Cosmology and String Theory
Lecture 3 on String Cosmology:
The wavefunction of the universe in the Landscape
Henry Tye 戴自海
Cornell University and Hong Kong University of Science and Technology
Asian Winter School, Japan January 2012 In the previous 2 lectures :
• We have seen, so far, the failure of finding a single Type IIA meta-stable de-Sitter vacuum. • We argue that the number of meta-stable de-Sitter vacua in string theory is a lot fewer than naively expected. • However, we probably still expect a good number of such local minima in the Landscape. We argue that tunneling can be much faster than naively expected. • In the He-3 superfluid system, experiments have shown that the exponent in the tunneling rate is about 6 orders of magnitude bigger than that predicted by theory. Eternal Inflation
• Suppose our universe sits in a meta-stable de-Sitter vacuum for time T. 2THEAUTHOR
What is eternal inflation ?
a(t) eHt ⇠ H2 1/G⇤ ⇠ Consider a patch of size 1/H. Suppose T 3/H. ⇠ 3 3 9 The universe would have grown by a factor of e · = e . If inflation ends in one patch, there are still many other (causally disconnected) patches which continue to inflate. So inflation never ends. Eternal Inflation
• A very simple and elegant scenario • With the original picture of Bousso-Polchinski and KKLT, it seems unavoidable that the wavefunction will sit at some point in the Landscape • Naively it implies that most of the universe is still undergoing eternal inflation today, so our observable universe is a tiny tiny part of the whole universe • No matter how suppressed, parts of the universe can tunnel to other vacua, both with higher and lower CC. So the whole Landscape is “populated”. Eternal Inflation
• We need a strong version of the Anthropic principle to explain our presence. • If there are say10500 vacua, there’ll be many vacua extremely similar to ours. Among these many very similar vacua, we cannot tell which one we live in. In this sense, we’ll lose predictive power. • The question of measure cannot really be resolved since there is no global time. Can we avoid eternal inflation ?
• Based on what we discussed in the previous 2 lectures, I argue that eternal inflation is not only avoidable, it is unlikely. • Since we know little about the potential of the Landscape, except that it is very large and complicated, is there much we can say about it ? • In this lecture, I like to argue “yes”. • This lecture is partly based on hep-th/0611148 and 0708.4374. Strategy : • Treat the landscape as a d-dimensional random potential • Use the scaling theory developed for random potential (disordered medium) in condensed matter physics • justify the key points of the above scenario • do a renormalization group analysis on the mobility of the wavefunction of our universe • calculate some of the properties of the landscape, e.g., the critical CC • argue why we should end up in a vacuum with an very small C.C. Condensed matter physics want to understand when a sample is conducting or insulating. When conducting, the charge carriers are mobile. When insulating, the charge carriers are localized. They study this behavior for a random potential.
For us, the landscape is the random potential. The wavefunction of a charge carrier becomes the wavefunction of the universe.
A localized or trapped wavefunction will imply eternal inflation.
We like to learn when it is trapped and when it is mobile We like to see :
Mobility at high CC and trapped at very low CC.
There is a critical CC, which is exponentially small.
At sites with CC larger than this critical value, the wavefunction of the universe is mobile.
At sites with CC smaller than this critical value, the wavefunction of the universe is isolated, with exponentially long lifetime.
The transition at this critical CC is sharp. Anderson localization transition
• random potential/disorder medium • insulation-superconductivity transition • quantum mesoscopic systems • conductivity-insulation in disordered systems • percolation • strongly interacting electronic systems • high Tc superconductivity • ...... Some references :
• P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. Lett. 109, 1492 (1958). • E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan, Scaling Theory of Localization : Absence of Quantum Diffusion in Two Dimensions, Phys. Rev. Lett. 42, 673 (1979). • B. Shapiro, Renormalization-Group Transformation for Anderson Transition, Phys. Rev. Lett. 48, 823 (1982). • P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57, 287 (1985). • M. V. Sadovskii, Superconductivity and Localization,'' (World Scientific, 2000). • Les Houches 1994, Mesoscopic Quantum Physics. is an unstable fixed point, implying that the transition between mobility and localization is a sharp one. Consider a specific (enveloping) wavefunction ⇤(r) e r / at site r = 0 with a typical | | ⇥ in the cosmic landscape (see Figure 1(b)). The wavefunction seems to be completely localized at the site if the localization length ⇥ a, where ⇥ measures the size of the ⇥ site and a is the typical spacing between sites. They are determined by local properties of the order of ⇤. Then one expects that for L ⇤, the e ective conductance becomes and ⇥ is known as the localization length. With resonance tunnelin⌅g, ⇥ can be somewhat of the order of ⇤. Then oneexpexonpecentstiallythat smforalLl: ⇤, the e ective conductance becomes bigger, although ⇤(r) may still ⌅be exponentially suppressed at distance a. At distance exponentially small: L/ scales around a, the conductance goes like gd(L) e (2.2) L/ ⇤ gd(L) e (2.2) For a small disorder,⇤ the medium is in a/a metallic state and the conductivity ⌅ is independent For a small disorder, the medium is in a metallic stateg(aand) t⇤he(acond) uctivite y ⌅ is independent (1.3) of the sample size if the siz e| is m| uch larger than the mean free path, L l. Conductance of the sample size if the size is much larger than th2 e meDefinean2a/ free a dimensionlesspath, L l .conductanceConductan gce in a d-dim. ⌅ Note that,isfordetermineda ⇥, 0 in⇤th(ais) casee just. by the usual⌅ Ohm’s law and for a d-dimensional hypercube, is determined in this case just by ⇤the usual Ohm’s| | la w andhypercubicfor a d-dimensional region of macroscopichypercube, size L Given wge=hag(vae:) (1.3) at scale a, what happens to g at distance scales L a ? That is, we have: ⇤ how does g scale ? The scaling theory of g is well studied in condensed 2 d matter physics. It d 2 g (L) = ⌅L (2.3) g (L) = ⌅L d (2.3) turns out that there isda critical conductance gc which is a function of d. If g(a) < gc, then L/ based on a simpleg(dLime) nes ionbasedal argu0onasmLaenst.im(Recpleandaldlimeththate lannsionindscdalap=earguis3,angm=iennsulati⌅t.(Area)(Recng me/Laldl ium.that⌅L.)Thein dwa=vefu3,nctiong = ⌅(Area)/L ⌅L.) ⌅ ⌅ ⌃ ⇤ ⇤ The conductanceisg tr=ulyg(alo) calizeatThelengthdc.onTduunnscalectanelingacisewagillm=tiakcgroscopic(eaexp) atonenlengthmeasuretiallyscalelongof thaane isddisorder.eaternalmicroscopicinWflatione mcomeasurees intoof the disorder. We (d 2) see that g(L) maply aendy. Ifupg(sewitha)e >thonatgec,ofgthe(Lth)econma2 duvyecrendtiyvitdi yuepisrenwithfint asymptotite on(ge(Lof) icthfor(eL/am2sv)forer yL)dias earL.ent asymptotand theic forms for L a. ⌅ ⌅ ⌃ ⌅ Here ⌅ is rescaledlansodscthatapegHereisisindimea⌅condnissionlesrucescting/mobials.ed so tlehatphaseg is. Thedimewnavsieonlesfunctis.on is free to move. Around the transition point, there is a correlation length that blows up at the phase transition. We Elementary scaling theory ofElloecalizationmentary sascalinsumesg ththateoryg ofofalodcalization-dimensionalassumeshypercutbhate g of a d-dimensional hypercube shall see that, at length scale a, this correlation length can be identified with ⇥. That is, ⇥ of size L satisfies the simpleofst sizdi eerLensatistial efiquesationthe simplof Conducting/mobilea renorest dmiali ezratientional (metalic)groupequation, where withof finitea renor conductivitymalization group, where plays the role of the correlation length. This is how, given g(a) (1.3) at scale a, g can grow d ln gd(L) large for large L.⇥ (g (L)) = d ln gd((2.4)L) d d d ln L ⇥ (g (L)) = (2.4) In this paper, we argue that, for large d, the crditicald conductandclne isLexponentially small, that is, the ⇥-function ⇥ (g) depends only on the dimensionless conductance g (L). Then d (d 1) d that is, the ⇥-function ⇥d(ggc) depe end s only on the dimensionless conductance gd(L). Then the qualitative behavior of ⇥d(g) can be analyzed in a simple⇧way by interpolating it between the 2 limiting forms given bthy eEq.qu(alitativ2.2) ande Eqbeha.(2.3vi).orFofor⇥thed(g)insulcanatingbe analyphasezed, (gin a simple0), it way by interpolating it between So it is not di⌃cult for g(a) > gc; given g(a) (1.3), we see that⇧fast tunneling happens follows from Eq.(2.2) and Ethq.(e2.42(2d)li1)thatmiting: forms given by Eq.(2.2) and Eq.(2.3). For the insulating phase, (g 0), it when 0 > e , or ⇧ follows from Eq.(2.2) and Eq.(2.4) that : g a lim ⇥d(g) ln (2.5) g 0 ⇧ g d > + 1 (1.4) ⇥ c ⇥ g lim ⇥d(g) ln (2.5) g 0 ⇧ g which is negativesofortheg wa0.vefunForctitheon cofonthductine univgeprshasee can(larmgeoveg),freitely⇥follinowsthefromquanEtuq.(mc2.3lan)dscape even for ⇧ and Eq.(2.4) thatan: exponenwhictiallyh sismallnegativg(a).eSinforcegthe 0.-fuFncortionthehasconaductinpositivge splophasee at(larthegelogcal),izitationfollows from Eq.(2.3) transition point (implylim ing⇥d(gan) undstab2⇧le fixed point), the transition be(2.6)tween the insulating and Eq.(g 2.4) that⇧: (localization) phase⇥⇤and the conducting (mobile) phase is sharp. We see that it is precisely which is positive for d > 2. Assuming the existence of two perturblimation⇥dexpan(g) siond s o2ver (2.6) the vastness of the cosmic landscape (as parameterizeg d by a lar⇧ge d ) that allows mobility the “coupling” g in the limits of weak and strong “couplings”, one c⇥⇤an write corrections to even if thewhicwavhefunis cptosionitiisveloforcalized d>. 2.ThiAsss muobmilitingy thalloewsexisthetenwcaeveoffuntwctioonpetrto urbsamationple expansions over Eq.(2.5) and Eq.(2.6) in the following form : the landscapthee v“coupery qulinickg”ly.gItinalsothemlieansmitsaofsemi-weakclassandicalstrdesongcrip“couption oflinthgse”,landonescapcane iswrite corrections to inadequate. g Eq.(2.5⇥d()gand0)E=q.(ln2.6)(1in+thbdeg +follow)ing form : (2.7) In the mobility ⇧phase, thegwc avefunction· ·an· d/or the D3-branes (and moduli) move d down ⇥the(glandscap) =e dto a2site wi+th a smalle r ⇥>. 0At that site, somge of(i2.8)ts neighboring sites d ⇧ ⌃ g · · · d ⇥d(g 0) = ln (1 + bdg + ) (2.7) have larger ⇥s, so the e⇤ective a increases, leading to⇧a larger e⇤ecgctive critical con· · ·ductance d Assuming these andgc. Tha smois hapothpmonotonens untilousthe ⇥cdon(gditi), itonis⇥(d1.4eas(g)yistonoplot)lon=thegedr ⇥sat-fu2 isnctionfied an+qdualitthe a-D3-bran d >e0is stuck (2.8) tively for all g and d, as shown in Figure 2. ⇧ ⌃ g · · · For d > 2, ⇥ (g) must have a zero: ⇥ (gc) = 0, where gc is the critical conductance. d Assuming thesde and a smooth monotonous ⇥d(g), it is easy to plot the ⇥-function qualita- The slope at the zero is positivtive,elysoforthisalzel rog andof ⇥dd(g,)ascorrshoespwnondsin toFiguan reunstable2. fixed point of Eq.(2.4). This means that a small positive or negative depar– 4 –ture from the zero will lead For d > 2, ⇥d(g) must have a zero: ⇥d(gc) = 0, where gc is the critical conductance. asymptotically to very di erent behaviors of the conductance. As g moves from g > gc to The slope at the zero is positive, so this zero of ⇥d(g) corresponds to an unstable fixed point g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge. of Eq.(2.4). This means that a small positive or negative departure from the zero will lead asymptotically to very di erent behaviors of the conductance. As g moves from g > gc to g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge. – 8 –
– 8 – the particle is scattered randomly and its wavefunction usually changes at the scale of the order l. However, the wavefunction remains extended plane-wave-like (Bloch wave- like) through the medium. If the wavefunction is initially localized at some site, it would quickly spread and move towards low potential sites. This mobility implies that the system is an unstable fixed point, implis a “condyingucthtor”.at the transition between mobility and localization is a sharp one. Anderson [19] showed that the wave function of a quantum particle in a random poten- tial can qualitatively change its nature if the randomness becomes large enough (strongly Consider a specific (enveloping) wavefunction ⇤(r) e r / at site r = 0 with a typical disordered). In this case, the wavefunction |bec| omes localized, so that its amplitude (enve- ⇥ in the cosmic landscape lop(seee) dFropsigurexpe onen1(bti)).ally withThediswtanavceeffromunctionthe censeetermsof localizato bteioncompr0: letely localized at the site if the localization length ⇥ a, where ⇥ measures the size of the ⇥⇥(r) exp( r r0 / ) (2.1) site and a is the typical spacing between sites. Th|ey are| ⇤ det e|rmin |ed by local properties and ⇥ is known as the localizationwhere isltheengtloch.alizatiWonithlengthresonan. This csiteuationtunnelinis shogw,n ⇥qucanalitatibveelysomin Figuewrehat1. When the particle wavefunction is completely localized at a single site, approaches a value bigger, although ⇤(r) may comstillparablbeeetoxptheonteypicalntiallyspacingsuppra beetssweeden siattes.diInstand 3,ceitawill. Atakte danisetxpanceonentially ⇥ scales around a, the conductanlongctimee gotoestulnnelike to a neighboring site. The lack of mobility implies that the system is an “insulator”. a/ Thegph(ay)sical me⇤(anaing) of Andersone localization is relatively simple: cohere(1.3)nt tunneling of particles is poss|ible onl| y between energy levels with the same energy. However, in case of strong2 randomnes2a/ s, l a and the states with the same energy are too far apart in space Note that, for a ⇥, 0 ⇤(a) e . ⌃ ⇤ |for tun|nelin g to be e ective. Since we are interested in large d, while condensed matter Given g = g(a) (1.3) at phscaleysics asystem, whats typicallyhappehnavse tod g3,atwedshisalltanceintrodscucealesa slighLtly reafined? Thatdefinitionis, here : ⇥ ⇤ how does g scale ? The scalin(1)gantheoryextendedofstateg isif w ella,stuas shodiedwn inin Figucondensere 1(a); d matter physics. It ⇧ (2) a weakly localized state if a, as shown in Figure 1(b); turns out that there is a critical conductance gc which is a function of d. If g(a) < gc, then (3) a localized state if < a, as shown in Figure 1(c); L/ g(L) e 0 as L (4)anda stronglthe lany locdscalizedapsetateis anwheninsulati a. Asngwmee shdallium.see, forThelargewda,vaefustronnctiongly localized ⌅ ⌅ ⌃ ⌅ is truly localized. Tunnelingstatewillcantakstille leadexptoonenmobitiallity lyif itlongsatisfieans thed econternaldition (in1.4flation). Whencomthis cesonditiintoon is not satisfied, we have (d 2) play. If g(a) > gc, the conductivity is finite (g(L) (L/a) ) as L and the (5) a truly localized state. ⌅ ⌃ landscape is in a conducting/mobiFor d 3,leapwhaseeakly.loThecalizedwstateavefunwillchtiavone lostis mobilifree toty alreadymove.(althArououghndit isthbeelieved ⇥ transition point, there is a thcorat,relationunder thelrienghgtht condthatitionsb, suplowercondus up cattivitthy ecanphasetake plactrane insithtion.e localizationWe re- shall see that, at length scalegiona)., thThisat corra stronelatglyionlocalizlengthed statecmaany sbtille lideadentotifiedmobilitwithy for lar⇥.geThatd is notis,surpr⇥ ising. For large d, the particle has many directions for coherent or resonant tunneling. In a ran- plays the role of the correlationdom pleotenngth.tial enThvironismisenhot, itw,is mgivuchenmorg(ealik)ely(1.3that) atsomsecaledirectiaon, galcanlows easgroywmotion. large for large L. The goal is to quantify this intuition. We shall continue to use the language of condensed matter physics : a is the microscopic scale while L is the macroscopic scale. In this paper, we argue that, for large d, the critical conductance is exponentially small, For fixed d, we expect a transition from a “conductor” to an “insulator” as the potential barriers rise and fast tunn(deling1) is suppressed. To learn more about this transition, we g e should focus on thc e⇧behavior of the conductivity, or, more appropriately, the conductance. It turns out that the behavior of such a transition can be described by a scaling theory So it is not di⌃cult for g(asim) >ilargtoc;thatgivuseendgin(athe) (th1.3eory), ofwcritie sceeal phthenatomefastna [20tun]. Innelinthe scgalhaping theorypensof the 2(d 1) transition between mobility and localization, one considers the behavior of the conductance when 0 > e , or as a function of the sample size L. Dimensionally, we shall choose some microscopic units a d > + 1 (1.4) ⇥ so the wavefunction of the universe can move freely in the– 6q–uantum landscape even for an exponentially small g(a). Since the -function has a positive slope at the localization transition point (implying an unstable fixed point), the transition between the insulating (localization) phase and the conducting (mobile) phase is sharp. We see that it is precisely the vastness of the cosmic landscape (as parameterized by a large d) that allows mobility even if the wavefunction is localized. This mobility allows the wavefunction to sample the landscape very quickly. It also means a semi-classical description of the landscape is inadequate. In the mobility phase, the wavefunction and/or the D3-branes (and moduli) move down the landscape to a site with a smaller ⇥. At that site, some of its neighboring sites have larger ⇥s, so the e⇤ective a increases, leading to a larger e⇤ective critical conductance gc. This happens until the condition (1.4) is no longer satisfied and the D3-brane is stuck
– 4 – is an unstable fixed point, implying that the transition between mobility and localization is a sharp one. Consider a specific (enveloping) wavefunction ⇤(r) e r / at site r = 0 with a typical | | ⇥ in the cosmic landscape (see Figure 1(b)). The wavefunction seems to be completely localized at the site if the localization length ⇥ a, where ⇥ measures the size of the ⇥ site and a is the typical spacing between sites. They are determined by local properties and ⇥ is known asofthethe orlodercalizationof ⇤. Thelenngtoneh.expWecithts thratesonanfor L ce ⇤tun, thneline e ectivg, ⇥e condcan uctancebe sombewecomeshat How does g scale ? ⌅ bigger, although exp⇤(ron)enmatiallyy stillsmall:be exponentially suppressed at distance a. At distance Given g at scale a, what is g at scaleL/ L g (L) e (2.2) scales around a, the conductance goeass Llik becomese larged ?⇤ For a small disorder, the medium is in a metallic state and the conductivity ⌅ is independent a/ of the sample size ifgth(ae )size is⇤m(auc)h largere than the mean free path, L l. Conduc(1.3)tance ⌅ of the order of ⇤. Then oneisedeterminedxpects thatin thforis Lcase just⇤,| thbyethee| usualectiveOhm’sconductancelaw and forbecaomesd-dimensional hypercube, 2 ⌅ 2a/ exponentiallyNotesmthalat,l: for a we⇥ha, v e:0 ⇤(a) e . ⇤ | | L/ d 2 Given g = g(a) (1.3) atgdscale(L) ae, what happegnds(Lto) =g⌅atL distance scales(2.2)L a ? That(2.3)is, ⇤ ⇤ how does g scale based? Theonscalina simpgletheorydimensionofalg arguis wmellent.stu(Recdiedall inthatcondensein d = 3,dgmatte= ⌅(Area)r ph/Lysics.⌅LIt.) For a small disorder, the medium is in a metallic state and the conductivity ⌅ is independent ⇤ The conduInsulating,ctance g localized,= g(a) at length scaleConducting,a is a m icroscopic measure of the disorder. We of the samtuprnsle sizoute ifththate siztheree is misucahtrapped,crilargertic aleternalthancon inflationductanthe meance gfreec whicpath,mobileh isL a ful.nctionConduofctadn.ceIf g(a) < gc, then see that g(L) may end up with one of the 2 ver⌅y di erent asymptotic forms for L a. is determinedg(L)in the isL/case just0 asbyLthe usualandOhm’sthe lanlawdscandapfore isaand-dimensionalinsulating mehypdeium.rcube,The wavefunction⌅ ⌅ Here ⌅ ⌅is re⌃scaled so that g is dimensionless. we have: is truly localized. TunnElemeelingntarywsicalinll takg the eexpory ofonenlocalizationtially longassumesandthateternalg of a din-dimensflationionalcomhypeserincutobe d 2 play. If g(a) > ofg ,sizthee L satiscgond(fiLdue)s c=theti⌅vitsimplL y isestfindi iteeren(tigal(Lequ) ation(L/aof a)renor(d 2)m)ali(2.3)aszatiLon group, whereand the c ⌅ ⌃ based onlana sdscimpaple edimeis innsaioncondal arguuctming/mobient. (Reclealpl hasethat.inThed =w3,avgef=und ln⌅c(Area)tigon(L)is/Lfree to⌅Lmo.) ve. Around the ⇥ (g (L)) = d ⇤ (2.4) The condutranctansictione g =poing(at,) attherelengthis sacalecorarelationis a microscopiclengthd thatmd easureblowofsd lnthupLe datisorder.the phaseWe transition. We see that shallg(L) maseey thendat,uatp withthlengthat ions, thescaleeof⇥th-fuaenc,2tthionviser⇥ycorr(dig) edepelatreendniont asymptots onlengthly on theiccanfordimensionlesbmesidforenLtifieds conaduwith. ctan⇥ce. gThat(L). Thenis, ⇥ d ⌅ d Here ⌅ isplreascysalthedesorolethatofgththiseedqucimeoralitativrelationnsionlese behas.levinorgth.of ⇥dTh(g)iscanisbhoe analyw, givzedenin ag(simplea) (1.3wa)y atby inscaleterpolatina, ggcanit betgroweewn Elementary scaling theorytheof2 lilomcalizationiting formsasgivsumesen bytEq.hat(2.2g of) ana d-dimensEq.(2.3).ionalFor thehypinsulercuatingbe phase, (g 0), it large for large L. ⇧ of size L satisfies the simplestfollodwi sefromrentialEq.e(qu2.2ation) and ofEq.(a 2.4renor) thatmali: zation group, where In this paper, we argue that, for large d, the critical conductance is exponentially small, g d ln gd(L) lim ⇥d(g) ln (2.5) ⇥d(gd(L)) = (gd 01) ⇧ gc (2.4) d lngLc e ⇥ which is negative for g 0. F⇧or the conducting phase (large g), it follows from Eq.(2.3) that is, the ⇥-function ⇥d(g) depends only on the⇧dimensionless conductance gd(L). Then So it is not di⌃cultand Eq.(for 2.4g()athat) >:gc; given g(a) (1.3), we see that fast tunneling happens the qualitative behavior of ⇥ (g) can be analyzed in a simple way by interpolating it between 2(d d1) lim ⇥d(g) d 2 (2.6) when 0 > e , or g ⇧ the 2 limiting forms given by Eq.(2.2) and Eq.(2.3). For the⇥⇤insulating phase, (g 0), it which is positive for d > 2. Assuaming the existence of two ⇧perturbation expansions over follows from Eq.(2.2) and Eq.(2.4) that : d > + 1 (1.4) the “coupling” g in the limits of w⇥eak and strong “couplings”, one can write corrections to Eq.(2.5) and Eq.(2.6) in thge following form : lim ⇥d(g) ln (2.5) g 0 ⇧ g so the wavefunction of th⇥e universe canc move freely in gthe quantum landscape even for ⇥d(g 0) = ln (1 + bdg + ) (2.7) which is annegativexpeonforentig ally0.smallFor theg(ac).onductinSincegthpehase -fu(larncgetion⇧g),hasit follagocpwsositfromive sElop·q.(· · 2.3e at) the localization ⇧ d and Eq.(2.4tran) thatsition: point (implying an unstab⇥dle(gfixed)p=oind t),2 the +transition d >be0tween the insulating(2.8) ⇧ ⌃ g · · · (localization) phase and thelim c⇥ond(gdu) ctind g (m2 obile) phase is sharp. We s(2.6)ee that it is precisely Assumingg these and ⇧a smo oth monotonous ⇥ (g), it is easy to plot the ⇥-function qualita- the vastness of the cosmic⇥⇤landscape (as parameterizedd by a large d) that allows mobility which is positive for d > 2. tivAsselyuformingall thg ande exdis, tasenshoce wnof tinwoFigupererturb2. ation expansions over the “coupevlineg”n ifg inththee wliamvefunits ofcFtworioneakd >isand2,lo⇥scalizedtr(gong) mdust“coup. haThivlinesagsmze”,obro:oneilit⇥dc(yangcallo) write= ws0, wherecorrectionsthe gwc aisvetheftouncriticticonal ctonoductansampclee. Eq.(2.5) andthe Elandq.(2.6scap) ine thveeryThefolquloslopwicinkeglyatform. theIt zealso:ro is mpositiveanse,asosthemi-is zeclassro of ⇥icald(g)descorrcripespondstiontoofanthunstablee landfisxedcappeoinist of Eq.(2.4). This means that a small positive or negative departure from the zero will lead inadequate. g asym⇥ptoticallyd(g 0)to=velnry di(1 er+enbtdbge+haviors) of the conductance. As(2.7)g moves from g > gc to In the mobility phase,⇧ the wavgefuc nction an· · d/or· the D3-branes (and moduli) move g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the down the landscape to a site with da smaller ⇥. At that site, some of its neighboring sites ⇥d(g mobilit)y=edge.d 2 + d > 0 (2.8) have larger ⇥s, so⇧the⌃e⇤ectiv e a ingcrease· · ·s, leading to a larger e⇤ective critical conductance Assuminggcthes. The isandhapa spmoensothunmonotontil the cousonditi⇥d(ong), (it1.4is )easisynoto lonplotgether sat⇥-fuisfinctioned anqdualitthea-D3-brane is stuck tively for all g and d, as shown in Figure 2. – 8 – For d > 2, ⇥d(g) must have a zero: ⇥d(gc) = 0, where gc is the critical conductance. The slope at the zero is positive, so this zero of ⇥d(g) corresponds to an unstable fixed point of Eq.(2.4). This means that a small positive or negative departure from the zero will lead – 4 – asymptotically to very di erent behaviors of the conductance. As g moves from g > gc to g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge.
– 8 – of the order of ⇤. Then one expects that for L ⇤, the e ective conductance becomes ⌅ exponentially small: L/ g (L) e (2.2) d ⇤ of the orofdertheof or⇤.derThofen ⇤one. Tehxepneconets theatxpecfortsLthat⇤for, thLe e ectiv⇤, the econde euctancective condbecuctanceomes becomes For a small disorder, the me⌅dium is in⌅a metallic state and the conductivity ⌅ is independent exponentiallyexponsmentiallyall: smofalthl:e sample size if the size is much larger than the mean free path, L l. Conductance L/ L/ ⌅ is determined gind(thL)is casee g j(ustL) byethe usual Ohm’s law and for(2.2)a d-dimensional(2.2)hypercube, ⇤ d ⇤ For a small disorder, the mewe dhaiumve:is in a metallic state and the conductivity ⌅ is independent For a small disorder, the medium is in a metallic state and the condd 2 uctivity ⌅ is independent of the sample size if the size is much larger than the mean freegd(pLath,) = L⌅L l. Conductance (2.3) of the sample size if the size is much larger than the mean free⌅ path, L l. Conductance is determined in this casebasedjust bony thea simusualple dOhm’simensionlawalandarguformenat.d-dimensional(Recall thathinypde⌅r=cub3,e,g = ⌅(Area)/L ⌅L.) is determined in this case just by the usual Ohm’s law and for a d-dimensional hypercube, ⇤ we have:we have: The conductance g = g(a) at length scale a is a microscopic measure of the disorder. We d 2 see that g(L)gmad(Ly) end= ⌅Lup with onedof2 the 2 very di erent asymptot(2.3) ic forms for L a. gd(L) = ⌅L (2.3) ⌅ based on a simple dimenHeresional⌅ isargurescmalenet.d so(Recthatallgthatis dimein ndsi=onles3, s.g = ⌅(Area)/L ⌅L.) of basedthe orondera simofple⇤.dimeTnhseionn alonearguemxenpect. t(Recs thalatl thatforinLd = 3,⇤g, =th⇤⌅e(Area)e e/Lctive⌅Lcond.) uctance becomes The conductance g = g(a) atEllengthementaryscalescalina isgathmeoryicroscopicof localizationmeasureasofsumes⌅the dtisorder.hat g of aWde-dimens⇤ional hypercube The conductancofe sizg =e Lg(satisa) atfielengths the simplscaleestadisi aeremniticalroscopicequationmeasureof a renorof mthalie zdatiisorder.on groupW,ewhere see thexpat gon(L)enmatiallyy end usmp withall: one of the 2 very di erent asymptotic forms for L a. see that g(L) may end up with one of the 2 very di erent asymptotL/ ic for⌅ms for L a. Here ⌅ is rescaled so that g is dimensionless. gd(L) ed ln gd(L) ⌅ (2.2) Here ⌅ is rescaled so that g is dimensionless. ⇥d(gd(L)) = (2.4) Elementary scaling theory of localization assumes that g of a d⇤-dimensd lnionalL hypercube Elementary scaling theory of localization assumes that g of a d-dimensional hypercube of sizFeorL satisa smfiesalthel dsimplisorder,thatestisd, ith theere⇥en-futimealncedtquioniumation⇥ (isg)ofdepina renoraendmes montallalilyzationiconthestategroupdimensionlesand, wherethes concondductanuctivitce g (Ly).⌅Thenis independent 2THEAUTHORof size L satisfies the simplest di erentid al equation of a renormalization group, where d of the sample siztheequifalitativtheesizbehaeviisordmofln⇥ucgd((ghL))clargeran be analythanzed intha esimplemeanwayfreeby interppath,olatinLg it betwl.eeCondun ctance What is eternal inflation ? ⇥ (g (L)) = d (2.4) the 2 limitingd dforms givend lnbyLEq.d(2.2ln)gdan(Ld)Eq.(2.3). For the insulating phase, (g⌅ 0), it is determined in this case just⇥d(bgdy(Lthe)) = usual Ohm’s law and for a d-dimensional(2.4) ⇧ hypercube, of the order of ⇤. Then onefolloewxspfromects Eq.that(2.2fora)(tand)L eEHtq.(⇤,2.4th)ethatdeln eL:ctive conductance becomes that wis,ethhae ⇥v-fue:nction ⇥d(g) depends only on the⇠⌅dimensionless conductance gd(L). Then exponentiallythatsmisal, thl: e ⇥-function ⇥d(g) depends only on the dimensionless gconductance gd(L). Then the qualitative behavior of ⇥d(g) can be analyz2ed in a simple wlimay b⇥yd(ing)terplnolating it between (2.5) H 1L//G ⇤ g 0 ⇧ d g2 the 2 limthiteingqufalitativorms give enbehabyviEq.or of(2.2⇥dg)(dang()Ldc)anEqb⇠.(ee 2.3analy). Fzoredgthedin(⇥Lainsulsimple) =ating⌅wLapyhase bcy ,in(terpg olatin0), (2.2)itg it between (2.3) Consider a patch of size 1/H. ⇤ ⇧ follows fromthe 2Eq.lim(2.2iting) andformswhicEq.(hgiv2.4is enn)ethatgativby Eq.:e for(2.2g ) an0.d FEqor.(the2.3).conFductinor theg insulphaseating(largephaseg), it, (follg ows0),fromit Eq.(2.3) For a small Supposedisorder,T the3/Hme.dium is in a metallic state⇧ and the conductivity ⌅ is independent ⇧ basedfollowons froma⇠sEq.im(p2.2le) dandimeEq.(ns2.4ion) althatargu: 3 3men9 t. (Recall that in d = 3, g = ⌅(Area)/L ⌅L.) of the sampThele siz universee if the wouldsizande is haveEq.(muc grown2.4h)largerthat by a: than factorth ofege me· =ane free. path, L l. Conductance ⇤ If inflation ends in one patch,lim there⇥d are(g) stillln many other ⌅ (2.5) The conductance g = gg(a0) at ⇧lengthg scalelimga⇥d(isg) a md ic2roscopic measure of th(2.6)e disorder. We is determined in this case just by the usual⇥ Ohm’s lacw andgfor a d-dimensional⇧ hypercube, (causally disconnected) patches which continuelim ⇥ tod( inflate.g) ln⇥⇤ (2.5) g 0 ⇧ g we whichave:hseiseSonthegativ inflationat egf(orL neverg) whicma ends.0.yhFisorendptheositiucvonpe ductinforwithd ⇥>g 2.ponhaseAsse ofu(larmithgeng egthc),e2itexvfollisetreoynwscedifofrom tewroEeq.(pnetrt2.3urbasymptot) ation expanicsionfors omvesr for L a. ⇧ d 2 ⌅ the “coupling”gd(gLin) =the⌅liLm itsn of weak and strong “couplings”, one(2.3)can write corrections to and Eq.(whic2.4) hthatis n:egative for g 0. FForor the conL ducting phase (large g), it follows from Eq.(2.3) Here ⌅ is rescaled so⇧that g isg deime nsionless. and Eq.(2.4) thatEq.(: 2.5) andlimEq.(⇥2.6d(g))⇠in thd e fol2 lowing form : (2.6) based on a simple dimensional argumg ent. (g(Rec) ⇧nalln(l g/gthatc) in d = 3, g = ⌅(Area)/L ⌅L.) Elementary scaling⇥⇤theory⇠ of localization assumes that⇤g of a d-dimensional hypercube lim ⇥d(g) d 2 g (2.6) Thewhicconhduiscptanositicevge for= gd(a>) at2. lengthAssumingscaletheaeisxgisatemnciecroscopicof t⇥⇧wdo(gp emrt0)easureurb=ationln of(1expanth+e bddsisorder.gion+s ov)erWe (2.7) of size L satisfies the simplest ⇥⇤di erential⇧equationgc of a renor· · · malization group, where see that g(L) may end up with one of the 2 very di erent asymptotic forms for L a. the “coupwhicling”h isg inpostheitivliemforits ofd w>eak2. andAssustrmonging “coupthe exlinisgsten”,ceoneof ctanw odwriteperturbcorrectionsation expanto sions over ⇥ (g ) = d 2 + > 0⌅ (2.8) HereEq⌅.(2.5is r)eandscaleEdq.(so2.6that) in gthise foldimelowninsgionlesforms.: d ⇧ ⌃ g · · · d the “coupling” g in the limits of weak and strong “couplingsd”,lnonegd(cLan)write corrections to Elementary scaling theory of localization assumes that g of a d-dimensional hypercube Eq.(2.5) and Eq.(Ass2.6um)ining ththese fole andlowina gsmoformg oth⇥d:monoton(gd(L))ous=⇥ (g), it is easy to plot the ⇥-function qualita- (2.4) ⇥d(g 0) = ln (1 + bdg + ) d d ln L (2.7) of size L satisfies the simpltivestelydfori eralelngti⇧alandequd,ationas gshoc ofwnainrenorFigu·m·re·ali2.zation group, where g ⇥ (g ) = d 2⇥d(gd + 0) = ln (1>+0 bdg + ) (2.8) (2.7) that is, the ⇥-fud ncFortiond >⇥2,d(⇥gd()g)depmdust⇧lnendgha(vLse)aonzegcdlyro: on⇥d(gthec) =· ·0,dimensionles· where gc is thescriticoncalduconcductantancece.gd(L). Then ⇧ ⌃⇥ (g ( L)) = g · d· · (2.4) The slope dat tdhe zero is positive, sod this zero of ⇥d(g) corresponds to an unstable fixed point the qualitative behavi⇥dor(g of ⇥d)(=g)ddclnan2Lbe analy+ zed din>a0 simple way by in(2.8)terpolating it between Assuming these and a smoof othEq.(monoton2.4). T⇧hisous⌃me⇥and(sg),th itatisa eassmgaly ltop·ositivplot· · thee or⇥n-fuegatnctionive deparqualittura-e from the zero will lead that is, the ⇥-function ⇥ (g) depends only on the dimensionless conductance g (L). Then tivelythfore al2l lig manditingd, asd asymfshoormswnptoticallyingivFiguenreto2bv.eyryEq.di e(r2.2ent b)ehanaviorsd Eqof .(the2.3cond). uctanceFord the. As ginsulmovesatingfrom gp>hasegc to, (g 0), it Assuming these and a smooth monotonous ⇥d(g), it is easy to plot the ⇥-function qualita- the qualitativFor d >e b2,eha⇥dvi(gor) mofustg⇥
– 8 –
A simple way to get a feeling of this
Recall that there is a bound state for a 1-dim attractive delta potential, but no bound state in the 3-dim case.
andA sphericalFl(x0, ξ)getscomplicatedforlarge square-well attractivel.OnecansolveEq.(A.6)numerically.Forfixed potential in QM : x = k R,asolutiontoEq.(A.6)yieldstheboundstatewithenergyE = k2(1 ξ2)/(2m). and Fl(x0, ξ)getscomplicatedforlarge0 0 l.OnecansolveEq.(A.6)numerically.Forfixed− 0 − In the d =1case,thereisalwaysatleastoneboundstate.Inthed =3case,there x = k R,asolutiontoEq.(A.6)yieldstheboundstatewithenergyE = k2(1 ξ2)/(2m). 0 0 is no bound state if x0 < π/2. For large (odd) d,onefindsthatthereisnoboundstate− 0 − In the d =1case,thereisalwaysatleastoneboundstate.Inthesolution if x0 = k0R is less than a critical value Pc, d =3case,there is no bound state if x < π/2. For large (odd)k0R B. Effect on tunneling due to a change in mass (or brane tension) In the quantum mechanical example in Appendix A, we see that binding is stronger for a more massive particle. Also, the exponential damping of a bound state wavefunction is stronger for a larger mass. It follows that tunneling is suppressed as the mass increases. Here we consider tunneling for a scalar field with similar results. Following [20], let us start with the effective Langrangian density 1 L = ∂ φ∂µφ U(φ)(B.1) 2 µ − λ µ2 2 ' U(φ)= φ2 + (φ a)+Λ (B.2) 8 − λ 2a − p " # –29– of the order of ⇤. Then one expects that for L ⇤, the e ective conductance becomes ⌅ exponentially small: L/ g (L) e (2.2) d ⇤ For a small disorder, the medium is in a metallic state and the conductivity ⌅ is independent of the sample size if the size is much larger than the mean free path, L l. Conductance ⌅ is determined in this case just by the usual Ohm’s law and for a d-dimensional hypercube, we have: d 2 gd(L) = ⌅L (2.3) based on a simple dimensional argument. (Recall that in d = 3, g = ⌅(Area)/L ⌅L.) ⇤ The conductance g = g(a) at length scale a is a microscopic measure of the disorder. We see that g(L) may end up with one of the 2 very di erent asymptotic forms for L a. ⌅ Here ⌅ is rescaled so that g is dimensionless. Elementary scaling theory of localization assumes that g of a d-dimensional hypercube of size L satisfies the simplest di erential equation of a renormalization group, where Figure 2: The -function of the dimensionless conductance g, d(g) versus ln g, for di erent values d ln gd(L) of g and d. The cases for⇥ (dg <(L2,)) =d = 2 and d = 3, 4, 5, 6 are shown.(2.4)For g 0, d(g) is linear in d d d ln L ⇧ ln g, and d(g) d 2 as g . There is no fixed point for d < 2. For d > 2, there is always that is, the ⇥-function ⇥⇧d(g) depends on⇧ly on⌃the dimensionless conductance gd(L). Then a fixed point (the zero of the -function at g = gc(d)) and the slope at the fixed point is positive. the quFalitativor largee behad, thviore ofsp⇥acd(igngs) canofbetheanalyfixedzed inpoia simplents forwaadjy bacy inenterpt dolatinvaluesg it arbetewappeen roximately equal in ln g. the 2 limiting forms given by Eq.(2.2) and Eq.(2.3). For the insulating phase, (g 0), it ⇧ follows from Eq.(2.2) and Eq.(2.4) that : g lim ⇥d(g) ln (2.5) The state of a mediumg 0 is su⇧pposgedlc y determined by disorder at microscopic distances Figur⇥e 2: The -function of the dimensionless conductance g, d(g) versus ln g, for di erent values whichofis thnegative orde eforr ofg typi0. cFalor thespacingconductina bgepthaseween(larneighge g),bitorifollngowssifteroms. EUsingq.(2.3) g(a) as an initial value ⇧ of g and d. The cases for d < 2, d = 2 and d = 3, 4, 5, 6 are shown. For g 0, d(g) is linear in and Eq.(and2.4in) tegratinthat : g Eq.(2.4), it is easy to find its properties for the 2 cases : ⇧ ln g, and d(g) d 2 as Figurg e. 2:ThereThe i s-funonctionfixed pofoithent fordimdension< 2. leForss condd > uctan2, thecre gis, alwd(ga)ysversus ln g, for di erent values lim ⇥d(g) d 2 2 d (2.6) For g(a) > gc, the condg uctivity⇧⇤ ⇧ = g( L)L tend⇧ s⌃for L to a constant non-zero • a fixed⇥⇤point (the zero of thofe g-fuandnctiond. ⇧Tath⌃egcas= esgc(ford)) dand< 2,theds=lop2eandat thde=fixed3, 4,p5oin, 6tareis pshoositwn.ive. For g 0, d(g) is linear in whichvalue.is positive for d > 2. AssForumilargeng the dex, isthtenecespofactwingso pertofurbtationhe fixedexpanpsionoins tsovefror adjacent d values are approximately equal in ln g. ⇧ the “coupling” g in the limits of weak and strong “couplings”, one clnangwrite, ancorrectionsd d(g) to d 2 as g . There is no fixed point for d < 2. For d > 2, there is always For g(a) < gc in the limit of L we get insulating behavior⇧; that is, ⇤ = 0.⇧ ⌃ Eq.(2.5• ) and Eq.(2.6) in the following form : ⇧ ⌃ a fixed point (the zero of the -function at g = gc(d)) and the slope at the fixed point is positive. We see that the behavior of d(g) close to its zero determines the critical behavior at the g For large d, the spacings of the fixed points for adjacent d values are approximately equal in ln g. transition.of theThereorderis a⇥sofdh(garp⇤.0)tranT= hlnseitionn(1one+forbdg +ge(xap))eclargerts thoratsmallerfor (2.7)Lthan g⇤c,. the e ective conductance becomes The⇧ stategc of a me·d· ·ium is supposedly⌅determined by disorder at microscopic distances For g g , (g) is linear in lnd g (see Figure 2), so we have the following approximation: exponce⇥nd(tiallygd ) =smd al2l: + d > 0 (2.8) ⇤ ⇧of⌃ the ord egr of· ·t·ypical spacing a between neighboring sites. Using g(a) as an initial value 1 g 1 g g L/ Assuming these and a smooth monotonous ⇥ (g), it is easy to plot the ⇥g-fudnction(Lc ) qualite a- (2.2) and integratind d(gg) Eq.(ln2.4), it isThe easystateto⇤ findof itsa mepropdiumertiesis sfuor(2.9)pptheosedl2 cyasesdete:rmined by disorder at microscopic distances tively for all g and d, as shown in Figure 2. ⌅ ⇥ gc ⌅ ⇥ gc For g(a) > g , the condofuctivitthe ordy ⇤e=r ofg(tLypi)Lc2ald spacingtends fora Lbetweentoneigha cobnoristanngt snonites.-zeUsingro g(a) as an initial value For d > 2,For⇥d(ga) smmustalhalvde isorder,a zero: ⇥d(gthc) =e 0,mec wherediumgc isistheincritiacmeal contallductanic statece. and the conductivity ⌅ is independent where 1/⇥ is positiv•e and is equal to the bracket part of Eq.(2.7) evaluated at gc. First we⇧ ⌃ The slope at tofhe zethroeis sampositivpe,vlealue.so thsizis ezeroifofth⇥de(g)sizcorreespisondsmuctoandanhulargernstableintegratinfixedthanpoigntthEq.(e me2.4an), itfreeis easpath,y toLfind lits. Condupropertiesctancfore the 2 cases : of Eq.(consider2.4). Thisthmeeancass theatwherea smallgp(ositiva) >e orgcn. egatSinceive depar d(gtur) >e fr0,omththe zfleroowwillis leadto large g, so we integrate ⌅ 2 d For g(a) < gc in the limitFoforLg(a) > wgce, gethteincondsulatinuctivitg behay ⇤vior= g; (tLhat)Lis, ⇤tend= 0.s for L to a constant non-zero of the order of ⇤asym. Tptoticallyhenisonetodeterminedveryedxi perecenttbsehithanviorsatthofisforthecasecondL juctanceust⇤b.,yAsththegemoevusuales efromctivgOhm’s>e gccondto lauctancew and forbeca domes-dimensional hypercube, Eq.(2.4) to find g(L•). We can approximate⌅ the in•tegral b⇧y first⌃ integrating Eq.(2.4) using ⇧ ⌃ g < gc, mobility is lost and theWwaevefuseenctionthatis trthulyelobcalizeehadv. Thisiorvsharpofalue. dtran(g)sitioncloseis thtoe its zero determines the critical behavior at the exponentially small:Eq.(2.9w)eunhatilv e:d(g) reaches d 2 at g⇥, then we use d(g) = d 2 (Eq.(2.6)) to reach large mobility edge. transition. ThereL/is a sharpFtranor g(saition) < gforc ding2(tha)elargerlimit orof sLmaller thweangegtc.insulating behavior; that is, ⇤ = 0. gd(L) e • gd(L) = ⌅L (2.2)⇧ ⌃ (2.3) For g ⇤gc, d(g) is linearWe sineelnthgat(sethe eFigurbeheav2ior), soofw e ha(gv)ecthlosee foltoloitswingzeroappdeteroximationrmines :the critical behavior at the ⇤ d For a small disorder, the mebaseddiumonisainsima pmele talldimeicnstatesionalandargutranthemenscondition.t. (RecuctivitThereall ythatis⌅aisshiniarpndepd =trane3,nsdenigtion=t for⌅(Area)g(a) /Llarger ⌅orLs.)maller than g . Insulating,– 8 –localized, 1 g 1 g g c – 9 – (g) ln c ⇤ (2.9) of the sample size if the sizThee isconmduucctrapped,htanlargerce geternal=thang (inflationa)thate lengthmean freescaleFConducting,ordpgaath,isgac,L mobilem di(cgroscopic)l.isCondulinearmineasurectlnangc(seeofe Fthigure deisorder.2), so weWhae ve the following approximation: ⇤ ⌅ ⇥ gc ⌅ ⇥ gc see that g(L) may end up with one of the 2 very⌅di erent asymptotic forms for L a. is determined in this case just by the usual Ohm’s law and for a d-dimensional hypercube, 1 g 1 g g where 1/⇥ is positive and is equal to the bracket part of Eq .((2.7g) ) evaluatedln at gc⌅. Firc st we (2.9) we have: Here ⌅ is rescaled so that g is dimensionless. d ⌅ ⇥ g ⌅ ⇥ g consider the case where g(a) > gc. Since d(g) > 0, the flow is to large cg, so we inc tegrate Elementary scaling thedory2 of localization assumes that g of a d-dimensional hypercube Eq.(g2.4d(L) )to=find⌅Lg( L). We can approximate the integral by(2.3)first integrating Eq.(2.4) using of size L satisfies the simplest di ewhererential1/e⇥quisationpositiofveaanrenord is mequalalizatito onthegroupbrack,etwherepart of Eq.(2.7) evaluated at gc. First we Eq.(2.9) until (g) reaches d 2 at g , then we use (g) = d 2 (Eq.(2.6)) to reach large based on a simple dimensional argument. (Recald l that inconsiderd = 3,theg⇥cas= e⌅where(Area)g/L(ad) > g⌅c.LSi .)nce d(g) > 0, the flow is to large g, so we integrate Eq.(2.4) to fidndlngg(dL().L)We can⇤approximate the integral by first integrating Eq.(2.4) using The conductance g = g(a) at length scale a is a microscopic⇥d(gd(mLeasure)) = of the disorder. We (2.4) Eq.(2.9) until d(lng)Lreaches d 2 at g , then we use (g) = d 2 (Eq.(2.6)) to reach large see that g(L) may end up with one of the 2 very di erent asymptoticd forms for L a.⇥ d ⌅ Here ⌅ is rescaled so thatthgatisisd, imethe n⇥s-fuionlesnctions. ⇥d(g) depends only on the dimensionles– 9 – s conductance gd(L). Then Elementary scaling ththeeoryqualitativof localizatione behaviorasofsumes⇥d(g)tchatan bge analyof a dz-dimensed in a simpleional hwypayebrycuinbterpe olating it between the 2 limiting forms given by Eq.(2.2) and Eq.(2.3). For the insulating phase, (g 0), it of size L satisfies the simplest di erential equation of a renormalization group, where – 9 – ⇧ follows from Eq.(2.2) and Eq.(2.4) that : d ln gd(L) g ⇥d(gd(L)) = lim ⇥d(g) ln (2.4) (2.5) d ln L g 0 ⇧ g ⇥ c that is, the ⇥-function ⇥dwhic(g) hdepis endnegativs onelyforong the0.dimensionlesFor the consductincondugcptanhasece(largd(geL).g),Thenit follows from Eq.(2.3) ⇧ the qualitative behavior ofand⇥d(Eq.(g) c2.4an)bthate analy: zed in a simple way by interpolating it between the 2 limiting forms given by Eq.(2.2) and Eq.(2.3). For theliminsul⇥d(gating) d phase2 , (g 0), it (2.6) g ⇧ ⇧ follows from Eq.(2.2) and Eq.(2.4) that : ⇥⇤ which is positive for d > 2. Assuming the existence of two perturbation expansions over g the “coupling”limg ⇥ind(theg) limlnits of weak and strong “couplings”, one(2.5)can write corrections to g 0 ⇧ g Eq.(2.5) and E⇥q.(2.6) in the folc lowing form : which is negative for g 0. For the conducting phase (large g), it follg ows from Eq.(2.3) ⇧ ⇥d(g 0) = ln (1 + bdg + ) (2.7) and Eq.(2.4) that : ⇧ gc · · · d lim ⇥d(g) ⇥ (gd 2 ) = d 2 + > 0(2.6) (2.8) g d d ⇥⇤ ⇧ ⇧ ⌃ g · · · which is positive for d > 2. Assuming the existence of two perturbation expansions over Assuming these and a smooth monotonous ⇥d(g), it is easy to plot the ⇥-function qualita- the “coupling” g in the litivmielyts offorwaleakl g andand sdtr, asongsho“coupwn inlinFigugs”,reone2. can write corrections to Eq.(2.5) and Eq.(2.6) in the following form : For d > 2, ⇥d(g) must have a zero: ⇥d(gc) = 0, where gc is the critical conductance. The slope at the zero is positivg e, so this zero of ⇥d(g) corresponds to an unstable fixed point ⇥d(g 0) = ln (1 + bdg + ) (2.7) of Eq.(2.4). Th⇧is means thgatc a small positiv· · · e or negative departure from the zero will lead asymptotically to very ddi erent behaviors of the conductance. As g moves from g > gc to ⇥d(g ) = d 2 + d > 0 (2.8) g <⇧gc,⌃mobility is los t andg the· · ·wavefunction is truly localized. This sharp transition is the mobility edge. Assuming these and a smooth monotonous ⇥d(g), it is easy to plot the ⇥-function qualita- tively for all g and d, as shown in Figure 2. For d > 2, ⇥d(g) must have a zero: ⇥d(gc) = 0, where gc is the critical conductance. The slope at the zero is positive, so this zero of ⇥d(g) corresponds–to8 an– unstable fixed point of Eq.(2.4). This means that a small positive or negative departure from the zero will lead asymptotically to very di erent behaviors of the conductance. As g moves from g > gc to g < gc, mobility is lost and the wavefunction is truly localized. This sharp transition is the mobility edge. – 8 – Now we have: g(a) g ⇥ (g g ) (d 2) c (2.18) d ⇥ c ⇤ g c ⇥ and for the critical exponent of localization length in Eq.(2.16), we get : 1 ⌅ (2.19) ⇤ d 2 which may be considered as the first term of the ⇤-expansion from d = 2 (where ⇤ = d 2), i.e., near “lower critical dimension” for localization. We see from Eq.(2.17) that gc depends on d, which is clearly model-dependent. For a Fermi gas with spin-1/2 particles in 3-dimensions (d = 3), = ⇧ 2, so g 1/⇧2 [23]. 3 c ⇤ Here we are interested in gc’s dependence on d for large d. For tunneling in the landscape to be fast when above the transition, it would mean that the critical conductance g for large d 102 should be exponentially small. That is, the Now we have: c ⇤ landscape should be in the conducting phase evegn(afor) angc exponentially small microscopic ⇥d(g gc) (d 2) (2.18) Noconduw wecthaancve:e g(a). Noting that⇥gc ⇤1 for d = 3., wegcsee that gc as a function of d should ⇥ g(a) gc⇥ anddecrefor thease ecxriticalponenetixpallyonen. Atglanceof⇥d(logcalizationofgcFi) gure(dle2nsuggestsgth2) in Eq.(t hat2.16th),e zwerose geoft :⇥d (as a function(2.18)of ⇥ ⇤ gc d) are approximately equally spaced in ln g, implyi ng that g⇥c should decrease exponentially 1 andas aforfuthenctioncriticalof d.expIf wonene let of localization⌅ length in Eq.(2.16), we get : (2.19) ⇤ d 2 1 ln gc = ln gc(⌅d) ln gc(d + 1) = k > 0 (2.19)(2.20) which may be considered as the first term of the ⇤-expansion from d = 2 (where ⇤ = d 2), ⇤ d 2 i.e., near “lower critical dimension” for localizat ion. whicwe hcanmaexpy bree cssongsci(dered) ind termsas theoffirstgc(3)term(forofdthe= 3)⇤-e,xpansion from d = 2 (where ⇤ = d 2), We see from Eq.(2.17) that gc depends on d, which is clearly model-dependent. For i.e., near “lower critical dimension” for localiz(dat3)ion.k 2 2 g 0, ⇥adF(ermig) gaslnw(gith/gspinc), -1/2repropartiducingcles gincEq.((3-dimensionsd) 2.5e ). Forg(cd(3)g= 3), 3,=⇥⇧d (g,)so gcd 1/2,⇧ r(2.21)[e23p].roducing ⇧ W⇧e see from Eq.(2.17) that gc depen⌅ds on d, which⇧is ⌃clearly mode⇧l-dep⇤e ndent. For Eq.(2.6).HereThewecritiare icnalterexpestedonenin gct’scdanependencebe expronessdedforinlargetermsd. of gc, 2 2 a AFermisimpgasle glwanceith spinat Figu-1/2reparti2 suggescles ints3-dimensionsthat k 1. (Tdo=b3),e more 3 =sp⇧e cifi, cs,o wgce lik1e/to⇧ kno[23].w For tunneling ingthe 0,land⇥d(sgcap) e lnto(gb/ge cfast), rewhenprod⇥ucingabovEq.(e the2.5tr).ansitionFor g , it ,w⇥ouldd(g)⇤medan that2, reproducing Herequanwtitatie areveinlytethreesteddep⇧ inendgce’sncde⇧eofpendencegc on d.1on2It+disgforlickellyargethatd.this speci⇧fic⌃qualitati⇧ve prop erty the critical conductanceEq.(2.6gc). forThelarcriti⌅ge=cald exp10onenshoult candbebexpre expessonened intialltermsy small.of gc, That is, the (3.2) is inseForntusinntiveelitongthinethdetailandls sofcapaneytoparb1⇤eticularfast(dwhenmo2)dabegl.coInve thethe trnextansitionsection, it,wwoulde shalmel eanxamthatine landscape should be in the conducting phase2even for an 1exp+ gonenc tially small microscopic thae scimpriticalle moconddeluctanceto obtaingc thfore qularangetitatd iv10e dep shoulendence⌅d=be ofexpgconenon dtiall. y small. That is, the (3.2) conductance g(a). Noting that gc 1 for⇤ d = 3., we se1e that(d g2)c gasc a function of d should In thelan⇤dsc-expapeansionshould, bfeorinsmallthe con⇤du=⇥ctidng phase2 > 0,evenonefor solvan expes onenfor tiallthey zerosmall ofmicroscthe op⇥-fuic nction decrease exponentially. InA theglance⇤-expofansionFigure , for2 smallsuggests⇤ = dthat2 th> e0,zoneerossolvof es⇥dfor(asthea fzerounctionof thofe ⇥-function (3.1) to obtaicondu3. nEcxptgancc o=nee g1n(/at2).ia⇤.llNotinyThSmalegnthatElq.(Cgrc3.2itical)1 rforeCcodonductv=er3.,s twheancese sev aluee thatofgcthaseacfuriticalnctionexpof doneshoulntd⌅ = 1/⇤ d) are approximately(3.1equall) to obtaiy spacn gedc =⇥in1/ln2⇤g. ,TimhenplyiEq.(ng3.2that) recogcveshoulrs thedvaluedecreofasetheecxpriticalonenexptialonelynt ⌅ = 1/⇤ in Eq.(2.19decre). aseWeexspeeonthentiatinallyEth.q.(isA2.19glanceis).aWseimplofseeFithgureeatmth2oisdelsuggestsis athsimplatthatermepothdelroedthzuceerosat rseofpthro⇥deduce(asapps tharoprefunctionappropriateiateoffeatufeaturesres as aConsiderfunctionaoflatticed. Ifmowedleeltof d-dimensional disordered system studied by Shapiro [22]. Since d) are approximatelyexpequallected. y spaced in ln g, implying that gc should decrease exponentially expected. What is the critical gc ? aswae fuarnctione intereofstedd. inIf wthComeeleasymptotitparing to cEq.(beha2.17vior), itofgivgcesas a function1/2. Forof d,=wh3,eraenthumericalis qualitativanalyesis of this ln gc = ln gc(d) ln gc(d + 1)2 = k > 0 (2.20) ComparinfeatugretoshouldEq.(b2.17e univ),ersitalgiv, theesde tails2 of1the/2.moFdorel dod⌅ es= not3, caonncumericalern us. Weanalymay viewsis of this model yields gc = 0.255⌅and ⌅ = 1.68. This value for the critical exponent ⌅ is higher this model as a simplthaneth inatterplningEolatiq.(= l2.19nong)that(dfor) dsatln=is3gfifr(edoms +allthe1)the=⇤-egeneralkxp>ansion0 p.ropHoewrtevieer,s ethxpisecvalueted.(2.20)is compatible model yiweeldcans gexpc =ress0.g255c(d) anin termsd ⌅ =ofc g1c.(3)68.c(forThi d =s 3)vcalu, e for the critical exponent ⌅ is higher than that in Eq.(2.19) forwithdth=e e3stimatefromofthe1.25 ⇤<-e⌅xp< 1ansion.75 [31],.suHoggestingwevthater, theisleadvalueing ⇤-expisansioncompisatibratherle we3.1canAexpSimplress eg Mo(d) delin terms of g (3) (for d(d=3)3)k , c crude for ⇤ = 1.gcc(d) e gc(3) (2.21) with the estimate of 1.25 < F⌅or – 12 – g 0, ⇥ (g) ln(g/g ), reproducing Eq.(2.5). For g , ⇥ (g) d 2, reproducing ⇧ d ⇧ c ⇧ ⌃ d ⇧ Eq.(2.6). The critical exponent can be expressed in terms of gc, 1 + g ⌅ = c (3.2) 1 (d 2)g c In the ⇤-expansion, for small ⇤ = d 2 > 0, one solves for the zero of the ⇥-function (3.1) to obtain gc = 1/2⇤. Then Eq.(3.2) recovers the value of the critical exponent ⌅ = 1/⇤ in Eq.(2.19). We see that this is a simple model that reproduces the appropriate features expected. Comparing to Eq.(2.17), it gives 1/2. For d = 3, a numerical analysis of this 2 ⌅ model yields gc = 0.255 and ⌅ = 1.68. This value for the critical exponent ⌅ is higher than that in Eq.(2.19) for d = 3 from the ⇤-expansion. However, this value is compatible where the ruwithnnthinge estimateof p isofgi1v.25en 1 + gc ⇤g =⇤ (p ) 1 (3.11) (3.15) p1 2(d 2)g p⇤ c – 13 – That is, for pLet< ups consider, p the0,zeroswhileof forp(p) atp p>=ppc,,wherep pc1.satisWfiees see that d = 1 at the fixed 2 ⇧ 2 ⇧ p point p1 = 0. For such a smallpc dlnpp,c =the(d mediu1)(1 mpc)isln(1insulpc)ating. So one ma(3.y12)conclude that For d 2, there are 3 fixed points, which we shall label as p = 0, p and p = 1 : besides ⇥ 1 2 3 the 2 fixed points, namely p1 = 0 and p3 = 1, there is another fixed point at p = p2 in between; that is, p3 > p2 > p1 or 1 > p2 > 0. For d = 2, it is easy to see that p2 = 1/2. For d = 3, Ref.[22] gives p = 0.16. For large d, we have 2 – 13 – (d 1) p e (3.13) 2 ⌃ Expanding (p) around p , we have, for ⇥ = p p 1, p c | c| ⌅ ln p ⇥ (p) ⇥ d + c = (3.14) p ⇤ 1 p ⇤ c ⇥ p where the critical exponent ⇤p for p is determined in terms of pc. For large d, we see that p2 is an unstable fixed point while p1 = 0 and p3 = 1 are stable fixed points. Using (3.13), we have, for large d, ⇤ (p ) 1 (3.15) p 2 ⇤ That is, for p < p , p 0, while for p > p , p 1. We see that d = 1 at the fixed 2 ⇧ 2 ⇧ p point p1 = 0. For such a small dp, the medium is insulating. So one may conclude that – 13 – 4 4 2 2 ln g ln g 0 0 !2 !2 !4 !4 4 4 4 2 ln g 0 2 2 !2 !4 4 Βg!g,p" Β !g,p" 0 g 0 2 Βg!g,p" !2 0 !2 !2 !4 !4 !4 0.0 0.0 0.0 0.5 p 0.5 0.5 p p 1.0 1.0 1.0 4 2 ln g 0 !2 !4 4 2 Β !g,p" g 0 Figure 2: The β-function βg(g,p)withinclusionofpercolationford =6andtheβ-function βp(p) for the running of p,fordifferent p and d.Theblacklinecorrespondstoβg(g,p)=0. !2 In general, we expect the brane motion in the landscape to include rolling, scatterings!4 and tunneling. Classical scattering in the landscape raisestheissueofpercolation,thatis,0.0 whether such scattering will shut down the long distance mobility. Including this effect, the conductance g also depends on a percolation probability p (0 p 1). Extending the 0.5 ≤ ≤ above analytic formula to include percolation [16] : p 1 p 1.0 β (g, p)=(d 1) 1+ − ln(1 p) (g +1)ln(1+1/g), (2.6) g − p − − Figure 2: TheFigureβ-function! 2: Theββg-function(g,p)withinclusionofpercolationfor" βg(g,p)withinclusionofpercolationford =6andthed =6andtheβ-function ββp-function(p) βp(p) for the runningfor the of p running,fordiff oferentp,fordip andfferentd.Theblacklinecorrespondstop and d.Theblacklinecorrespondstoβg(g,p)=0.βg(g,p)=0. –8– In general,In we general, expect thewe expect brane motion the brane in the motion landscape in the tolandscape include torolling, include scatterings rolling, scatterings and tunneling.and tunneling. Classical scattering Classical inscattering the landscape in the raiselandscapestheissueofpercolation,thatis, raisestheissueofpercolation,thatis, whether suchwhether scattering such scattering will shut down will shut the long down distance the long mob distanceility. Including mobility. this Including effect, this effect, the conductancethe conductanceg also dependsg also on depends a percolation on a percolation probability probabilityp (0 p p1).(0 Extendingp 1). Extending the the ≤ ≤ ≤ ≤ above analyticabove formula analytic to formula include to percolation include percolation [16] : [16] : 1 p 1 p βg(g, p)=(βd (g,1) p)=(1+d −1) ln(11+ −p) ln(1(g +1)ln(1+1p) (g +1)ln(1+1/g), /g)(2.6), (2.6) g − −p − p − − − ! ! " " Figure 2: The β-function βg(g,p)withinclusionofpercolationford =6andtheβ-function βp(p) for the running of p,fordifferent p and d.Theblacklinecorrespondstoβg(g,p)=0. –8– –8– In general, we expect the brane motion in the landscape to include rolling, scatterings and tunneling. Classical scattering in the landscape raisestheissueofpercolation,thatis, whether such scattering will shut down the long distance mobility. Including this effect, the conductance g also depends on a percolation probability p (0 p 1). Extending the ≤ ≤ above analytic formula to include percolation [16] : 1 p β (g, p)=(d 1) 1+ − ln(1 p) (g +1)ln(1+1/g), (2.6) g − p − − ! " –8– g 0, ⇥ (g) ln(g/g ), reproducing Eq.(2.5). For g , ⇥ (g) d 2, reproducing ⇧ d ⇧ c ⇧ ⌃ d ⇧ Eq.(2.6). Theis an critiunstabclealfixeexpd poinonent, implt cyinang thbate theexprtransitionessed binetwtermseen mobilitofy gancd, localization is a sharp one. r / Consider a specific (enveloping) wavefunction1⇤+(r)gc e | | at site r = 0 with a typical ⇥ in the cosmic landscape (see Figur⌅ e=1(b)). The wavefunction seems to be completely (3.2) 1 (d 2)gc localized at the site if the localization length ⇥ a, where ⇥ measures the size of the ⇥ In the site⇤-eandxpansiona is the t,ypficoral smallspacing b⇤et=weend sites.2 Th>e0,y areonedetesolvrmineesd byforlocalthepropzeroerties of the ⇥-function and ⇥ is known as the localization length. With resonance tunneling, ⇥ can be somewhat (3.1) to obtaibiggenr,galthc =ough1/2⇤⇤(.r) TmahyenstillEbq.(e e3.2xpon)enrteiallycovseurpprs etssheed vataluedistanofce tha. eActridisticaltance exponent ⌅ = 1/⇤ in Eq.(2.19sc).alesWaroue sndeea,ththeatconthduisctanisceagoseimpls like e model that reproduces the appropriate features a/ expected. g(a) ⇤(a) e (1.3) | | ComparinNote gthat,toforEq.(a 2.17⇥, ), ⇤it(a)giv2 ese 2 a/ 2. 1/2. For d = 3, a numerical analysis of this ⇤ 0 | | ⌅ g 0, ⇥ (g) Givln(gen/gg),=regp(aro)d(ucing1.3) atEq.(scale2.5a)., whatFor ghappen, s⇥to(gg) at ddistance2, rescproalesducingL a ? That is, model⇧ yidelds⇧ gc =c 0.255 and ⌅ = 1.68.⇧Thi⌃ sd valu⇧ e for the crit⇤ ical exponent ⌅ is higher Eq.(2.6). Thehowcritidoceals gexpsconenale ?t Thecan bscaline exprg etheoryssed inoftermsg is wofellgcstu, died in condensed matter physics. It than that inturnsEoutq.(2.19that there) foris adcri=tical3 cfronomductanthece g ⇤whic-exph isansiona function. Hoof dw. Ifevger,(a) – 4 – at a small ⇥ site. This transition takes place at a critical ⇥c. The mobility shuts o⇤ for The Quantum Landscape ⇥ < ⇥c, and the universe ends up atataasitesmallwith⇥asismte.allTh⇥is Consider a particle moving in a discordered (random potential) medium. The phase of its wavefunction varies randomly. The distance over which it fluctuates by 2 defines the – 5 – mean free path l, the microscopic length scale of interest here. When the disorder is small, – 5 – Behavior of the wavefunction manifold to another. Its motion in the landscape involves repeated fast tunneling, quantum hopping, scattering and ordinary rolling, may be even some slow-roll. The potential for the inflation in this scenario is illustrated in Fig. 4. In this figure, Figure 4: Cartoon for the tunneling path of the universe in the string landscape. The universe rolls, percolates and tunnels down the landscape. TunnelingisfastatsiteswithΛ > Λc,and exponentially slow at sites with Λ < Λc,whereΛc is exponentially small compared to the Planck scale. It is important to note that the actual path is multi-dimensional and the scale of this picture is not accurate. we (over)simplify the picture of tunneling/percolation path of our universe to be one- dimensional. This picture is similar in looks to that proposed in Ref.[28], where a saltatory relaxation (i.e., relaxation by jumps) of the cosmological constant takes place as the uni- verse tunnels repeatedly down the potential. Here we give an explicit dynamical realization of this behavior that happens in the landscape due to its vastness. Despite the presence of classically local minima, the wavefunction may simply roll past them since these minima are either too weak to trap the wavefunction, or strong enoughtotrapthewavefunction, but the trapping is so weak that the wavefunction can easily tunnel out. (For a D3-brane moving down such a random potential, its mobility would be further enhanced by its ki- netic energy as well as bulk radiation, since these tend to further suppress binding. The DBI action may significantly enhance the tunneling rate [25].Inaddition,asshownin Eq.(3.17), the tunneling is faster when the brane is higher upinthelandscape.Onthe other hand, brane radiation, cosmological expansion of the universe and other damping mechanisms would tend to counteract, providing an interesting non-trivial dynamics.) The –20– the medium is insulating for p < p2. Following Eq.(3.13), we see that the attractive region p > p 0 for the insulating phase is exponentially small. 2 ⇥ For p > p2, depending on the initial microscopic values of g and p, the medium is in an insulating phase with g 0 when g(p) < g (p), or, in a conducting phase when ⇤ c g(p) > g (p) : (g) d 2, where g (p) is given by Eq.(3.9). Since p 0 as d , we c ⇤ c 2 ⇤ ⇤ ⌅ see that the condition for the conducting/mobile phase is quite insensitive to disorder at the mesoscopic scale. We see that there are universality properties that one may extract from the cosmic landscape that is most likely insensitive to any of the details. However, a better knowledge of its structures can be important for finding out which universality class the renormaliza- tion group properties of the cosmic landsthcape meedbiumelonis ingssuto.lating for p < p2. Following Eq.(3.13), we see that the attractive region p > p 0 for the insulating phase is exponentially small. 2 ⇥ For p > p2, depending on the initial microscopic values of g and p, the medium is 4. A Critical Cosmological Constinanant insulating phase with g 0 when g(p) < g (p), or, in a conducting phase when ⇤ c the medium is insulating for p < p g.(pF)ol>logw(ipng) :E q.((g)3.13d), w2,e sewheree thatg (thep) isattrgivacentivbye Eregionq.(3.9). Since p 0 as d , we 2 c ⇤ c 2 ⇤ ⇤ ⌅ Now we likep2to>spee h0oforw athecrinsuliticalating pcanhaseseemise exptherge.atonentheIttiallcondisyitionreassmall.onforablethe contoduasctinsumeg/mobthatile phasetunnisequilingte insensitive to disorder at ⇥ c For p > p , depending on thethinite mialesosmiccopicrosscalecopi.c values of g and p, the medium is takes place from a site with2 1 > 0 only to any other sites with which is smaller but semi- in an insulating phase with g 0 whenWe seg(epthat) < tghere(p),aror,e uinniversalita condyuctingpropertiesphasethatwhoneen may extract from the cosmic positive, that is, 0. For the di⇤scussion to be apc plicable to general uncompactified g(p) > g1(p) : (g) d 2, wherelang dsc(p)apisegivthatenisbymosEq.(t li3.9kely).inseSinncseitivp e to an0 asy ofdthe details., we However, a better knowledge c ⇥ ˆ⇥ ⇤ c 2 ⇤ ⇤ ⌅ spatial dimensions,see that theletthconde meitiondesdiumignateforis thinesuthclatinoneduofmgciforasststinstrug/mps So we see that the avoidance– 15of–eternal inflation in the early universe is probably automatic when the universe is in the conducting component of the landscape, i.e., when ⇤ > ⇤c. So it is reasonable to assume that no eternal inflation happens during the mobile phase. – 15 – Big assumption • Tunneling or evolving to AdS vacua ? History of our Universe - A Speculation Picture Taking a crude look at the whole Gell-mann • Parts can be replaced by better parts later, as our understanding gets better, with more input from data. Quantum Creation of the Universe “Tunneling” from Nothing Time Lorentzian deSitter space "1/2 Radius ! Euclidean 4 S Nothing Vilenkin, 1982, 1983 Hartle and Hawking, 1983 BRIEF ARTICLE BRIEF ARTICLE THE AUTHOR THE AUTHOR (1) P = e3⇡/G⇤ (1) • This implies thatP the =moste likely3⇡/G ones⇤ are the ones with extremely small CC. However, the size of the de-Sitter space goes like (2) 1/⇤1/2 Classical effects must come into play for such a large universe. That means we should include decoherence effect. 1 1 What is wrong ? How to fix it ? • Tunneling is to a huge super-macroscopic universe, with nothing in it. • Interacting with the environment suppresses tunneling. • There must be some gravitational radiation. • Its interaction with the system (cosmic scale factor a) introduces decoherence. • Smaller CC means larger universe means more radiation implies more decoherence thus suppressing the tunneling. The improved Euclidean action SE,dC vs SE,0 The Improved Wavefunction breakdown of semiclassical approximation for large ! S E,dC Tunneling probability The Improved Wavefunction 0 −S F F ! P ! eP E,dCe = e P eF ⇥ ⇥ 12n 3 12nd 3 d F = FG = 2l4 2 4 s G ls S E,0 SE,0 is unbounded from below, but the interaction with the environment has made S = F E,dC S is unboundedbounded fromfrobmelow.below, but the interaction Firouzjahi,E,0 Sarangi and H.T., hep-th/0406107, 0505104 with the environment has made S = F E,dC bounded from below. Tunneling from Nothing, including decoherence Time Lorentzian Gravitational Perturbative Mode 1!loop Effect Euclidean Nothing History ? • The Universe is a spontaneous creation from NOTHING. • It starts with a vacuum energy somewhere below the string scale. • It evolves in the landscape, producing inflation. • It then reaches a vacuum site in the landscape with a small CC and an exponentially long lifetime. • This is where we live. History? A Quantum Fluctuation from Nothing (No classical space or time) ↓ Universe moves in the Landscape (and inflates) ↓ Brane Inflation (Branes moving slowly towards each other) (Universe grew exponentially) ↓ Branes collided to heat up the universe 10-30 sec. ↓ Hot Big Bang Epoch (Nucleosynthesis around 10 sec. ) ↓ Matter-dominated Epoch (Star/galaxy formation begins at 1012 sec. ) ↓ Today’s Universe size ~ 1 1018 sec.