Table Of ContentWave Function of a Brane-like Universe
Aharon Davidson, David Karasik, and Yoav Lederer
Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
(e-mail: [email protected])
Within the mini-superspace model, brane-like cosmology means performing the variation with re-
specttotheembedding(Minkowski)timeτ beforefixingthecosmic(Einstein)timet. Thedeparture
from Einstein limit is parameterized by the ’energy’ conjugate to τ, and characterized by a classi-
callydisconnectedEmbryonicepoch. Incontrastwithcanonicalquantumgravity,thewave-function
of the Universe is (i) τ-dependent, and (ii) vanishes at the Big Bang. Hartle-Hawking and Linde
proposals dictate discrete ’energy’ levels, whereas Vilenkin proposal resembles α-disintegration.
In this paper, the Universe is viewed as a curvedfour- involving the familiar potential
9 dimensional bubble [1] floating in a higher dimensional
9 flat background. To discuss the quantum disintegration V(a)=4a2 1 H2a2 . (6)
−
9
of sucha Brane-likeUniverseand derivethe correspond-
1 (cid:0) (cid:1) δ
ing time-dependent and Big-Bang resistant wave func- Quantization means replacing P i and imposing
n →− δa
tion, we restrict ourselves to the framework of the mini- theWheeler-DeWitt(WDW)equation Ψ(a)=0onthe
Ja superspace model [2]. wave function of the Universe. H
2 Assuming the Universe to be homogeneous, isotropic, The non-standard procedure wouldbe to allow both
and closed, the Friedman-Robertson-Walker (FRW) line a(t•) andτ(t) to serveastwo independent canonicalvari-
1 element can be written in the form ables. Thevariationwithrespecttoτ(t)resultsinasim-
v
ple conservationlaw. Owing to (a,a˙,τ˙), the ’energy’ω
3 ds2 =σ2 (τ˙2 a˙2)dt2+a2(t)dΩ2 , (1) L
0 − − 3 conjugate to τ is conserved. Imposing the cosmic gauge
10 wσ2here4GdΩis23adnenoromt(cid:2)easlitzhaetiomneftarcictoor.fOanuenmitay3(cid:3)-sstpihlleerxee,racnisde τe˙r2g−y’a˙c2o=nse1rvoantliyonatetqhuiastsiotangien,twoeacgaennreeraarlirzaendgeevtohleut’eionn-
0 ≡ 3π equation
of course the gauge freedom of fixing τ(t). However, the
9
9 more general form helps us keep track of the way the a˙2+1=ξH2a2 , (7)
/ FRW manifold is embedded within a 5-dim Minkowski
c
q spacetime with ξ(a) being a root of
-
r ds2 =σ2 dτ2+da2+a2dΩ2 . (2) ω2
g 5 − 3 ξ(ξ 1)2 = . (8)
: − H6a8
v Apedagogicalcaseo(cid:2)fsufficientcomplexity(cid:3)involvesapos-
i Eq.(7) is recognized as the Regge-Teitelboim (RT) [3,4]
X itive cosmological constant Λ. The corresponding mini-
equation of motion, with Einstein limit approached as
Lagrangian,defined by I = dt, is given by
r L ω 0, that is ξ 1 (the physical brunch is identified
a
→ →
aa˙2 R with ξ 1). This comes with no surprise, given that
= +a(H2a2 1) τ˙2 a˙2 , (3) RT cano≥nicalvariablesareinfactthe embedding coordi-
L − √τ˙2 a˙2 − −
(cid:18) − p (cid:19) nates. Recalling that classicalRT-cosmology[4] involves
where H2 16πGΛ. only one independent equation of motion, the equation
The st≡anda3rd mini-superspace prescription is to im- arising by varying with respect to a(t) is superfluous.
po•setheso-calledcosmicgauge,namelyτ˙2 a˙2 =1,and The 2-momentum P =(δL,δL) is given by
treat a(t) as a single canonical variable. T−his way, the α δτ˙ δa˙
variationwith respect to a(t) gives rise to Raychaudhuri
a˙ 2 aτ˙
equation, and the complementary evolution equation P = +1 H2a2 , (9a)
τ
"(cid:18)√τ˙2−a˙2(cid:19) − #√τ˙2−a˙2
a˙2+1=H2a2 (4)
a˙ 2 aa˙
P = +3 H2a2 . (9b)
a
is then nothing but the Arnowitt-Deser-Misner (ADM) −"(cid:18)√τ˙2−a˙2(cid:19) − #√τ˙2−a˙2
Hamiltonian constraint = 0. Let P = 2aa˙ be the
momentum conjugate toHa, the Hamiltonia−n takes the The t-derivativeswhichenter eq.(9)convenientlyfurnish
form a time-like unit 2-vector
= 1 P2+V(a) , (5) nα τ˙ , a˙ , n2+1=0 . (10)
H −4a ≡ √τ˙2 a˙2 √τ˙2 a˙2
(cid:18) − − (cid:19)
(cid:0) (cid:1)
1
Invoking now a (τ˙,a˙)-independent matrix Virasoro-typeconstraints. Theso-calledmomentumcon-
δΨ
straintequationyA =0,whichistriviallysatisfiedat
ρα = 2a(H2a2−1) 0 , (11) ,iδyA
β 0 2a(H2a2 2) themini-superspacelevel,isaccompaniedbyabifurcated
(cid:18) − (cid:19)
WDW equation
Pα can be put in the compact form
δ δ
(ρ λI) 1 AB Ψ=λΨ ,
Pα = 1(nρn)nα+ραnβ , (12) δyA − − δyB (18)
2 β δyδA(cid:0)(ρ−λI)−2(cid:1)AB δyδBΨ=Ψ .
after subtracting a(H2a2 1)(n2 + 1)nα. A naive at-
tempt to solve nα(ρ,P), a−s apparently dictated by the Given the diag(cid:0)onal ραβ spe(cid:1)cified by eq.(11), and up to
Hamiltonian formalism, and substitute into the con- all sorts of order ambiguities, the mini-superspace wave
straint n2 +1 = 0, falls short. The cubic equation in- function Ψ(τ,a) obeys
volveddoesnotadmitasimplesolution,andtheresulting ∂2Ψ
constraint is anything but quadratic in the momenta. =ξ(ξ 1)2H6a8Ψ , (19a)
− ∂τ2 −
The way out, that is linearizing the problem, involves
the definition of an independent quantity λ, such that ∂2Ψ =a2 2+(ξ 1)H2a2 2( 1+ξH2a2)Ψ , (19b)
−∂a2 − −
nρn+2λ=0 . (13) where the λ (cid:2) ξ dictionary rea(cid:3)ds
↔
Off the Einstein limit, λ is not an eigenvalue of ρα, and λ a (ξ+1)H2a2 2 . (20)
β ≡ −
we can solve for nα(ρ,P,λ) to find The separation of (cid:2)variables is accom(cid:3) plished by substi-
tuting Ψ(τ,a) = ψ(a)χ(τ). Eq.(19b) then tells us that
α
nα = (ρ λI)−1 Pβ . (14) ξ = ξ(a). In turn, eq.(19a) can admit a solution only
− β
h i provided ξ(a) is such that ξ(ξ 1)2H6a8 = ω2 is a con-
−
Thisallowsustofinallyconvertthecombinedconstraints stant. This is how the conserved ’energy’ ω, introduced
n2+1=0 and nρn+2λ λ(n2+1)=0 into by eq.(8), enters the quantum game.
−
Altogether,the τ-dependent wavefunctionofthe Uni-
P(ρ λI) 2P +1=0 ,
− − (15) verse acquires the familiar form
P(ρ λI) 1P +λ=0 .
−
(cid:26) − Ψ(τ,a)=ψ(a)e iωτ , (21)
−
The first equation is the derivative with respect to λ of
withtheτ-dependencedroppingoutattheEinsteinlimit.
the other. This suggests that λ be elevated to the level
The radial component ψ(a) satisfies the residual WDW
ofacanonicalnon-dynamicalvariableintheforthcoming
equation
Hamiltonian formalism.
Needlesstosay,theaboveseemstobeatipofabigger ∂2
+V(a) ψ =0 , (22)
iceberg, a mini-superspace version of Brane-like gravity. −∂a2
(cid:18) (cid:19)
Indeed, carrying out the (say) 10-dim RT embedding of
wherethe modifiedpotential,depictedinfig.(1),is given
the 4-dim ADM formalism, we have recently derived the
explicitly by
quadratic Hamiltonian [5]
V(a)=a2 2+(ξ 1)H2a2 2(1 ξH2a2) . (23)
H= 12N PA (ρ−λI)−1 ABPB +λ +NiyA,iPA , (16) (cid:2) − (cid:3) −
h i
(cid:0) (cid:1)
where the novel Lagrange multiplier λ accompanies the
standard non-dynamical variables, the lapse function N
andtheshiftvectorNi. ToshedlightonthematrixρAB,
one infers that
ρAB
=(hiahjb hijhab)yA yB + (3)+6H2 ηAB ,
2√h − |ab |ij R
(cid:16) (cid:17)
(17)
with (3) denotingthe3-dimRicciscalarconstructedby
R
means of the spatial 3-metric h =η yAyB.
ij AB i j
| | δ
The quantum theory dictates P i . The FIG. 1. Wheeler-DeWitt potential for a Brane-like Uni-
A →− δyA verse(solidcurve),andforEinstein Universe(dashedcurve).
corresponding wave function Ψ(τ,a) is subject to two
2
V(a) admits a barrier provided ωH ≤ 3√23, which we exp 2i aL√ Vda′ π = 1 . (30)
nowadoptasthecaseofinterest. Thebarrierisstretched (cid:20) (cid:18)Z0 − − 4(cid:19)(cid:21) ±
between a < a < a , where a are the two positive
L R L,R Theresultis’energy’(nottobeconfusedwiththeenergy
roots of H2a3 a+ω = 0. For ωH 1, the classical
− ≪ E = 0) quantization. To be specific, for ωH 1, we
turning points are located at ≪
invokeeq.(26)and after some algebraderivethe discrete
1 ’energy’ spectrums
aL ω , aR H−1(1 ωH) . (24)
≃ ≃ − 2
2
ωHH,L (4n 1) , (31)
At long distances, only a slight deviation from the origi- n ≃ 3 ±
r
nal potential is detected, namely
such that ωL =√3ωHH.
min min
V(a ω) 4a2(1 Hω H2a2) . (25) Having a non-zero ground state ’energy’ ωmin is re-
≫ ≃ − − markable. It is the closest one can get to Einstein limit
But at short distances, a serendipitous well (with a sur- ω =0. But what exactly do we mean by a ground state,
plus of ‘kinetic energy’ at the origin) makes its appear- andwhydoestheEinsteinlimitmakesense? Asuccessful
ance (presumably Euclidean) theory of boundary conditions
must explain why is low ω preferable to high ω.
V(a ω) ω2 3ω4/3a2/3+4a2 . (26)
≤ ≃− −
The emerging classically disconnected Embryonic epoch
is the essence of brane-like quantum cosmology.
A theory of boundary conditions is still to be con-
structed. The situation is even more complicated in a
scheme where the Big-Bang is classically alive and can-
notbe tradedfor a Euclideanconic-singularity-freepole.
The Riemann tensor gets pathological as a 0, leaving
→
us with no alternative but to interpret ’nothing’ [6] as
FIG. 2. Hartle-Hawking ground state (n = 0) wave func-
Ψ(τ,a=0)=0 . (27) tion of a brane-like Universe (notice its vanishing at the Big
Bang). The dashed curve being the underlying brane-like
This way,followingDeWitt [7]argument,we ’neutralize’ Wheeler-DeWitt potential.
the Big Bang singularity by making the origin quantum
mechanically inaccessible to wave packets.
Vilenkin(V)[10]proposalonthe otherhandischarac-
At this stage,while sticking to the full Lorentzianpic- •
terized by an outgoing wave function
ture, namely Ψ = ψ(a)e iωτ even under the potential
−
barrier,our discussion bifurcates with respect to the left 1 a
ψ (a>a ) exp i √ Vda . (32)
over boundary condition: V R ∼ √ V − ′
− (cid:20) ZaR (cid:21)
Following Hartle-Hawking (HH) [8] or Linde (L) [9] The WKB behavior of the wave function can then be
•
proposals, where Hermiticity (real ω) is the name of the tracedbackallthewaytotheoriginwhereitissupposed
game,the naiveWKB wavefunctionunderthe barrieris to vanish. The consistency condition then reads
given by
aL π 1+4θ2
exp 2i √ Vda , (33)
1 a − ′− 4 ≃ 1 4θ2
ψHH,L(aL <a<aR) exp √Vda′ , (cid:20) (cid:18)Z0 (cid:19)(cid:21) −
≃∓√V (cid:20)±ZaL (cid:21) aR√Vda′
where θ e aL is the opacity coefficient. The
(28) ≡
latter equatioRn can only be satisfied by a complex ’en-
respectively. Thecorrespondingnucleationprobabilityis ergy’ ω =ω˜− 21iΓ. It should be noticed how the Hartle-
Hawking (Linde) discrete spectrum, that is ω˜ ωHH(L)
aR → n
e±2ZaL √Vda′ e±3H42(1− 23Hω) . (29) Afolltloowgeetdhebry, tΓhe→dis0in,teisgrraetcioovneorefdVfiolernkθin≪bu1bb(lθe ≫high1l)y.
P ∼ ≃ resembles α-decay.
The matching at a = a yields a symmetric (antisym- Euclidization is next. In the first glance it may look
R
metric) combination of equal strength outgoing and in- like the Lorentzian and the Euclidean regimes share the
going waves. The a = a matching into the Embryonic one and the same Embedding spacetime, and that Eu-
L
zone would contradict the Big-Bang boundary condition clidizationcanbeformulatedinthelanguageofthemini-
eq.(27) unless superspace light-cone. However, a simple investigation
3
reveals that a closed Euclidean FRW metric cannot be of time in canonical quantum gravity, and (ii) How to
embedded within a flat Minkowski spacetime. It calls ’neutralize’, quantum-mechanically, the Big-Bang prob-
for a flat Euclidean background, attainable by means of lem at the Lorentzian level. All this without upsetting
Wick rotationτ iτ (withthe correspondingcosmic theleadingwave-functionproposals. Itremainstobeun-
E
gauge being τ˙ 2→+a±˙ 2 =1). derstoodthoughhowtoconverttheemergingclosedUni-
E E
We are not in a position to tell whether Euclidean verseintoanopenone(followingperhapsHawking-Turok
gravity is only a technical tool, serving to explain cer- [11]prescription),howdoesinflationenterthegame(pre-
tain quantum and/or thermodynamic aspects of the sumably along Linde [12] or Vilenkin [13] trails), and
Lorentzian theory, or perhaps has life of its own. This whether there exists some leftover experimental crumb.
way or the other, the emerging picture is of a Euclidean Atanyrate,severalmodelindependentfeatures,notably
manifold sandwiched between two Lorentzian regimes. the classically disconnected Embryonic epoch, are to be
regarded as the finger-prints of the underlying theory.
Brane-likeUniversegravityconstitutesacontrolleddevi-
ation (automatic energy/momentum conservation) from
Einsteingravity,with the latter regardedas the classical
ground-state limit.
ACKNOWLEDGMENTS
It is our pleasure to thank Professors E. Guendelman
andR.Brusteinforvaluablediscussionsandenlightening
remarks.
FIG. 3. The Euclidean regime sandwiched between the
Embryonicand theExpandingLorentzian epochs.
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E
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E
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demonstrate (i) How to overcomethe problem (absence)
4
