbe
non-loal,asitis. Thenon-loaland ontextual
quan-tum potential auses the quantum eets. It has no
parallel inlassialphysis.
The funtion A plays a dual role in the Bohm-de
Broglie interpretation: it gives thequantum potential
and the probability density distribution of positions,
but this last role isseondary. Ifin somemodelthere
isnonotionofprobability,weanstillgetinformation
from thesystem using the guidane relations. Inthis
ase,A 2
doesnotneedtobenormalizable. The
Bohm-de Broglie interpretation is not, in essene, a
proba-bilistiinterpretation. Itis straightforwardtoapply it
toasinglesystem.
Thelassiallimitanbeobtainedinaverysimple
way. We only have to nd the onditions for having
Q= 0 5
. The question onwhy in a real measurement
weseeaneetive ollapseofthewavefuntion is
an-swered by noting that, in a measurement, the wave
funtion splits in a superposition of non-overlapping
branhes. Hene the point partile (representing the
partilebeing measuredplusthe marosopi
appara-tus) will enter into one partiular branh, whih one
depends on theinitial onditions, and it will be
inu-ened by the quantum potential related only to this
branh, whihis the onlyonethat is notnegligiblein
the region where the point partile atually is. The
other empty branhes ontinue to exist, but they
nei-ther inuene on the point partile nor on any other
partile[15℄. Thereisaneetivebutnotrealollapse.
TheShrodingerequation isalwaysvalid. There isno
need to havea lassial domain outside the quantum
systemto explaina measurement, neither is the
exis-teneofobserversruialbeausethisinterpretationis
objetive.
Forquantumeldsinatspaetime,weanapplya
similarreasoning. Asanexample,taketheShrodinger
funtionalequationforaquantumsalareld:
ih
(;t)
t =
Z
d 3
x
1
2
h 2
Æ 2
Æ 2
+(r) 2
+U()
(;t): (5)
Writingagainthewavefuntional as =Aexp(iS=h),weobtain:
S
t +
Z
d 3
x
1
2
ÆS
Æ
2
+(r) 2
+U()+Q()
=0; (6)
A 2
t +
Z
d 3
x Æ
Æ
A 2
ÆS
Æ
=0; (7)
d
where Q() = h 2
1
2A Æ
2
A
Æ 2
is the orresponding
(un-regulated) quantum potential. The rst equation is
viewed as a modied Hamilton-Jaobi equation
gov-erning theevolutionof someinitial eld onguration
throughtime,whihwillbedierentfrom thelassial
oneduetothepreseneofthequantumpotential. The
guidanerelationisnowgivenby
=
_
= ÆS
Æ
: (8)
3
It has been shownthat undertypial haotisituations, and onlywithinthe Bohm-deBroglie interpretation,a probability
dis-tribution P 6=A 2
wouldrapidlyapproahthe valueP = A 2
[32,33 ℄. Inthisase, the probabilitypostulatewouldbeunneessary,
and weouldhavesituations, inveryshorttimeintervals,wherethismodiedBohm-deBroglieinterpretationwoulddierfromthe
Copenhageninterpretation.
4
Thenon-loalityofQbeomesevidentwhenwegeneralizetheausalinterpretationtoamanypartilessystem.
5
Itshouldbeveryinterestingtoinvestigatetheonnetionbetweenthisbohmianlassiallimitandthephenomenonofdeoherene.
Theseondequationis theontinuityequationforthe
probabilitydensityA 2
[(x);t
0
℄ofhavingtheinitialeld
ongurationattimet
0
givenby(x).
AdetailedanalysisoftheBohm-deBroglie
interpre-tationof quantum eldtheory isgivenin Ref. [34℄for
theaseofquantum eletrodynamis.
III The Bohm-de Broglie
In-terpretation of
Minisuper-spae Canonial Quantum
Cosmology
In this setion, we summarize therules of the
Bohm-deBroglieinterpretationofquantum osmologyinthe
aseofhomogeneousminisuperspaemodels. Whenwe
are restrited to homogeneous models, the
supermo-mentum onstraintof GR is identially zero, and the
shiftfuntion anbesettozerowithoutloosinganyof
the Einstein's equations. The hamiltonian is redued
togeneralminisuperspaeform:
H
GR
=N(t)H (p (t);q (t)); (9) where p
(t) and q
(t) represent the homogeneous
de-grees of freedom oming from ij
(x;t) and h
ij (x;t).
TheminisuperspaeWheeler-DeWittequationis:
H (^p
(t);q^
(t)) (q)=0: (10)
Writing =Rexp(iS=h),andsubstitutingitinto(10),
weobtainthefollowingequation:
1 2 f (q ) S q S q +U(q
)+Q(q
)=0; (11)
where Q(q )= 1 R f 2 R q q ; (12) and f (q
)and U(q
)are theminisuperspae
parti-ularizationsoftheDeWitt metriG
ijk l
[36℄andofthe
salarurvaturedensity h 1=2
R (3)
(h
ij
)ofthespaelike
hypersurfaes, respetively. The ausal interpretation
appliedtoquantumosmologystatesthatthe
trajeto-ries q
(t) are real, independently of any observations.
Eq. (11) is the Hamilton-Jaobi equation for them,
whihisthelassialoneamendedwithaquantum
po-tential term(12), responsible forthe quantum eets.
Thissuggeststodene:
p = S q ; (13)
wherethemomenta arerelatedtotheveloities inthe
usualway: p =f 1 q : (14)
Toobtainthequantumtrajetorieswehavetosolvethe
followingsystemofrstorderdierentialequations:
S(q ) q =f 1 N q t : (15)
Eqs. (15)are invariant under time
reparametriza-tion. Hene, even at the quantum level, dierent
hoiesofN(t)yieldthesamespaetimegeometryfora
givennon-lassialsolutionq
(t). Thereisnoproblem
of time in the ausal interpretation of minisuperspae
quantumosmology.
Letusnowapplythese rules,asexamples,to
min-isuperspae models with a free massless salar eld.
Takethelagrangian:
L= p ge R w ; ; : (16)
Forw= 1wehaveeetivestringtheorywithoutthe
Kalb-Rammondeld. Forw= 3=2wehavea
onfor-mally oupled salar eld. Performing the onformal
transformationg =e g
weobtainthefollowing
la-grangian:
L= p
g
R (!+ 3 2 ) ; ; ; (17)
where the bars have been omitted. We will dene
C w (!+ 3 2 ).
III.1. The isotropi ase
Weonsider nowtheRobertson-Walkermetri
ds 2 = N 2 dt 2 + a(t) 2 1+ 4 r 2 [dr 2 +r 2 (d 2 +sin 2 ()d' 2 )℄; (18)
where thespatialurvaturetakesthevalues0,1, 1.
Insertingthisinthelagrangian(17),andusingtheunits
whereh==1,weobtainthefollowingation:
S= 3V 4l 2 p Z Na 3 2 _ a 2 N 2 a 2 +C w _ 2 6N 2 + a 2
dt ; (19)
whereV isthetotalvolumedividedbya 3
ofthe
spae-likehypersurfaes,whiharesupposedtobelosed,and
l
p
isthePlanklength. V dependsonthevalueofand
onthetopologyofthe hypersurfaes. For=0itan
beaslargeaswewantbeausetheirfundamental
poly-hedra an have arbitrary size. In the ase of = 1
and topology S 3
, V = 2 2 . Dening 2 = 4l 2 p 3V , the
hamiltonianturnsouttobe:
p
a =
aa_
2
N
; (21)
p
= C
w a
3
_
6 2
N
: (22)
Usually the sale fator has dimensions of length
be-ause we use angular oordinates in losed spaes.
Hene we will dene a dimensionless sale fator ~a
a=. In that ase the hamiltonian beomes, omitting
thetilde:
H = N
2
p 2
a
a +6
p 2
C
w a
3 a
: (23)
As appears as an overall multipliative onstant in
thehamiltonian,weansetitequaltoonewithoutany
loss of generality, keeping in mind that the sale
fa-tor whih appears in the metriis a, not a. Wean
further simplify thehamiltonianbydening ln(a)
obtaining
H = N
2exp(3)
p 2
+
6
C
w p
2
exp(4)
: (24)
where
p
=
e 3
_
N
; (25)
p
= C
w e
3
_
6N
: (26)
The momentum p
is a onstant of motion whih
wewill all
k. We will restritourselvesto the
physi-allyinterestingase,due toobservations,of=0and
C
w >0.
Thelassialsolutionsin thegaugeN =1are,
=
r
6
C
w +
1
; (27)
where
1
is anintegrationonstant. Intermofosmi
timetheyare:
a = e
=3 r
6
C
w
kt 1=3
; (28)
=
r
2
3C
w
ln (t)+
2
: (29)
Thesolutionsontratorexpandforeverfroma
singu-larity, depending on the sign of
k, without any
ina-tionaryepoh.
Let us now quantize the model. With a
partiu-lar hoie of fator ordering, we obtain the following
Wheeler-DeWittequation
6
C
w
e 4
=0 : (30)
Employingtheseparationof variablesmethod, we
ob-tainthegeneralsolution
(;)= Z
F(k)A
k ()B
k
()dk ; (31)
wherekisaseparationonstant,and
B
k ()=b
1 exp(i
r
C
w
6
k)+b
2 exp( i
r
C
w
6 k) ;
(32)
and
A
k
()=a
1
exp(ik)+a
2
exp( ik) ; (33)
Wewill nowmakegaussian superpositionsof these
solutionsand interprettheresultsusing theausal
in-terpretationofquantummehanis. ThefuntionF(k)
is:
F(k)=exp
(k d) 2
2
: (34)
Wetakethewavefuntion:
(;)= Z
F(k)[A
k ()B
k
()+A
k ()B
k
()dk℄ ;
(35)
witha
2 =b
2 =0.
Performingtheintegrationinkweobtainfor (we
willdene
q
Cw
6
andomitthebarsfromnowon)
= p
exp
(+) 2
2
4
exp[id(+)℄+exp
( )
2
2
4
exp[ id( )℄
: (36)
InordertoobtaintheBohmiantrajetories,wehaveto
alulate the phaseS of the abovewave funtion and
substituteitintotheguidaneformula
p
= S
; (37)
p
= S
; (38)
where
p
=
e 3
_
N
; (39)
p
=
e 3
_
N
: (40)
We will work in the gauge N = 1. These equations
onstituteaplanarsystemwhihanbeeasilystudied:
_
=
2
sin(2d)+2dsinh( 2
)
exp(3)
2[os(2d)+osh( 2
)℄
; (41)
_
=
2
sin(2d) 2dos(2d) 2osh( 2
)
exp(3)
2[os(2d)+osh( 2
)℄
:
(42)
Theline=0dividesongurationspaeintwo
sym-metri regions. The line = 0 ontains all
singu-lar points of this system, whih are nodes and
en-ters. The nodes appear when the denominator of the
above equations, whih is proportional to the norm
of the wavefuntion, is zero. No trajetory an pass
through these points. They happen when =0 and
os(d) =0,or =(2n+1)=2d,n an integer,with
separation =d. The enter points appear when the
numerators are zero. They are given by = 0 and
= 2d[otan(d)℄= 2
. They are interalated with
the node points. As j j! 1 these points tend to
n=d,andtheirseparationsannotexeed=d. Asone
an see from the abovesystem, the lassialsolutions
(a(t)/t 1=3
)arereoveredwhenjj!1orjj!1,
theotherbeingdierentfrom zero.
There are plenty of dierent possibilities of
evolu-tion,dependingontheinitialonditions. Nearthe
en-terpointsweanhaveosillatinguniverseswithout
sin-gularitiesand withamplitudeof osillationoforder 1.
Fornegative valuesof , the universe arise lassially
from asingularitybut quantum eets beome
impor-tant foringit to reollapseto another singularity,
re-overinglassialbehaviournearit. Forpositivevalues
of , the universe ontratslassially but when is
smallenoughquantumeetsbeomeimportant
reat-ing aninationary phasewhih avoidsthesingularity.
The universe ontrats to a minimum size and after
reahing this point it expands forever, reovering the
lassial limit when beomes suÆiently large. We
an see that for negativewehavelassial limitfor
smallsalefatorwhileforpositivewehavelassial
limitforbigsalefator.
III.2. The anisotropi ase
To exemplify the quantum isotropization of the
Universe, let us take now, instead of the F
riedman-Robertson-Walkerof Eq. (18), the homogeneous and
anisotropiBianhiIline element
ds 2
= N
2
(t)dt 2
+exp[2
0
(t)+2
+ (t)+2
p
3 (t)℄dx 2
+
exp[2
0
(t)+2
+ (t) 2
p
3 (t)℄dy 2
+
exp[2
0
(t) 4
+ (t)℄dz
2
: (43)
d
This line element will be isotropi if and only if
+
(t)and (t)are onstants. InsertingEq. (43)into
the lagrangian (17), supposing that the salareld
depends only on time, disarding surfae terms, and
performing a Legendre transformation, we obtainthe
followingminisuperspaelassialhamiltonian
H = N
24exp(3
0 )
( p 2
0 +p
2
+ +p
2
+p 2
); (44)
where (p ;p ;p ;p ) are anonially onjugate to (
0 ;
+
; ;), respetively, and we made the trivial
redenition! p
C
w =6 .
Weanwritethishamiltonianinaompatformby
dening y
= (
0 ;
+
; ;) and their anonial
mo-mentap
=(p
0 ;p
+ ;p ;p
),obtaining
H = N
24exp(3y 0
)
p
p
; (45)
where
is the Minkowski metri with signature
( +++). Theequationsofmotionaretheonstraint
re-spettothelapsefuntion N H p p
=0; (46)
andtheHamilton'sequations
_ y = H p = N
12exp(3y
0 ) p ; (47) _ p = H y
=0: (48)
The solution to these equations in the gauge N =
12exp(3y 0 )is y = p t+C
; (49)
wherethe momentap
areonstantsdueto the
equa-tionsof motionand theC
areintegrationonstants.
We an see that the only way to obtain isotropy in
these solutions is by making p
1 = p
+
= 0 and p
2 =
p =0,whihyieldsolutionsthatarealwaysisotropi,
theusualFriedman-Robertson-Walker(FRW)solutions
with a salar eld. Hene, there is no anisotropi
solution in this model whih an lassially beome
isotropi during the ourse of its evolution. One
anisotropi, always anisotropi. If we suppress the
degreeoffreedom, theuniqueisotropi solutionisat
spaetime beause in this ase theonstraint(46)
en-foresp
0 =0.
Todisuss theappearane ofsingularities,weneed
theWeylsquaretensorW 2
W
W
. Itreads
W 2 = 1 432 e 120 (2p 0 p 3 + 6p 0 p 2 p + +p 4 +2p 2 + p 2 +p 4 + +p 2 0 p 2 + +p 2 0 p 2 ): (50) d
Hene, the Weyl square tensor is proportional to
exp( 12
0
) beause the p's are onstants (see Eq.
(48)) and the singularity is at t = 1. The
lassi-al singularity an be avoided only if we set p
0 = 0.
But then, due to equation (46), we would also have
p
i
= 0, whih orresponds to the trivial ase of at
spaetime. Hene, theunique lassialsolution whih
isnon-singularisthetrivialatspaetimesolution.
The Dira quantization proedure yields the
Wheeler-DeWitt equation, whih in the present ase
reads 2 y y (y
)=0 : (51)
Let us now investigate spherial-wavesolutions of
Eq. (51). Theyread
3 = 1 y f(y 0
+y)+g(y 0 y) ; (52) where y q P 3 i=1 (y i ) 2 .
The guidane relations in the gauge N =
12exp(3y
0
)are(seeEqs. (47))read
p
0 =
0 S=Im
0 3
3
= y_ 0 ; (53) p i = i S=Im
i 3
3
=y_ i
; (54)
where S isthephaseofthewavefuntion. Intermsof
f andg theaboveequationsread
_ y 0 = Im f 0 (y 0
+y)+g 0 (y 0 y) 0 0 ; (55) _ y i = y i y Im f 0 (y 0
+y) g 0 (y 0 y) f(y 0
+y)+g(y 0
y)
; (56)
where theprime meansderivative with respet to the
argument of the funtions f and g, and Im(z) is the
imaginarypartoftheomplexnumberz.
FromEq. (56)weobtainthat
dy i dy j = y i y j ; (57)
whih implies that y i (t) = i j y j
(t), with no sum in j,
where the i
j
are real onstants and 1 1 = 2 2 = 3 3 =
1. Hene,apart somepositivemultipliativeonstant,
knowingaboutoneof they i
meansknowingaboutall
y i
. Consequently,weanreduethefourequations(55)
and(56)toaplanarsystembywritingy=Cjy 3
j,with
C > 1, and working only with y 0
and y 3
, say. The
planarsystemnowreads
_ y 0 = Im f 0 (y 0 +Cjy 3
j)+g 0 (y 0 Cjy 3 j) f(y 0 +Cjy 3
j)+g(y 0 Cjy 3 j) ; (58) _ y 3 = sign(y 3 ) C Im f 0 (y 0 +Cjy 3 j) g 0 (y 0 Cjy 3 j) f(y 0 +Cjy 3
j)+g(y 0 Cjy 3 j) : (59)
Note that if f = g, y 3
stabilizes at y 3
= 0 beause
_ y 3
as well asall other time derivatives of y 3
are zero
at thisline. As y i (t)= i j y j
(t), all y i
(t) beome zero,
andtheosmologialmodelisotropizesforeveroney 3
reahesthisline. Ofourseoneanndsolutionswhere
y 3
never reahesthis line, but in this ase there must
be some region where y_ 3
= 0, whih implies y_ i
= 0,
and this is an isotropi region. Consequently,
haveanisotropiphase,whih anbeomepermanent
inmanyases.
IV The Bohm-de Broglie
In-terpretation of Superspae
Canonial Quantum
Cos-mology
Inthissetion,wewillquantizeGeneralRelativity
The-ory(GR)withoutmakinganysimpliationsorutting
of degrees of freedom. The matter ontent is a
mini-mallyoupledsalareldwitharbitrarypotential. All
subsequentresultsremainessentiallythesameforany
matter eld whih ouples uniquely with the metri,
notwiththeirderivatives.
Thelassial hamiltonian of full GR with a salar
eld isgivenby:
H= Z
d 3
x(NH+N j
H
j
) (60)
where
H = G
ijk l
ij
k l
+ 1
2 h
1=2
2
+
+h 1=2
1
(R (3)
2)+ 1
2 h
ij
i
j
+U()
(61)
H
j
= 2D
i
i
j +
j
: (62)
d
In these equations, h
ij
is the metri of losed
3-dimensional spaelike hypersurfaes, and ij
is its
anonialmomentum givenby
ij
= h 1=2
(K ij
h ij
K)=G ijk l
( _
h
k l D
k N
l D
l N
k );
(63)
where
K
ij =
1
2N (
_
h
ij D
i N
j D
j N
i
); (64)
isthe extrinsiurvature ofthehypersurfaes(indies
are raisedand loweredby the3-metri h
ij
and its
in-verseh ij
). Theanonialmomentumofthesalareld
isnow
=
h 1=2
N
_
N
i
i
: (65)
Thequantity R (3)
isthe intrinsiurvatureof the
hy-persurfaesandhisthedeterminantofh
ij
. Thelapse
funtion N andtheshiftfuntion N
j
aretheLagrange
multipliers of thesuper-hamiltonian onstraint H0
and the super-momentum onstraint H j
0,
respe-tively. TheyarepresentduetotheinvarianeofGR
un-der spaetime oordinate transformations. The
quan-tities G
ijk l
anditsinverse G ijk l
(G
ijk l G
ijab
=Æ ab
k l )are
givenby
G ijk l
= 1
2 h
1=2
(h ik
h jl
+h il
h jk
2h ij
h k l
); (66)
G
ijk l =
1
h 1=2
(h
ik h
jl +h
il h
jk h
ij h
k l
); (67)
whihisalled theDeWittmetri. ThequantityD
i is
thei-omponentoftheovariantderivativeoperatoron
thehypersurfae,and=16G= 4
.
Thelassial4-metri
ds 2
= (N 2
N i
N
i )dt
2
+2N
i dx
i
dt+h
ij dx
i
dx j
(68)
andthesalareldwhiharesolutionsoftheEinstein's
equations an be obtainedfrom the Hamilton's
equa-tionsofmotion
_
h
ij =fh
ij
;Hg; (69)
_
ij
=f ij
;Hg; (70)
_
=f;Hg; (71)
_
=f
;Hg; (72)
for somehoie ofN andN i
, andif weimpose initial
onditionsompatiblewiththeonstraints
H0; (73)
H
i
0: (74)
It is a feature of the hamiltonian of GR that the
4-metris (68) onstruted in this way, with the same
initial onditions,desribethe samefour-geometryfor
anyhoieofN andN i
. Thealgebraoftheonstraints
lose in the following form (we follow the notation of
fH (x);H (x 0
)g = H i
(x)
i Æ
3
(x;x 0
) H
i
(x 0
)
i Æ
3
(x 0
;x)
fH
i
(x);H (x 0
)g = H (x)
i Æ
3
(x;x 0
) (75)
fH
i (x);H
j (x
0
)g = H
i (x)
j Æ
3
(x;x 0
)+H
j (x
0
)
i Æ
3
(x;x 0
)
d
Toquantize this onstrainedsystem,wefollowthe
Diraquantizationproedure. Theonstraintsbeome
onditionsimposed onthepossiblestatesof the
quan-tum system,yieldingthefollowingquantumequations:
^
H
i
j > = 0 (76)
^
Hj > = 0 (77)
Inthemetriandeldrepresentation,therstequation is
2h
li D
j Æ (h
ij ;)
Æh
lj +
Æ (h
ij ;)
Æ
i
=0; (78)
whihimpliesthatthewavefuntional isaninvariant
underspaeoordinatetransformations.
The seond equation is the Wheeler-DeWitt
equa-tion[35,36℄. Writingitunregulated in theoordinate
representationweget
h 2
G
ijk l Æ
Æh
ij Æ
Æh
k l +
1
2 h
1=2 Æ
2
Æ 2
+V
(h
ij
;)=0; (79)
where V isthelassialpotentialgivenby
V =h 1=2
1
(R (3)
2)+ 1
2 h
ij
i
j
+U()
: (80)
d
This equation involves produts of loal operators at
thesamespaepoint,heneitmustberegularized.
Af-ter doingthis, oneshouldnd afator orderingwhih
makesthetheoryfreeofanomalies,inthesensethatthe
ommutatorof the operator versionof theonstraints
loseinthesamewayas theirrespetivelassial
Pois-sonbrakets(75). Hene,Eq. (79)isonlyaformalone
whihmustbeworkedout[28,29,30℄.
Letus now seewhat is theBohm-deBroglie
inter-pretation ofthesolutionsof Eqs. (76)and (77)in the
metriandeldrepresentation. Firstwewritethewave
funtionalinpolarform =Aexp(iS=h),whereAand
S are funtionals of h
ij
and . Substituting it in Eq.
(78),wegettwoequationssayingthatAandS are
in-variantundergeneralspaeoordinatetransformations:
2h
li D
j ÆS(h
ij ;)
Æh
lj +
ÆS(h
ij ;)
Æ
i
=0; (81)
2h
li D
j ÆA(h
ij ;)
Æh
lj +
ÆA(h
ij ;)
Æ
i
=0: (82)
ThetwoequationsweobtainforAandS when we
substitute =Aexp(iS=h)intoEq. (77)willofourse
anyase,oneoftheequationswillhavetheform
G
ijk l ÆS
Æh
ij ÆS
Æh
k l +
1
2 h
1=2
ÆS
Æ
2
+V +Q=0; (83)
where V is the lassial potential given in Eq. (80).
Contrarytotheothertermsin Eq. (83),whihare
al-readywelldened,thepreiseformofQdependsonthe
regularizationandfatororderingwhiharepresribed
forthe Wheeler-DeWitt equation. In theunregulated
formgiveninEq. (79),Qis
Q= h 2
1
A
G
ijk l Æ
2
A
Æh
ij Æh
k l +
h 1=2
2 Æ
2
A
Æ 2
: (84)
Also,theotherequationbesides(83)inthis aseis
G
ijk l Æ
Æh
ij
A 2
ÆS
Æh
k l
+ 1
2 h
1=2 Æ
Æ
A 2
ÆS
Æ
=0: (85)
Let us nowimplement the Bohm-de Broglie
inter-pretationforanonialquantumgravity. Firstofallwe
notethatEqs. (81)and(83),whiharealwaysvalid
ir-respetiveofanyfatororderingoftheWheeler-DeWitt
equation, are like the Hamilton-Jaobi equations for
GR, supplemented by an extra termQ in the ase of
quantum eld theory in atspaetime, wewill
postu-late that the 3-metri of spaelike hypersurfaes, the
salareld,andtheiranonialmomenta alwaysexist,
independentonanyobservation,andthattheevolution
ofthe3-metriandsalareldanbeobtainedfromthe
guidanerelations
ij
= ÆS(h
ab ;)
Æh
ij
; (86)
=
ÆS(h
ij ;)
Æ
; (87)
with ij
and
given by Eqs. (63) and (65),
re-spetively. Likebefore,thesearerstorderdierential
equationswhihanbeintegratedtoyieldthe3-metri
andsalareldforallvaluesofthetparameter. These
solutions depend on the initial valuesof the 3-metri
and salareld at someinitialhypersurfae. The
evo-lutionoftheseeldswillofoursebedierentfromthe
lassialoneduetothepreseneofthequantum
poten-tialtermQinEq. (83). Thelassiallimitisonemore
oneptuallyverysimple: itisgivenbythelimitwhere
the quantum potential Q beomes negligible with
re-spetto thelassialenergy. Theonly dierenefrom
the previous ases of the non-relativisti partile and
quantum eldtheory in atspaetimeis thefat that
theequivalentofEqs. (3)and (7) foranonial
quan-tum gravity, whih in the naive ordering is Eq. (85),
annot be interpreted as a ontinuity equation for a
probabilitydensityA 2
beauseofthehyperbolinature
of theDeWitt metriG
ijk l
. However,evenwithout a
notionofprobability,whihinthisasewouldmeanthe
probabilitydensitydistributionforinitialvaluesofthe
3-metri andsalareld in aninitial hypersurfae,we
anextratalotofinformationfromEq. (83)whatever
isthe quantum potentialQ,aswill seenow. Afterwe
gettheseresults,wewillreturntothisprobabilityissue
inthelastsetion.
Firstwenotethat,whateveristheformofthe
quan-tum potentialQ, itmustbeasalardensity ofweight
one. This omes from the Hamilton-Jaobi equation
(83). FromthisequationweanexpressQas
Q= G
ijk l ÆS
Æh
ij ÆS
Æh
k l 1
2 h
1=2
ÆS
Æ
2
V: (88)
As S is aninvariant(seeEq. (81)), thenÆS=Æh
ij and
ÆS=Æmustbeaseondranktensordensityandasalar
density, bothof weight one, respetively. Whentheir
produts are ontratedwith G
ijk l
and multiplied by
h 1=2
,respetively,theyformasalardensityofweight
one. As V is also asalardensityof weightone,then
Qmustalsobe. Furthermore,Qmustdependonlyon
h
ij
and beause it omes from the wave funtional
whih depends only on these variables. Of ourse it
anbenon-loal(weshowanexampleintheappendix),
i.e.,dependingonintegralsoftheeldsoverthewhole
spae,butitannotdependonthemomenta.
Now we will investigate the following important
problem. From the guidane relations (86) and (87),
whihwillbewrittenintheform
ij
ij
ÆS(h
ab ;)
Æh
ij
0; (89)
and
ÆS(h
ij ;)
Æ
0: (90)
weobtainthefollowingrstorderpartialdierential
equations:
_
h
ij
=2NG
ijk l ÆS
Æh
k l +D
i N
j +D
j N
i
(91)
and
_
=Nh 1=2
ÆS
Æ +N
i
i
: (92)
Thequestionis,givensomeinitial3-metriand salar
eld, whatkindofstruturedoweobtainwhenwe
in-tegrate this equations in the parameter t? Does this
strutureform a4-dimensionalgeometry withasalar
eld for any hoie of the lapse and shift funtions?
NotethatifthefuntionalSwereasolutionofthe
las-sialHamilton-Jaobiequation,whihdoesnotontain
thequantumpotentialterm,thentheanswerwouldbe
in theaÆrmativebeausewewouldbein thesopeof
GR.ButSisasolutionofthemodiedHamilton-Jaobi
equation (83), and we annotguarantee that this will
ontinueto betrue. Wemayobtainaomplete
dier-entstrutureduetothequantum eetsdrivenbythe
quantum potential term in Eq. (83). Toanswerthis
question we will move from this Hamilton-Jaobi
pi-ture of quantum geometrodynamis to a hamiltonian
piture. This is beausemanystrong results
onern-ing geometrodynamis were obtainedin this later
pi-ture[25,37℄. Wewillonstrutahamiltonianformalism
whihisonsistentwiththeguidanerelations(86)and
(87). It yields thebohmian trajetories (91)and (92)
iftheguidanerelationsaresatisedinitially. Onewe
havethishamiltonian,weanusewellknownresultsin
theliteraturetoobtainstrongresultsaboutthe
Bohm-deBroglieviewofquantumgeometrodynamis.
ExaminingEqs. (81)and (83),we an easilyshow
[20℄thatthehamiltonianwhihgeneratesthebohmian
trajetories, onethe guidane relations(86)and (87)
aresatisedinitially,isgivenby:
H
Q =
Z
d 3
x
N(H+Q)+N i
H
i
(93)
wherewedene
H
Q
H+Q: (94)
The quantities H and H
i
are the usual GR
super-hamiltonianandsuper-momentumonstraintsgivenby
Eqs. (61)and(62). Infat,theguidanerelations(86)
and H
i
0beauseS satises(81)and(83).
Further-more,theyareonservedbythehamiltonianevolution
givenby(93)[20℄.
We now have ahamiltonian, H
Q
, whih generates
the bohmian trajetories one the guidane relations
(86)and(87)areimposedinitially. Inthefollowing,we
aninvestigateifthetheevolutionoftheelds driven
byH
Q
formsafour-geometrylikeinlassial
geometro-dynamis. FirstwereallaresultobtainedbyClaudio
Teitelboim [37℄. In this paper, he shows that if the
3-geometriesandeldongurationsdenedon
hyper-surfaesareevolvedbysomehamiltonianwiththeform
H = Z
d 3
x(N
H+N i
H
i
); (95)
andifthisevolutionanbeviewedasthe\motion"ofa
3-dimensionalutina4-dimensionalspaetime(the
3-geometriesanbeembeddedinafour-geometry),then
the onstraints
H 0 and
H
i
0 must satisfy the
followingalgebra
f
H(x);
H(x 0
)g = [
H i
(x)
i Æ
3
(x 0
;x)℄
H i
(x 0
)
i Æ
3
(x;x 0
) (96)
f
H
i (x);
H(x 0
)g =
H (x)
i Æ
3
(x;x 0
) (97)
f
H
i (x);
H
j (x
0
)g =
H
i (x)
j Æ
3
(x;x 0
)
H
j (x
0
)
i Æ
3
(x;x 0
) (98)
d
Theonstantin(96)anbe1dependingifthe
four-geometry in whih the 3-geometries are embedded is
eulidean ( = 1) or hyperboli( = 1). These are
theonditionsfortheexisteneofspaetime.
Theabove algebrais the sameasthe algebra(75)
of GRif wehoose= 1. Butthehamiltonian (93)
is dierent from the hamiltonian of GR only by the
preseneofthequantum potentialtermQinH
Q . The
Poisson braket fH
i (x);H
j (x
0
)g satises Eq. (98)
be-ausetheH
i ofH
Q
denedinEq. (93)isthesameasin
GR.AlsofH
i (x);H
Q (x
0
)gsatisesEq. (97)beauseH
i
is the generatorofspatial oordinate transformations,
andasH
Q
isasalardensityofweightone(remember
that Q must be a salar density of weightone), then
it mustsatises thisPoisson braket relationwith H
i .
What remains to be veried is if the Poisson braket
fH
Q (x);H
Q (x
0
)gloses asin Eq. (96). Wenowreall
the result of Ref. [25℄. There it is shown that a
gen-eral super-hamiltonian
HwhihsatisesEq. (96),isa
salardensityofweightone,whosegeometrialdegrees
of freedom are givenonly by thethree-metri h
ij and
itsanonialmomentum,and ontainsonlyeven
pow-ers and no non-loal term in the momenta (together
with theother requirements, theselast twoonditions
arealso satisedbyH
Q
beauseit isquadrati in the
momentaand thequantumpotentialdoesnotontain
anynon-loaltermonthemomenta),then
Hmusthave
thefollowingform:
H=G
ijk l
ij
k l
+ 1
2 h
1=2
2
+V
G
; (99)
where
V
G
h
1=2
1
(R (3)
2
)+ 1
2 h
ij
i
j +
U()
:
(100)
Withthisresultweannowestablishtwopossible
se-narios forthe Bohm-deBrogliequantum
geometrody-namis,depending ontheformof thequantum
poten-tial:
IV.1. Quantum geometrodynamis
evo-lution is onsistent and forms a non
de-generate four-geometry
In this ase, the Poisson braket fH
Q (x);H
Q (x
0
)g
must satisfy Eq. (96). Then Q must be suh that
V +Q=V
G
withV givenby(80)yielding:
Q= h 1=2
(+1)
1
R (3)
+ 1
2 h
ij
i
j
+ 2
(
+)+
U()+U()
: (101)
Thenwehavetwopossibilities:
IV.1.1. The spaetimeis hyperboli(= 1)
InthisaseQis
Q= h 1=2
2
(
+)
U()+U()
: (102)
Hene Q is like a lassial potential. Its eet is to
renormalizetheosmologialonstantandthelassial
salareld potential, nothingmore. The quantum
ge-ometrodynamis isindistinguishablefrom thelassial
one. It is not neessaryto require the lassial limit
Q=0beauseV
G
=V +Qalreadymaydesribe the
lassialuniversewelivein.
IV.1.2. The spaetimeis eulidean (=1)
InthisaseQis
Q= h 1=2
2
1
R (3)
+ 1
2 h
ij
i
j
+ 2
(
+)+
U()+U()
: (103)
d
Now Q not only renormalize the osmologial
on-stant and the lassial salar eld potential but also
hange the signature of spaetime. The total
poten-tial V
G
= V +Q may desribe someera of the early
universe when it hadeulidean signature, but notthe
present era, when it is hyperboli. The transition
be-tweenthesetwophasesmusthappeninahypersurfae
whereQ=0,whihisthelassiallimit.
We an onlude from these onsiderations that
if a quantum spaetime exists with dierent features
from the lassial observed one, then it must be
eu-lidean. Inother words, thesolerelevantquantum
ef-fetwhih maintainsthenon-degeneratenature ofthe
four-geometryofspaetimeisitshangeofsignatureto
aeulideanone. Theother quantumeets are either
irrelevantorbreakompletelythespaetimestruture.
ThisresultpointsinthediretionofRef. [38℄.
IV.2. Quantum geometrodynamis
evo-lution is onsistent but does not form a
non degenerate four-geometry
In this ase, the Poisson braket fH
Q (x);H
Q (x
0
)g
does not satisfy Eq. (96) but is weakly zero in some
other way. Some examples are given in Ref. [40℄.
They are real solutionsof the Wheeler-DeWitt
equa-tion, where Q = V, and non-loal quantum
poten-tials. Itisveryimportanttousetheguidanerelations
to lose thealgebra in these ases. It means that the
hamiltonian evolution with the quantum potential is
onsistentonlywhen restritedto thebohmian
traje-tories. For other trajetories, it is inonsistent.
Con-luding,whenrestritedtothebohmiantrajetories,an
algebra whih doesnot lose in general may lose, as
shown in theaboveexample. Thisis animportant
re-markontheBohm-deBroglieinterpretation of
anon-ial quantum osmology, whih sometimes is not
no-tied.
the "struture onstants" of the algebra that
hara-terizes the \pre-four-geometry" generated by H
Q i.e.,
the foam-likestruture pointedlongtime agoin early
worksofJ.A. Wheeler[35,42℄.
V Conlusion and Disussions
TheBohm-deBroglieinterpretationofanonial
quan-tum osmology yields a quantum geometrodynamial
piturewherethebohmianquantumevolutionof
three-geometries may form, depending on the wave
fun-tional,aonsistentnondegeneratefourgeometrywhih
mustbeeulidean(butonlyforaveryspeialloalform
ofthequantumpotential),andaonsistentbut
degen-erate four-geometryindiating the presene of speial
vetorelds and the breaking of the spaetime
stru-tureasasingleentity(inawiderlassofpossibilities).
Hene, in general, and always when the quantum
po-tential is non-loal, spaetime is broken. The
three-geometries evolved under the inuene of a quantum
potentialdonotingeneralstiktogethertoformanon
degenerate four-geometry, a singlespaetime with the
ausal struture of relativity. This is not surprising,
asitwasantiipated longago[42℄. Amongthe
onsis-tent bohmian evolutions, the more general strutures
thatareformedaredegeneratefour-geometrieswith
al-ternativeausal strutures. We obtainedthese results
taking aminimally oupled salar eld asthe matter
soure ofgravitation,but it anbe generalizedto any
matter soure with non-derivative ouplings with the
metri,likeYang-Mills elds.
Asshownintheprevioussetion,anondegenerate
four-geometryanbeattainedonlyifthequantum
po-tential have the spei form (101). In this ase, the
sole relevantquantum eet willbeahange of
signa-ture of spaetime, somethingpointingtowards
ial evolution but with degenerate four-geometry, we
have shown that any real solution of the
Wheeler-DeWitt equation yields a struture whih is the
ide-alizationofthestronggravitylimitofGR.Thistypeof
geometry,whih isdegenerate, hasalreadybeen
stud-ied[41℄. Duetothegeneralityofthispiture(itisvalid
for any real solutionof theWheeler-DeWittequation,
whihisarealequation),itdeservesfurtherattention.
It maywellbethat thesedegeneratefour-metriswere
theorretquantumgeometrodynamialdesriptionof
theyounguniverse. Itwouldbealso interestingto
in-vestigateifthesestrutureshavealassiallimit
yield-ingtheusualfour-geometryoflassialosmology.
For non-loal quantum potentials, we have shown
thatapparentlyinonsistentquantumevolutionsarein
fat onsistentifrestritedtothebohmiantrajetories
satisfyingtheguidanerelations(86)and(87). Thisis
apointwhih issometimesnottakeninto aount.
If we want to be strit and impose that quantum
geometrodynamisdoesnot break spaetime, then we
willhavestringentboundaryonditions. Assaidabove,
anondegeneratefour-geometryanbeobtainedonlyif
thequantumpotentialhavetheform(101). Thisisa
se-vererestritiononthesolutionsoftheWheeler-DeWitt
equation.
These restritionsonthe form ofthe quantum
po-tential do not our in minisuperspae models [19℄
beause there the hypersurfaes are restrited to be
homogeneous. The only freedom we have is in the
timeparametrizationofthehomogeneoushypersurfaes
whih foliate spaetime. There is a single onstraint,
whih of ourse always ommute with itself,
irrespe-tiveofthequantum potential. Thetheoremprovenin
Ref. [25℄, whih was essentialin all the reasoning of
the last setion, annot be used here beause
minisu-perspae models do not satisfy one of their
hypothe-ses. In setion 3 we studied quantum eets in suh
minisuperspae models and we showed that they an
avoidsingularities, isotropizethe Universe, andreate
inationaryepohs. Itshouldbeveryinterestingto
in-vestigateifthesequantumphasesoftheUniversemay
haveleft sometraeswhih ouldbedetetednow,as
intheanisotropiesoftheosmimirowavebakground
radiation.
Aswehaveseen,in theBohm-deBroglieapproah
we an investigate further what kind of struture is
formed in quantum geometrodynamis by using the
Poisson braket relation (96), and the guidane
rela-tions (91) and (92). By assuming the existene of
3-geometries,eldongurations,andtheirmomenta,
in-dependentlyonanyobservations,theBohm-deBroglie
interpretation allowsus to use lassial tools, like the
hamiltonian formalism,tounderstand thestrutureof
quantumgeometry. Ifthisinformationisuseful,wedo
not know. Already in the two-slit experimentin
non-relativisti quantum mehanis, the Bohm-de Broglie
tilehas passedthrough: ifit arriveat theupperhalf
of the sreen it must have ome from the upper slit,
and vie-versa. Suh information we do not have in
the many-worlds interpretation. However, this
infor-mation is useless: we an neither hek it nor use it
inotherexperiments. Inanonialquantumosmology
thesituation maybethesame. TheBohm-deBroglie
interpretation yields alot of information about
quan-tum geometrodynamis whih we annot obtainfrom
the many-worlds interpretation, but this information
maybeuseless. However,weannotanswerthis
ques-tionpreiselyifwedonotinvestigate further,andthe
toolsareatourdisposal.
Wewouldliketoremarkthat allthese resultswere
obtainedwithoutassuminganypartiularfator
order-ingandregularizationoftheWheeler-DeWittequation.
Also,wedidnotuseanyprobabilistiinterpretationof
thesolutionsof theWheeler-DeWittequation. Hene,
it is a quite general result. However, we would like
to make some omments about the probability issue
inquantumosmology. TheWheeler-DeWittequation
whenappliedtoaloseduniversedoesnotyielda
prob-abilisti interpretation for their solutions beause of
itshyperboli nature. However,it hasbeensuggested
many times [21, 43,44, 45, 46℄ that at the
semilassi-al level we an onstruta probabilitymeasure with
thesolutionsof theWheeler-DeWittequation. Hene,
forinterpretationswhereprobabilitiesareessential,the
problemof ndingaHilbert spae forthe solutionsof
theWheeler-DeWittequationbeomesruialif
some-onewantstogetsomeinformationabovethe
semilas-siallevel. Ofourse,probabilitiesarealsousefulinthe
Bohm-deBroglieinterpretation. Whenweintegratethe
guidanerelations(91)and (92),theinitialonditions
arearbitrary,anditshould benietohavesome
prob-abilitydistributiononthem. However,aswehaveseen
along this paper, we an extrat a lot of information
fromthefullquantumgravitylevelusingtheBohm-de
Broglieinterpretation, withoutappealing toany
prob-abilistinotion.
Itwouldalsobeimportanttoinvestigatethe
Bohm-de Broglie interpretation for other quantum
gravita-tionalsystems,likeblakholes. Attemptsinthis
dire-tion have been made, but within spherial symmetry
in empty spae[47℄, where wehaveonly anite
num-ber of degreesof freedom. It should be interesting to
investigatemoregeneralmodels.
Theonlusions ofthis paperareof ourselimited
by manystrongassumptions wehavetaitly made,as
supposing that a ontinuous three-geometry exists at
thequantum level(quantum eets ouldalsodestroy
it),orthevalidityofquantizationofstandardGR,
for-getting other developments like string theory.
How-ever,even ifthis approah isnot theappropriateone,
it is nieto see how far we ango with the Bohm-de
deBroglieinterpretationmayatleastberegardedasa
nie\gauge"[48℄tobeusedinquantumosmology,as,
probably, it will prove harder, or even impossible, to
reahthedetailedonlusionsofthispaperusingother
interpretations. Furthermore, if the ner view of the
Bohm-deBroglieinterpretationofquantumosmology
anyieldusefulinformationintheformofobservational
eets,thenwewillhavemeanstodeidebetween
inter-pretations, somethingthat will be veryimportant not
only for quantum osmology, but for quantum theory
itself.
Aknowledgments
WewouldliketothankCNPqofBrazilfornanial
support.
Referenes
[1℄ N. Bohr, Atomi Physis and Human Knowledge
(Si-ene Editions, New York, 1961); N. Bohr; Phys. Rev.
48,696(1935).
[2℄ W.Heisenberg,ThePhysialPriniplesoftheQuantum
Theory(Dover,NewYork,1949).
[3℄ J. von Neumann, Mathematial Foundations of
Quan-tum Mehanis(PrinetonUniversityPress, Prineton,
1955).
[4℄ R. Omnes, The Interpretation of Quantum Mehanis
(PrinetonUniversityPress,Prineton, 1994).
[5℄ H. D. Zeh,Found. Phys.1, 69(1970); E. Joos and H.
D.Zeh,Z.Phys.B59,223(1985);W.H.Zurek,Phys.
Rev.D26,1862(1982);W.H.Zurek,Phys.Today 44,
36(1991).
[6℄ C. Kiefer, Class. Quantum Grav. 18, 379 (1991); D.
Giulini,E.Joos,C.Kiefer,J.Kupsh,I.O.Stamatesu
and H. D. Zeh, Deoherene and the Appearane of a
Classial World in Quantum Theory (Springer-Verlag,
Berlin,1996).
[7℄ V.F Mukhanov, inPhysial Originsof Time
Asymme-try,edbyJ.J.Halliwell,J.Perez-MeraderandW.H.
Zurek(CambridgeUniversityPress,1994).
[8℄ H. D. Zeh, in Deoherene and the Appearane of a
Classial World in Quantum Theory (Springer-Verlag,
Berlin,1996).
[9℄ M.Gell-MannandJ.B.Hartle,inComplexity, Entropy
andthePhysisofInformation,ed.byW.H.Zurek
(Ad-disonWesley,1990).
[10℄ J. P. Paz and W. H. Zurek, Phys. Rev. D 48, 2728
(1993).
[11℄ G.C.Ghirardi,A.RiminiandT.Weber,Phys.Rev.D
34470(1986);G.C.Ghirardi,P.PearleandA.Rimini,
Phys.Rev.A42,78(1990).
[12℄ R.Penrose,inQuantumImpliations: Essaysin
Hon-our of David Bohm, ed. by B. J. Hiley and F. David
[13℄ TheMany-WorldsInterpretationofQuantum
Mehan-is,ed.byB.S.DeWittandN.Graham(Prineton
Uni-versityPress,Prineton,1973).
[14℄ D. Bohm,Phys.Rev.85,166(1952);D. Bohm,B.J.
HileyandP.N.Kaloyerou,Phys.Rep.144,349(1987).
[15℄ P. R. Holland, The Quantum Theory of Motion: An
Aount of the de Broglie-Bohm Causal Interpretation
of Quantum Mehanis (Cambridge University Press,
Cambridge,1993).
[16℄ J.C.Vink,Nul.Phys.B369,707(1992).
[17℄ Y.V.Shtanov,Phys.Rev.D54,2564 (1996).
[18℄ A.Valentini,Phys.Lett.A158,1,(1991).
[19℄ J. A. de Barros and N. Pinto-Neto, Int. J. of Mod.
Phys.D7,201(1998).
[20℄ N. Pinto-Neto and E. S. Santini, Phys. Rev. D 59,
123517,(1999).
[21℄ J.Kowalski-GlikmanandJ.C.Vink,Class.Quantum
Grav.7,901(1990).
[22℄ E.J.Squires,Phys.Lett.A162,35(1992).
[23℄ J.A.deBarros,N.Pinto-NetoandM.A.Sagioro-Leal,
Phys.Lett.A241,229(1998).
[24℄ R.ColisteteJr.,J.C.FabrisandN.Pinto-Neto,Phys.
Rev.D57,4707(1998).
[25℄ S. A. Hojman, K. Kuhar and C. Teitelboim, Ann.
Phys.96,88(1976).
[26℄ E. Cartan, Annales Sientiques de l'Eole Normale
Superieure40,325(1923)and41,1(1924).
[27℄ N.C.TsamisandR.P.Woodward,Phys.Rev.D36,
3641 (1987).
[28℄ K. Maeda and M. Sakamoto, Phys.Rev.D 54, 1500
(1996).
[29℄ T.Horiguhi,K.MaedaandM.Sakamoto,Phys.Lett.
B344,105(1995).
[30℄ J.Kowalski-Glikmanand K.A. Meissner, Phys.Lett.
B376,48(1996).
[31℄ J. M. Levy Leblond, Ann. Inst. Henri Poinare 3, 1
(1965).
[32℄ D.BohmandJ.P.Vigier,Phys.Rev.96,208(1954).
[33℄ A.Valentini,Phys.Lett.A156,5(1991).
[34℄ P.N.Kaloyerou,Phys.Rep.244,287(1994).
[35℄ J.A. Wheeler, in Battelle Renontres: 1967
Le-tures in Mathematial Physis, ed. by B. DeWitt and
J.A.Wheeler(BenjaminNewYork,1968).
[36℄ B.S.DeWitt,Phys.Rev.160,1113(1967).
[37℄ C.Teitelboim,Ann.Phys.80,542(1973).
[38℄ EulideanQuantumGravity,ed.byG.W.Gibbonsand
S.W.Hawking(WorldSienti,London,1993).
[39℄ C.Teitelboim,Phys.Rev.D25,3159(1982).
[40℄ N. Pinto- Neto and E. Sergio Santini,
`Ge-ometrodin^amiaqu^antianainterpreta~aodeBohm-de
Broglie: oespao-tempoqu^antiodevesereulideano?',
[42℄ J.A.Wheeler,Ann.Phys.2,604(1957);J.A.Wheeler,
inRelativity, Groups and Topology, ed. by B. DeWitt
andC.DeWitt(GordonandBreah,NewYork,1964);
G.MPattonandJ.AWheeler,inQuantumGravity.An
Oxford Symposium, ed.by C.J. Isham, R.Penroseand
D.Siama(ClarendonPress, Oxford,1975).
[43℄ T.Banks,Nul.Phys.B249,332(1985).
[44℄ T. P. Singh and T. Padmanabhan, Ann. Phys. 196,
296(1989).
[45℄ D. Giulini and C. Kiefer, Class. Quantum Grav. 12,
403(1995).
[46℄ J.J.Halliwell, inQuantum Cosmology and Baby
Uni-verses,ed.byS.Coleman,J.B.Hartle,T.PiranandS.
Weinberg(WorldSienti,Singapore,1991).
[47℄ M. Kenmoku, H. Kubotani, E. Takasugi and Y.
Ya-mazaki,reportgr-q9810039.
[48℄ WethankthisimagetoBrandonCarter.
[49℄ K.Kuhar,Phys.Rev.D50,3961 (1994).
[50℄ J.LoukoandS.N.Winters-Hilt,Phys.Rev.D54,2647
(1996).
