Hawking Radiation Large Distance (Much Bigger Than Planck Length)

Resolving the information paradox

Samir D. Mathur

CERN 2009 Lecture 1: What is the information paradox ?

Lecture 2: Making black holes in string theory : fuzzballs

Lecture 3: Dynamical questions about black holes

Lecture 4: Applying lessons to Cosmology, open questions Two basic points: The Hawking ‘theorem’ : If

(a) All quantum gravity effects are conﬁned to within a given distance like planck length or string length

dn dn (b)√ TheN nvacuum√n + 1is unique√N√Nn √√nn++11 (n√+N1) √n + 1ndn (175)(n + 1) n (175) −√N n √n + ≈1 − √Ndt ∝√≈n + 1 dt(n∝+ 1) n (175) − ≈ 1 dt ∝ 1 gravity gravity ωR = [ l 2 mψm + mφn] = ω (176) Then there WILLωR be= inf[ormationl 2 m ψlossm + mφn] = ωR (17R6) R − − − 1 R − − − gravity ωR = [ l 2 mψm + mφn] = ω (176) m = n + n + 1, n = nm =nnL + nR + 1, n = nLR (1n7R7) (177) String theory: Bound statesL ofR quantaR − in− stringL−− Rtheory have a ‘large’− size λ m n + m m = 0, N = 0 (178) λ mψn + mφm= =n 0,+ nN =+0 ψ1, nφ= n n (178) (177) | − | L | R− | L − R This size grows withλ = the0, nmumber= l, ofn branes= 0λ, = 0inN, =them0ψ bound= l, staten = 0,, N(17=9)0 (179) ψ − − making the state a ‘horizon sizedλ quantumgramvitψy n + fuzzball’mφm = 0, gravNity = 0 (178) ωI |= ω−I |ωI = ωI (180) (180) 0 λψ= 0,

10 10

10 Lecture 1

What exactly is the black hole information paradox?

(arXiv 0803.2030) The information problem: a ﬁrst pass

Hawking radiation Large distance (much bigger than planck length)

How can the Hawking radiation carry the information of the initial matter ?

If the radiation does not carry the information, then the ﬁnal state cannot be determined from the initial state, and there is no Schrodinger type evolution equation for the whole system.

So we lose quantum theory ... We would like the information to come out of the hole in the details of the Hawking radiation ...

This would save us from having to contemplate more complicated solutions of the puzzle (remnants, wormholes, baby Universes, modiﬁcations of quantum mechanics ...)

Is it possible for this to happen ? The central issue

When a piece of coal burns away, a large number of photons are produced.

The information of the coal is hidden in ‘delicate correlations’ between these photons’

Can this happen in a black hole ? Hawking’s computation: semiclassical (gravity classical, matter quantum)

Get no correlations among radiation quanta

What if we use quantum gravity ?

Quantum gravity effects will be small

But we need ‘subtle correlations’, so maybe the delicate effects of quantum gravity will generate just the needed correlations ?

This is incorrect !!

The effects of quantum gravity have to change the state of the radiation by order unity .... Basic point:

SUBTLE correlations can carry information, as in the case of the burning coal

But this CANNOT be achieved by SMALL corrections to the Hawking emission process

We need an ORDER UNITY change in the evolution of ﬁeld modes at the horizon Plan of the talk

(A) Particle creation in curved space, black holes

(B) Entangled nature of the produced pairs

(D) Formulating Hawking’s argument as a ‘theorem’

If (a) All quantum gravity effects are conﬁned to within a given distance like planck length or string length

(b) The vacuum is unique

Then there WILL be information loss (A) Particle creation in curved space and black holes S1 y y : (0, 2πR) (175) → ClVˆ [l] Vˆ (176)

N = n1n5 (177) dn √N n √n + 1 √N √n + 1 (n + 1) n (178) − ≈ dt ∝ 1 ω = [ l 2 m m + m n] = ωgravity (179) R R − − − ψ φ R m = n + n + 1, n = n n (180) L R L − R λ m n + m m = 0, N = 0 (181) | − ψ φ | λ = 0, mψ = l, n = 0, N = 0 (182) − gravity ωI = ωI (183) 0 ψ 0 ψ 0 (184) | & | & ' | & ≈ What helps is that we will usually detecnt1,pna2r,tnic3les inns4ome region which is far away (fr1o8m5) the region where spacetime is curved, for example1/ant1ans2ynm3 ptotic infinity in a black hole g(e1o8m6)etry. There is a natural choice of coordinates at infinityα, in which the metric looks like η . We (n1n5) lp (1µν87) can still make boosts that keep the metric in this form, but the change of time coordinate What helps is that we will usually detect particles in some region which is far away from the under these boosts does not channg1en5the vacukummk. =Wnh1na5t happne5ns is that positive fr(e1q8u8e)ncy region where spacetime is curved, for example at asymptotic infinity in a black hole geometry. modes change to other positivne !fr=eqnuencynm!! =odnes, giving tkhme ex=pnec! tne!d change of the e(n1e8r9g)y of There is a natural choice opf coo1rdinate1s at i5nfinity, in whkich thpe 1metric looks like ηµν. We a quantum when it is viewed from a moving frame. can still make boosts that keep th√e metric in this!form, but the change of time coordinate But even though this maSymibcreo a=n2aπtur2a√lnc1hno5ice of coSormdicirnoa=tes4,πg√ivni1nng5 a natural defi(n1i9ti0o)n of under these boosts does not change the vacuu2m. What happens is that positive frequency particlmeso,dwesecmhaanygesttioll oatshkerwphoysitwiveecfarenqnuoetncuysemsooRdmese, ogtivhienrg otthheeerxcpuercvteildincehaarngceooorfdtihneatee(n1es9ry1gs)yteomf and itas qcuoarrnetsupmonwdhienng itpaisrtviicelwese.d Tfrohme paominotviinsgtfhAraatmew.e have to know the following physics at Sbek = = Smicro (192) some poinBtu: twehvaent itshothueghentheirsgymacyarbrieeda nbaytuthraelsech2poGaicretiocflecso?orTdhinisatiensf,ogrimviantgioanniastnuroatl gdievfiennitbioyntohfe definitpioanrtiocfletsh,ewpeamrtaicylestmilloadseks;wrhaythweer,cwanennoetekudsmteoksok=mnonew1onttphheereontehregryc-umrvoimlineenatrumcootrednisnoarte(fo1s9ry3ts)hteemse particlaensdtaittsesc.orFroersptohnedpinhgyspiacratlicfileesl.dsTthheatpwoientcoisnstihdaetr,wwe ehavsesutmo eknthowat tthhee fpoallrotwicinlegs pdheyfisnicesdaitn ! 1 the flastocmoeorpdoiinnat:tewshyastteismthweietnhemrgeytrciacrrηieµdν bayreththesee!opnaerstiwclhesi?chTghiivseinthfoermexaptieocnteisdnpohtygsiivcea(nl19eb4ny)etrhgey definition of the particle modes; rather, we need to know the energy-momentum tensor for these of the state, an energy which shows up for examp4leπ in the gravitational attraction between these pparttiicclleess.tates. For the physical fields tha∆tEwe=consider, we assume that the particles d(e1fi9n5e)d in n1n5L Sotthheeflraetisconoordainmabteigsuyisttyeminwhitohwmpeatrrticicηleµsν aree tdheefionneeds awthicnhfigniivtey,thbeutexipf etchteerdepihsyssoicmaleerneegrigoyn of the state, an energy which shows up forgexam0 ple in the gravitational attraction (b1e9t6w)een of spacetime which is curved, then wavemodes t→hat travel through that region and back out these particles. g nonzero (197) to spatial infinity can have a nontrivial number of particles at the end, even though they may So there is no ambiguity in how particles are defined at infinity, but if there is some region ∆TCF T = ∆Tgravity (198) have sotfarstpeadcewtiimthe nwohipcharitsiccluesrveexdc,ittehdeninwatvheemmodaetsththeatsttarratv.elWthhraotugwhe tnheaetdrengoiwonisantdo bgeatcksoomute S GM 2 physictaol sfpeealtiinagl ifnofirntithyeclaennghtahveanadnotinmtreivsiaclalneu−sminbveorel−ovfedpairntictlheissaptrtohceesesndo,f epvaerntitchleoucgrheatt(hi1oe9yn9.m) ay Particle creation in cu∼rved sp2ace have started with no particles excited inetShbekm aetGtMhe start. What we need now is to g(e2t00s)ome ∼ 3.1 pPhaysritcaicl lfeelcinrgeafotriothne:lepnghtyhsaincdaltipmiecstcuarlees involvieωdt in this process of particle creation. Take a scalar field withΨ = Lagrangianψ(x)e− (201)

Let us first get a simpler picture of why partic1les canµ get created when spacetime is curved. 3.1 Particle creation: physical pLic=tur∂eµφ∂ φ (202) We know that each fourier mode of a quantum 2field behaves like a harmonic oscillator, and if we areLient tuhsefiersxtcigteetdastsaimteplner pfoicrtutrheisoof swcihlylatpoarrttichleens cwaen hgaevtecrneaptaedrtiwclheesninsptahciestifmoue riisercumrvoedde.. | Expand! the field in fourier modes, and look at a particular mode. Thus tWhe kanmopwlithuadteeoafchthfiosufroiLeteurr imtheero damplitudemeoodfea, qw ofuha ithisncthu modemw10efice albedll bae,hhaavessaliLkeagarahnargmiaonniocf otshceillfaotromr, and if we are in the excited state n for this oscillator then we have n particles in this fourier mode. | ! 1 1 Thus the amplitude of thisThefo uLagrangianrier Lmo=d feor, aw˙ 2 h isich wωe2aca2ll a, has a Lagrangian of the form (3.30) 2 − 2 1 1 L = a˙ 2 ω2a2 (3.30) But as we move to later times, the spacetime 2can−di2stort, and the frequency of the mode can change, so that we will get But as we move to later times, the spacetime can distort, and the frequency of the mode can ! 1 1 ! change, so that we will get So we haveL a= harmonica˙ 2 oscillatorω 2a2. Wavefunction in ground (3.31) state gives vacuum,2 excited− 2 states mean particles in that ! 1 2 1 !2 2 mode L = a˙ ω a (3.31) We picture this situation in fig.1. Fig.1(a) show2s th−e 2potential where the frequency is ω. Let us requireWtehpaitcthuerreethairsesnitouaptaiorntiicnlefisgp.1r.esFeingt.1(ina)tshhioswfosuthrieerpomteondteia.l Twhherne wtheewfrielql uheanvcey tihs eω.vaLceutuums wavefurneqcutiiorenth0atfhoerrtehairsehnaormpaorntiicleosscpirlleasteonrt.inNtohwissfuopurpioersemwoedec.hTanhgene twheewpiolltehnavtiealthteo vtahceuuomne | ! for frewquaveenfcuyncωti!o;nth0is fpoorttehnitsiahlarinmosnhiocwonscinllafitogr..1(Nbo)w. Fsuoprptohsies wneewchpanogtentthiealp, ottheentviaalcutoumthestoantee | ! is a difffoerrfernetquweanvceyfuωn!;cthioisnpforotemnttiahleinonsehofworn firneqfiuge.1n(cby).ωF,oarntdhiws eneswkeptcohtenittiianl,fitgh.e1(vbac).uum state Firistasduipffperoesnet twhaavteftuhnectcihoannfgroemofthfreeqonueenfocryffrreoqmuenωcytoω,ωa!nwdawsevsekreytcshloiwt .inTfihge.1n(bw).e will find ! that the vFaicrustumsupwpaovseefuthnacttitohne wchilalnkgeeepofcfhreaqnugeincgyafsrotmheωpottoeωntiwalacshvaenrygessl,owin. sTuhchenawweawy itllhfiatndit remainthsatthtehveavcaucuumumstwaatveefuorncwtihoantwevilelrkpeeoptecnhtainagl iwnge ahsatvheeaptoatennytigailvcehnantigmese,.inInsupcahraticwualyarthwaht eitn remains the vacuum state for whatever pote!ntial we have at any given time. In particular when we reach the final potential with frequency ω , t!he vacuum wavefunction of fig.1(a) will have becomweethreeacvhactuhue mfinwalapveoftuenntcitailownitohf ffirgeq.1u(ebn)c.y Tωh,isthfeacvtacfuoullmowwsafvreofmunctthioen‘aodfiafibg.a1t(iac) twhiellorheamve’, become the vacuum wavefunction of fig.1(b). This fact follows from the ‘adiabatic theorem’, which describes the evolution of states when the potential changes slowly. which describes the evolution of states when the potential changes slowly.

7 7 Figure 1: (a) The potential characterizing a given fourier mode, and the vacuum wavefunction for this potential. (b) If the spacetime distorts, the potential changes to a new one, with its own vacuum wavefunction. (c) If the potential changes suSupposeddenly, w thee h aspaceve the expands.new pot Thenential thebut frthequencye old wa ofve ftheunc modetion, w changeshich will not be the vacuum wavefunction for this changed potential; thus we will see particles.

Now consider the opposite limit, where thOriginale poten modetial c, hvacuumanges statefrom the one in fig.1(a) to the one in fig.1(b) very quickly. Then the wavefunction has had hardly any time to evolve, and we get the situation in fig.1(c). The potential is that for frequency ω!, but the wavefunction If the frequency changes slowly, is still the vacuum wavefunction for frequenthecy vacuumω. T hstateis i sgoesno otverth eto vtheac uum wavefunction for ! frequency ω , but we can expand it in terms onefwt hvacuume wav estatefunctions n ! which describe the level | !ω n excitation of the harmonic oscillator for frequency ω! If the frequency changes very 0 ω = c 0 ω! + c 1 ω! + c 2 ω! + . . . (3.32) | ! 0| ! quickl1| !y,the wa2v|efunction! remains the old one, so the state is NOT the Actually since the wavefunction that we havvacuume is sy fmorm theet nericw upotentialnder reflections a a, we will → − get only the even levels n in our expansion | !

0 ω = c 0 ! + c 2 ! + c 4 ! + . . . (3.33) | ! 0| !ω 2| !ω 4| !ω

This is like the expansion (3.27), and a little more eﬀort shows that the coeﬃcients cn will be of the form that will give the exponential form (3.26). Thus under slow changes of the potential the fourier mode remains in a vacuum state, while if the changes are fast then the fourier mode gets populated by particle pairs. But what is the timescale that distinguishes slow changes from fast ones? The only natural timescale in the problem is the one given by frequency of the oscillator

∆T ω−1 ω!−1 (3.34) ∼ ∼ where we have assumed that the two frequencies involved are of the same order. If the potential changes over times that are small compared to ∆T , then in general particle pairs will be produced. We can now put this discussion in the context of curved spacetime. Let the variations of the metric be characterized by the length scale L; i.e., the length scale for variations of gab is L in the space and time directions, and the region under consideration also has length L ∼ ∼ in the space and time directions. We assume that the metric varies signiﬁcantly (i.e. δg g) ∼ in this region. Then the particles produced in this region will have a wavelength L and the ∼ number of produced particles will be order unity. Thus there is no other ‘large dimensionless number’ appearing in the physics, and the length scale L governs the qualitative features of particle production.

8 An example of such a metric variation would be if we take a star with radius 6GM (so it is not close to being a black hole), and then this star shrinks to a size 4GM (still not close to a black hole) over a time of order GM. Then in this process we would produce order unity ∼ number of quanta for the scalar field, and these quanta will have wavelengths GM. After ∼ the star settles down to its new size, the metric becomes time independent again, and their is no further particle production. As it stands, this particle production is a very small effect, from the point of view of energetics. In the above example, the length GM is of order kilometers or more, so the few quanta we produce will have wavelengths of the order of kilometers. The energy of these quanta will be very small, much smaller than the energy M present in the star which created the changing metric. So particle production can be ignored in most cases where the metric is changing on astrophysical length scales. We will see that a quite different situation emerges for the black hole, where particle production keeps going on until all the mass of the black hole is exhausted.

3.2 Particle production in black holes How do frequencies change in the black hole geometry ? The metric of a Schwarzschild hole is often written as 2GM dr2 ds2 = (1 )dt2 + + r2(dθ2 + sin2 θdφ2) (3.35) − − r 1 2GM − r This metric looks tTheime metricindepe looksnden ttime, so windependent,e might think at first that there should be no particle production. If webutha thisd a cotivmerse ionlndeyp theend outsideent geo ofm ethetry holefor a star, there would indeed be no particle production. What is different in the black hole case? The point is that the coordinate system in the abovThee m efulltri cgeometrcoversyo isnl ygivaenp abryt theof tPhenre sosepacetime – the part outside the horizon r = 2GM. Once wdiagram,e look a andt th ewefu seell m thatetri cthew ehorizonwill no ist see a time independent geometry. The full geometry is tra dniulltio surfacenally described by a Penrose diagram, which we sketch in fig.2. The region of this diagram where the particle production will take place is indicated by the box with dotted outline around the horizon. From the Penrose diagram we can easily see which point is in the causal future of which point, but since lengths have been ‘conformally scaled’ we cannot get a good idea of relative lengths at different locations on the diagram. Thus in fig.3 we make a schematic sketch of the boxed region in fig.2. The horizontal axis is r, which is a very geometric variable in the problem 2 – the value of r at any radius is given by writing the area of tFhigeure22-:sTphhe Peenrreose aditagratmhfaorta bplacok ihnolet(waithsou4t tπherback.reaction effects of Hawking evaporation). Null rays are straight lines at 45o. Thus we see that the horizon is a null surface. Hawking radiation collects at The line at r = 0 is the ‘center’ of the black hole; thus this fiustureanulrl ienfignitoy. n of high curvature (the singularity) after the black hole forms. The line r = 2GM is isthdireecthedorradiziaollyno.utwSaprdsaitniathle aitntefimpntittoyesciaspeo. nThus from the metric (3.35) we will have the right, at r . 2GM dr2 → ∞ 0 = ds2 = (1 )dt2 + (3.36) − − r 1 2GM The vertical axis in fig.3 is called τ; it is some time coordinate that we have introduced to− r which gives complement r. At large r we let τ t, where t is the Schwarzschild time. The metr2iGcMwill → dr = (1 )dt (3.37) not be good everywhere in the coordinates (r, τ); it will degenerate at the horizon. T−hisr will So if we are on the horizon r = 2GM then we get dr = 0, i.e. the particle stays on the horizon. not matter since all we want to do with the help of this figureWishatsihf tohewparhticolewstargteedosldighetlsyiocustsidneethaerhotrihzoen, and tried to fly radially outwards? Now it can escape, so after some time the particle will reach out to a larger radius, say r 3GM. ∼ horizon evolve to smaller or larger r values. This null geodesic starts out near the horizon, but ‘peels off’ towards infinity. Similarly, consider a massless particle that starts a little inside the horizon and tries to fly A massless particle that is at the horizon and trying its rabdiealslytouttowarfldsy. Tohius timne iet vcaenrnotmescaapneathgeehsoletoor even remain where it started; this escape, but stays on the horizon. This can be seen as follownsu.ll gTeohdeesic m‘peealssosffl’easnsd fapllsairnttiocwalredsfsomlalloerwr.sThae figure shows the geodesic reaching the radius r GM which is inside the hole, though still comfortably away from the singularity. ∼ null geodesic. Let us allow no angular part to its momentum toNoewnwseusereethatthinathisavlilcintihtyeof mtheohmorizeon,ttuhemre is a ‘stretching’ of spacetime going on. A small region near the horizon gets ‘pulled apart’ with the part inside the horizon moving deeper in, and the part outside the horizon moving out. We will make this more precise later, but we can now see that the metric indeed has a time dependence which can cause particle 9 creation. Moreover, this stretching goes on as long as the black hole lasts, since whenever we

10 S1 y y : (0, 2πR) (175) → ClVˆ [l] Vˆ (176)

1/n1n2n3 (186) α (n1n5) lp (187) n1n5 k mk = n1n5 n5 (188) ! np! = n1 n1! = n5, k mk = np! n1! (189)

Smicro = 2π√2√n1n5 !Smicro = 4π√n1n5 (190) R2 (191) A S = = S (192) bek 2G micro

k mk = n1np (193) ! 1 ! (194) 4π ∆E = (195) n1n5L g 0 (196) → g nonzero (197)

∆TCF T = ∆Tgravity (198) S GM 2 e− e− (199) ∼ 2 eSbek eGM (200) ∼ iωt Ψ = ψ(x)e− (201) 1 Why do frequencies change in the black hole geometry? L = ∂ φ∂µφ (202) 2 µ The horizon is a null geodesic τ (203) Geodesics starting just outside eventually escape outwards 10 Geodesics staring just inside fall into the hole

r=0 r =2GM r Thus there is a ‘stretching’ going on at the horizon S1 y y : (0, 2πR) (175) → ClVˆ [l] Vˆ (176)

1/n1n2n3 (186) α (n1n5) lp (187)

n1n5 k mk = n1n5 n5 (188) ! np! = n1 n1! = n5, k mk = np! n1! (189) √ ! An example ofSsmucichroa=m2etπric 2v√arnia1tnio5n would bSemificwroe=ta4keπ√a nst1anr5with radius 6GM(1(9so0)it is not close to being a black hole), and thenRt2his star shrinks to a size 4GM (still not c(lo1s9e1t)o a black hole) over a time of order GM. Then in this process we would produce order unity ∼ A number of quanta for the scalar fiSebledk, =and the=seSqmuiacnrota will have wavelengths GM.(1A9f2t)er 2G ∼ the star settles down to its new size, the metric becomes time independent again, and their is no further particle production. k mk = n1np (193) As it stands, this particle prod!uction i1s a very small effect, from the point of view of energetics. In the above example, the lengt!h GM is of order kilometers or more, so t(h1e9f4e)w quanta we produce will have wavelengths of the order of kilometers. The energy of these 4π quanta will be very small, much smal∆leEr th=an the energy M present in the star which c(r1e9a5te)d the changing metric. So particle production nca1nn5bLe ignored in most cases where the metric is changing on astrophysical length scales. g 0 (196) → We will see that a quite different sigtuantoionnzeermoerges for the black hole, where partic(le19p7r)o- duction keeps going on until all the mass of the black hole is exhausted. ∆TCF T = ∆Tgravity (198) S GM 2 3.2 Particle production in blaec−k hoele−s (199) ∼ 2 The metric of a SSpacelikchwarzes csliceshild hforol etheis eoblackSftbekn holewreitGtMen as (200) ∼ iωt 2GMΨ = ψ(x)ed−r2 (201) ds2 = (1 )dt2 + + r2(dθ2 + sin2 θdφ2) (3.35) − − r 1 1 2GM L = ∂ φ−∂µφr (202) 2 µ This metric looks time independent, so we mτight think at first that there should be no p(a2r0ti3c)le production. If we had a time independent geometry for a star, there would indeed be no particle producOutsidetion. W thehat horizon,is differ econstantnt in the black hole case? The point is that the coordinate time is spacelike system in the above metric covers only a pa1r0t of the spacetime – the part outside the horizon r = 2GM. Once we look at the full metric we will not see a time independent geometry. The full geometry isInsidetrad ithetio nhorizon,ally de sconstantcribed b ry a Penrose diagram, which we sketch in fig.2. The is spacelike region of this diagram where the particle production will take place is indicated by the box with dotted outline around the horizon. From the Penrose diagram we can easily see which point is in the causal future of which point, but since lengths have been ‘conformally scaled’ we cannot get a good idea of relative lengths at different locations on the diagram. Thus in fig.3 we make a schematic sketch of the r=0 r =2GM r boxed region in fig.2. The horizontal axis is r, which is a very geometric variable in the problem – the value of r at any radius is given by writing the area of the 2-sphere at that point as 4πr2. The line at r = 0 is the ‘center’ of the black hole; thus this is a region of high curvature (the singularity) after the black hole forms. The line r = 2GM is the horizon. Spatial infinity is on the right, at r . → ∞ The vertical axis in fig.3 is called τ; it is some time coordinate that we have introduced to complement r. At large r we let τ t, where t is the Schwarzschild time. The metric will → not be good everywhere in the coordinates (r, τ); it will degenerate at the horizon. This will not matter since all we want to do with the help of this figure is show how geodesics near the horizon evolve to smaller or larger r values. A massless particle that is at the horizon and trying its best to fly out never manages to escape, but stays on the horizon. This can be seen as follows. The massless particle follows a null geodesic. Let us allow no angular part to its momentum to ensure that all the momentum

9 timelike everywhere we could use it to deﬁne time evolution, and everything would look time independent: the slices don’t change, and the metric with this choice of time direction will look time independent. But this vector will not be timelike everywhere; it will become null on the horizon and be spacelike inside the horizon. We have taken extra care to make our slices not approach the singularity – we let them follow a r = constant path to an early enough stage where the singularity had not formed, and then took them in to r = 0. This feature of the slices is not directly related to the production of particles in Hawking radiation, but we have done the slicing in this way so that the evolution stays in a domain where curvature is everywhere low and so classical gravity would appear to be trustworthy. To summarize, the central point that we we see with these diﬀerent ways of exhibiting the slices is that the geometry of the black hole is not really a time independent one, and particle production can therefore be expected to happen.

On a Penrose diagram, these slices look like this ...

Figure 6: The slices drawn on the Penrose diagram. Later slices go up higher near future null inﬁnity and will thus capture more of the Hawking radiation.

4.1 The wavemodes Let us now look at the wavemodes of the scalar ﬁeld in the black hole geometry. We will look at non-rotating holes, so the metric is spherically symmetric, and we can decompose the modes of the scalar ﬁeld φ into spherical harmonics. Most of the Hawking radiation turns out to be in the lowest harmonic, the s-wave, so we will just focus our attention on this l = 0 mode; the physics extends in an identical way to the other harmonics. We will suppress the θ, φ coordinates, drawing all waves only in the r, t plane.

14 Evolution of wavemodes: To leading order, we can evolve wavemodes by letting the lines of constant phase be null geodescics

mode in part of the mode is outside, vacuum state part inside

Figure 8: (a) The region around the horizon is a vacuum. (b) An outgoing wavemode on an initial spacelike surface is evolved by letting the phase be constant on outgoing null geodescis. vacuum modes after this scattering. These outgoing modes then show up in the circled region of fig.8(a). We are interested in the further evolution of these modes. The outgoing field mode is drawn in more detail in fig.8(b), where we have caught the mode on a spacelike surface which we will call our ‘initial slice’. We follow our above described method of evolving the wavemodes by letting the phase be constant along outgoing radial null geodesics. These null geodesics look like straight lines on the Penrose diagram, so at first it might seem that the wavelength of the mode is not changing as we follow the mode out towards infinity. This is not true, since in the Penrose diagram the actual distances between points is large when the points are near infinity. (In drawing the Penrose diagram we squeeze the spacetime in a ‘conformal’ way so that all of spacetime fits in a finite box; this automatically squeezes points near infinity by a large amount.) What we really want to see is how the wavelength of the mode changes as the mode is evolved. So in fig.9 we sketch the evolution in the r τ diagram that we discussed above. The − initial slice is drawn again, with the outgoing wavemode on it. The lines of constant phase are drawn too, but now they do not look like straight lines. We had seen that the horizon itself is an outgoing null geodesic, that stays at all times at r = 2GM. The rays starting slightly outside the horizon eventually ‘peel off’ at go to spatial inifinity, while those starting slightly inside ‘peel off’ and fall in towards small r. Thus the wavemode will get distorted as it evolves. What we want to do now is to ‘catch’ the wavemode on a later spacelike slice. By following the null rays, we can obtain the phase of the wavemode all along this later slice. We can see that there is quite a distortion between the wavemode as seen on the initial spacelike slice and the

17 On the schematic diagram we can see that the wavemode stretches, and this will create particles ...

Figure 9: A wavemode which is a positive frequency mode on the initial spacelike surface gets distorted when it evolves to a later spacelike surface; the mode will not be made of purely positive frequencies after the distortion.

wavemode as it is ‘caught’ on this later slice, and the changes come because the null geodesics just inside and just outside the horizon evolve in quite diﬀerent ways. But if the wavemode is distorted, there can be particle creation. We will now look at the distortion in much more detail, and discuss the nature of this particle creation.

5.1 The coordinate map giving the expansion Consider the vicinity of the horizon sketched in fig.8(a). The local geometry is approximately flat space, and the field modes are in the vacuum state. Let us use null coordinates y+, y− to describe the spacetime here (recall that the angular S2 is suppressed throughout). The outgoing modes, which are of interest to us, are then of the form iky− ψinitial e (5.40) ∼ We will assume that k > 0. In the expansion of the field φˆ the positive frequency modes −iky− multiply the annihilation operators aˆk. We will write the negative frequency modes as e ; these will multiply the creation operators. Now let us see what coordinates would be good on the late time spacelike slice, sketched in fig.9. Consider the outer part of the slice Sout. This part is in a region which is close to flat Minkowski spacetime.We had discussed above that particles were well defined in such a region of spacetime, and this definition required us to use positive frequency modes based on the usual coordinates on Minkowski space. So we just use the standard definition of null coordinates here X+ = t + r, X− = t r (5.41) −

18 Our goal is to see where the distortion is large enough to create particles. On the initial − slice near the horizon, we take a fourier mode eiky . Consider this mode for the range GM < − y− < 0. Recall that y− is negative outside the horizon, and zero on the horizon. Also, for The stretching is logarithmicy− ! GM, th, easre iws neo comesigniﬁca nneart dist othertion horizon,of null rays, so that we get y− X−. What − ≈ so it is nonlinear we now wish to show is that even though there is a logarithmic distortion of coordinates for smaller GM ! y− < 0, there is no particle production for most of this range of y−; in fact − particle production will be relevant only for the few oscillations of the wavemode near y− = 0.

The part of the mode outside escapes to inﬁnity,

The part inside falls to the singularity

Figure 10: The fourier mode on the initial spacelike slice is evolved in the eikonal approximation, and ‘caught’ on the timelike surface r = constant near inﬁnity. (From the behavior of the mode on this surface we can immediately obtain what it looks like on any spacelike surface near inﬁnity.) The wavelength of oscillations become longer and longer as we go up the surface, with the last oscillation to emerge from the horizon extending all the way to t = ∞.

Thus we now look at the range GM < y− < 0, where we assume that the logarithmic − − maps (5.45) is a good approximation. Consider the fourier mode eiky on the initial slice. Let the wavelength of this mode be much smaller than GM 2π λ = = #GM, # 1 (5.49) k # Thus the number of oscillations of the wavemode in our range GM < y− < 0 is large − 1 # oscillations = 1 (5.50) # $ After the mode evolves to the late time slice we have to look at the wavelength in the X− coordinate system. Consider one oscillation of the wavemode, which in the y− coordinate

21 can make out of the fundamental constants c, !, G is

1 ! 2 G −33 lp = 10 cm (2.8) c3 ∼ ! " As we will see below, if it were indeed true that all quantum gravity effects were confined to within a length scale like lp (or any other fixed length scale) then we would get information loss, and quantum mechanics would need to be changed. But how can we get any other natural length scale for quantum gravity effects? If we collide two gravitons then it is true that quantum gravity effects should start when the wavelengths of the gravitons become order lp. But a black hole is made up of a large number of quanta N; the larger the black hole the larger this number N. Then we have to ask if quantum gravity effects extend over distances lp or over distances α ∼ N lp where α is some appropriate constant. In string theory we find that the latter is true, and that N, α are such that the length scale of quantum gravity effects becomes of the order of the radius of the horizon. This changes the process by which the radiation is emitted, and the radiation can emerge in a pure state.

2.3 The plan of the review There exist many reviews on the subject of black holes, and there are also reviews of the ‘fuzzball’ structure emerging from string theory [3]. What I will do here is a bit diﬀerent: I will try to give a detailed pictorial description of the information problem. We will study the black hole geometry in detail, and see how wavemodes evolve to create Hawking radiation. Then we will discuss the ‘mixed’ nature of the quantum state that is created in this radiation process. Most importantly, we will discuss why the argument of Hawking showing information loss is robust, and can only be bypassed by a radical change in one of the fundamental assumptions that we usually make about quantum gravity. We will close with a brief summary of black holes in string theory and the fuzzball nature of the black hole interior.

3 Particle creation in curved space

The story of HawkinIng therad iaquantumtion rea lfield,ly be gannihilationins with th operatorse understa nmdultipling oyf particle creation in curved spacetime. (positivFor revei efrwequencys see [4 ].modes) Par tandicles craeationre des coperatorsribed in t emrmultipls ofy an underlying quantum field, say a snegativcalar fiee lfrdequencyφ. We c amodesn write a covariant action for this field, and do a path In curviendtesgpralc.eBtiumt eh,oowndtohweeodthefienrehpaanrdti,cltehs?erIeniflsantospcaacneowneiceaxlpdaenfidntihteiofinelodfoppaerrtaitcolresa.s We can choose any coordinate t for time, and decompose the field into positive and negative frequency modes with respect to this time ˆt. Let th1e po1sitive fi"kr·e"xq−uiωetncy†m−oi"kd·"xe+siωbte called flatf(x space); then their φ = aˆ"ke + aˆ" e (3.9) √V √2ω ∗ k complex conjugates give negative fre#q"k uency mod$es f (x). The field op%erator can be expanded as where V is the volumeôf the spatial box where †we∗have taken the field curto lviedve, spaceand ω = φ(x) = aˆnfn(x) + aˆnfn(x) (3.11) $k 2 + m2 for a field with mass m. The vacuum is the state annihilated by all the aˆ | | n & ! " # Then we can define a vacuum state as one that aîs a0nn=ih0ilated by all the annihilation ope(r3a.1t0o)rs "k| # † and the aˆ create particles. aˆn 0 a = 0 (3.12) "k Particle creation ha| ppens! when the stretching mixes the positive frequency modes with the negative frequency modes ... The creation operators generate particles; for example a 1-particle state would be 4 † ψ = aˆ 0 a (3.13) | ! n| ! We have added the subscript a to the vacuum state to indicate that the vacuum is defined with respect to the operators aˆn. But since there is no unique choice of the time coordinate t, we can choose a different one t˜. We will then have a different set of positive and negative frequency modes, and an expansion ˆ ˆ ˆ† ∗ φ(x) = bnhn(x) + bnhn(x) (3.14) n ! " # Now the vacuum would be defined as ˆbn 0 b = 0 (3.15) | ! † and the ˆbn would create particles. † The main point now is that a person using the operators aˆ, aˆ would think that 0 a was a | ! vacuum, but he would not think that the state 0 b was a vacuum – he would find it to contain † | ! † particles of the type created by the aˆn. Let us see how one finds exactly how many aˆ particles there are in the state 0 b. The mode functions fn are normalized using an inner product defined | ! as follows. Take any spacelike hypersurface, with volume element dΣµ (thus the vector dΣµ points normal to the hypersurface and has a value equal to the volume of the surface element). Then µ ∗ ∗ (f, g) i dΣ (f∂µg g ∂µf) (3.16) ≡ − − $ Under this inner product we will have

∗ ∗ ∗ (fm, fn) = δmn, (fm, f ) = 0, (f , f ) = δmn (3.17) n m n − Now from the two diﬀerent expansions of φˆ we have

† ∗ ˆ ˆ† ∗ aˆnfn(x) + aˆnfn(x) = bnhn(x) + bnhn(x) (3.18) n n ! " # ! " # Taking the inner product with fm on each side, we get

ˆ ∗ ˆ† ˆ ˆ† aˆm = (hn, fm)bn + (hn, fm)bn αmnbn + βmnbn (3.19) n n ≡ n n ! ! ! !

5 5.3 Wavepackets

In fig.11(a) we depict a wavemode with a definite wavenumber k0. This wavemode has an infinite spatial extent. For physical arguments it is more convenient to have a wavemode that is localized in some region of space. Such a wavemode can be obtained by appropriately superposing wavemodes of different k. But we also wish to retain some properties of the mode arising from the fact that the wavenumber was k0. Thus we use only a small band of k around the value k0 ∆k k ∆k < k < k + ∆k, << 1 (5.57) 0 − 0 k Wavepackets | 0| This makes a wavetrain that ‘sort of’ has the wavenumber k0 but which decays after a certain nuWmbee rcanof obrscieakllati oupns athend iwas thvuemodes localiz eintod. O uwar dviepackscussioetsns a re mostly qualitative, so we will allow ourselves to use wavetrains that are only a few oscillations long; this means that we will nowithoutt take ∆k losingto be vtheery sessentialmall, but fphorysicsour pictorial understanding it will be enough to have k k0 in the rough neighborhood of k0.

But to make a wavepacket we need at least a few complete Figure 11: (a) A fourier mode with given wavelength λ = 2π (b) Appropriately superposing fourier modes oscillations of the wave k0 with wavenumbers near k0 we can make a wavepacket.

Let us now use the above discussion together to make the point that we are after. In fig.12 we mWakee awillwav brepaeakcke tupou tourof ainitialfew o swacillvaemodetions th aintot are wanotvepacktoo cloets,se to the horizon. This wavepacket evolves to a wavepacket near spatial infinity without significant distortion, since the ossocill athattion swmea kcaning tstudyhe wav exactlepacketys uwhichffer an paralmot soft u ntheifor mwastvremodeetching under the evolution. Thus leadsthere iston oparsignticleifica ncrt peationarticle production from the part of the wavemode where y− !. | | " 5.4 Modes straddling the horizon So far we have seen what part of the wavemode does not create particles. The part at y− ! GM does not get deformed. The part GM ! y− ! deforms logarithmically but can be − − # − broken up into wavepackets each of which suffers ‘nearly uniform stretching’, so again we do not get particle creation. A similar analysis can be performed for the domain y− > 0 which is inside the horizon. We can now turn to the part of the wavemode that does create particle pairs. Consider the wavemode on the initial surface and look at the domain of y− which covers a few oscillations on either side of the horizon y− = 0. Thus we have

y− ! (5.58) | | ∼ With just a few oscillations in this range, we cannot break this part of the wavemode further into wavepackets. Thus we must evolve it as a whole to the late time surface and see what it becomes. The evolution is described in ﬁg.13. On the initial slice we have regularly spaced oscillations. If we look at at surface just a little later, they are still pretty much like regularly spaced

23 A wavepacket that is localized outside the horizon stretches, but the stretching is approximately constant over the wavepacket Thus positive frequencies do not mix with negative frequencies, and there is no particle creation

Figure 12: If we look at the oscillations that are not too close to the horizon, then we can make a wavepacket out of them that evolves to a wavepacket at inﬁnity. Suppose we can make a localized wavepacket such that in the region occupied by the wavepacket the ‘stretching’ of space is approximately uniform. Then there will be no mixing of positive and negative frequencies and therefore no particle production.

oscillations, since there has not been much deformation; thus so far there is no significant particle production. On slices that are much later, we see that the mode has deformed significantly: there are a few oscillations on the part S− of the surface that is inside the horizon, then a large gap untill we reach a region on S+, where we find oscillations again. Note that on this late time slice the deformation of these oscillations of the wavemode is − very nonuniform. We have a positive frequency mode on the initial surface eiky , but on the − late time surface we will get an admixture of positive frequency modes eiKX as well as negative −iKX− frequency modes e . The same happens for the part of the mode on S−. Thus there will be particle creation. The most important part of our entire discussion comes now. We know from eq.(3.28) that when we create particles by deforming spacetime then the vacuum state changes to a state of 1 ˆ†ˆ† † − 2 Pij γij bi bj ˆ the form e 0 . But in the present case we can break the creation operators bk into | ! ˆ† † two sets: those on S+ which we call bk and those on S− which we call cˆk. When we compute the state on the late time surface it turns out to have the form

P γˆb† cˆ† e k k k 0 (5.59) | ! We do not derive this result here; the derivation can be found for example in [2, 5, 6, 7]. But

24 If a wavepacket sits across the horizon, then we will get particle creation. The mode gets cut in two parts ...

Figure 13: A fourier mode on the initial spacelike surface is evolved to later spacelike surfaces. In the initial part of the evolution the wavelength increases but there is no signiﬁcant distortion of the general shape of the mode. At this stage the initial vacuum state is still a vacuum state. Further evolution leads to a distorted waveform, which results in particle creation.

this is the crucial result for the physics of information, so we will now spend some time in understanding it.

25 (B) Entangled nature of the produced pairs wavelengths. Consider a wavemode with wavelength shorter than the one shown in fig.13. Then this mode will evolve for a longer time before it suffers a nonlinear deformation. This situation in depicted in fig.14. On the initial slice we have drawn two wavemodes of different wavelengths. The one with the longer wavelength becomes distorted first, and creates the quanta labeled b1 and c1 on the late time slice. The wavemode with shorter wavelength evolves for a longer time before becoming distorted, and creates the quanta labeled b2, c2.

this is the crucial result for the physics of information, so we will now spend some time in understanding it.

5.5 The nature of the created pairs Consider again fig.13. On the initial surface the wavemode had a very short wavelength. On later time surfaces the wavelength has been stretched to a longer one, though there is no particle production because the stretching is almost uniform over the oscillations under consideration. The wavelength keeps getting longer as we go to later time slices, till the deformation becomes Figure 14: On the initial nspoanc-eulinkeifoslricme waendhavpeadrteipcilcetsedatrweocfroeuariteerdm. odBeus:t tthheelroengiesr ownavlyeleonngteh lmenogdtehisscale in the geometry drawn with a solid line and the shorter wavelength mode is drawn with a dotted line. The mode with longer – the scale GM – and one can see easily that when particles are produced the wavelength wavelength distorts to a nonuniform shape first, and creates an entangled pairs b1, c1. The mode with shorter of the mode has become GM. At this point the wavemode has also moved to distances wavelength evolves for some more time before suffering the∼same distortion, and then it creates entangled pairs b2, c2. The! naturGMe offro them tcrheeatedhori zpairson, and further deformation stops. Thus the wavelength of the produced quanta is GM. These are the Hawking radiation quanta, so we see that this radiation has a ∼ The state of the firsttepmapirerba1t,uc1reis oλf −th1 e for1m. The exact temperature is [2] ∼ ∼ GM γˆb†cˆ† ψ 1 = Ce 1 1 0 1 (5.61) The temperature scale| ! is set by the | ! T = (5.60) ˆ† natural length scale in the geometry 8πGM Here b1 is an operator that creates a quantum in the localized wavepacket depicted as b1 in fig.14, and similarly cˆ† Scoretahteswtahveemquoadnetuenmdsofitstheevowluatvieopnacwkietth laabwealevdelecn.gthBecaGusMe ,wbeuhtawvehat was its wavelength 1 1 ∼ broken up wavemodes inotno tlhoecailnizietidalwsalviceepathckaettsw,ewheacdandrdaewfinn?e aOsnortthoisf lioncitailalvascliucuemthe0reb arine modes of all possible | ! 1 the region occupied by this mode b . If we are in this vacuum state then there are no quanta But the important1 fact about the state is that the b, c quanta ˆ† 25 in this region, if we actarwe iinth anb1 entangledonce the staten we have one quantum with this wavepacket, if we act ˆ†ˆ† with b1b1 then we have two quanta of this type, and so on. Doing the same for the modes on S− we can write the state (5.61) as

γˆb†cˆ† ψ = Ce 1 1 0 b 0 c (5.62) | !1 | ! 1 | ! 1

26 wavelengths. Consider a wavemode with wavelength shorter than the one shown in fig.13. Then this mode will evolve for a longer time before it suffers a nonlinear deformation. This situation in depicted in fig.14. On the initial slice we have drawn two wavemodes of different wavelengths. The one with the longer wavelength becomes distorted first, and creates the quanta labeled b1 and c1 on the late time slice. The wavemode with shorter wavelength evolves for a longer time before becoming distorted, and creates the quanta labeled b2, c2.

A similar state is produced by the wavemode which started oﬀ with a shorter wavelength on the initial slice. We get particle pairs described by

A similar state is produced by the waveγmˆb†ocˆ†de which started oﬀ with a shorter wavelength ψ 2 = Ce 2 2 0 b 0 c (5.63) on the initial slice. We get partic|le!pairs describ|ed! b2y| ! 2

The pairs b , c for different k lie in regions th†a†t do not overlap, so the overall state on the Figure 14k: Okn the initial spacelike slice weγˆbhcâve depicted two fourier modes: the longer wavelength mode is ψ 2 = Ce 2 2 0 b 0 c (5.63) A simlialater tsitmateesliisceprisodthueceddirebcyt tphreodwuacvteomf tohdeeswtahteicshψstkart2ed o2ff with a shorter wavelength drawn with a solid line and the sh|ort!er wavel|en!g|th! m| o!de is drawn with a dotted line. The mode with longer on the initial slice. We get particle pairs described by b , c waveTlehnegtphairdsisbtko,rctks ftoor adiffneorneunntikforlime inshraepgeionfisrstth,aatnddocnroetatoevseralnape,nstoanthgeleodvepraailrlssta1te 1o.n Tthhee mode with shorter ψ = ψ 1 ψ 2 ψ 3 . . . (5.64) wlavteelteinmgethsliecveoilsvetshefodrirseocmt eprmodourecttiomf† et†hbeesftoarteessuψffering the same distortion, and then it creates entangled pairs | ! γ|ˆb !cˆ ⊗ | ! ⊗ | k! ⊗ b , c . ψ = Ce 2 2 0 0 | ! (5.63) W2e h2ave presented a simplifi2ed discussion obf2 thec2created pairs; more technical details can be | ! ψ = ψ| ! |ψ! ψ . . . (5.64) found in [2, 5, 6, 7]. For a more accu|ra!te d|es!c1ri⊗pt|io!n2 w⊗e|sh!3ou⊗ld use a large number of oscillations The pairs bk, cTkhfeorstdaiffteeroenf tthkelifie risntrpegaiiornbs1t,hca1tidsoofnottheovfeorrlamp, so the overall state on the in makinWgeeahcahvewparveesepnatcekdeta s(iwmephliafiveed udsisecdusjsuiostn aoffethwe),craenadtedthpeanirws;emwoirlel theacvhenitcoalcdoentsaiidlsercamnabney late time slice is the direct product of the states ψ k † † wavenfouumndbeinrs[2in, 5e,a6c,h7]o.fFtohreainmtoerrevaalcscuornatSe±|deo!svcerripwtihonichwethsheowuγalˆdbveucˆpseacakleatrsgeexntuemndb.erBouf tostchilelaatbioonvse ψ 1 = Ce 1 1 0 (5.61) approinximmaaktiengdeesaccrhipwtiaovnepψhaacks=eatl(lψwtehehaevsesψeunsceed ojufψswt|haaf!tew.w.)e., annededthteon uw|ne!dwerilsltahnavdetthoecoents(ai5dn.e6grl4em)maennyt † 1 2 3 of quwanavtaen.ûmbers in each|of!the|int!erv⊗al|s o!n S⊗±|ov!er ⊗which the wavepackets extend. But the above Here b1 is an operator that creates a quantum in the localized wavepacket depicted as b1 in We have praepspernotxeimd aatesidmespclriifipetdiondihs†acsusaslilotnheofestsheneccereoaf twehdapt awiresn; emedorteo tuencdhenrisctanl ddetthaeilesnctaannglbeme ent fig.14, and similarly cˆ creates the quantum of the wavepacket labeled c1. Because we have found in [25,.65, o6f,Tq7hu].aenFteoanr. taamngorledacncuartau1treedeosfcriψption we should use a large number of oscillations broken up wavemodes into lo|ca!lized wavepackets, we can define a sort of local vacuum 0 b1 in in making each wavepacket (we have used just a few), and then we will have to consider many | ! Conts5hid.e6errethgTeihosnetaetoenctcψaunp1gieleddbnyattuhrise mofodψe b1. If we are in this vacuum state then there are no quanta wavenumbers in each of the|in!tervals on S± ove|r w! hich the wavepackets extend. But the above in this region, if we act with ˆb† once then we have one quantum with this wavepacket, if we act approximate dCeosncrsidpetriotnhehsatsataellψth1 e essence o1f what we need tγo2understand the entanglement † † | ! ˆ† † ˆ†ˆ† † † ψˆ1ˆ = C 0 b1 0 c1 + γb1 0 b1 cˆ1 0 c1 + b1b1 0 b1 cˆ1cˆ1 0 c1 + . . . of quanta. with| b!1b1 then w| e! ha⊗v|e !two qua| n!ta⊗of t|h!is typ2e,2 an|d! so⊗on. |D!oing the same for the modes on ! ˆ† † γ ˆ†ˆ† † † " S− we caψn1 w=riteCthe0 bs1tate0 (c51 .+61γ)b1a0sb1 cˆ1 0 2c1 + b1b1 0 b1 cˆ1cˆ1 0 c1 + . . . | =! C 0 b1| ! 0⊗c1| +! γ 1 b1 | !1 ⊗c1 +|γ! 2 b1 2 2 c|1 !+ .⊗. . | ! (5.65) | !! ⊗ | ! | ! ⊗ | ! | ! ⊗ | ! " 5.6 The entangled nature of ψ 2 γˆb†cˆ† = # C 0 b1 0 c1 + γ 1 b1 1 c1 + γ 2 b1 1 2 c1 + . . .$ (5.65) Thewhe rentanglede n b mea nnaturs thate wof|e! hthea|v⊗e |!staten! quant|a!of⊗tyψ|p!e1 b=1 iCn et|h!e s⊗ta|t0e!be1tc0. c1 (5.62) | ! 1 | ! | ! | ! Consider the state ψ 1 Twhheeirme pnorbtamnteafnesattuh#raet owfe thhaivsesntaqteuainstathoafttytphee bb1 inantdhecs1taetxeceittac.tion$s are ‘entangled’. To || !! 1 1 understaTnhdetihmispionrtmanotrefeadteutraeil,ofletthuisssttaakte ais stihmapt lteh2eexba1mapnlde ocf1 aenxceitnattaionngsleadrest‘aetnet.angled’. To † † γ † † 26 † † ψ u=ndeCrstan0d tbhis in0mcor+e dγeˆbtail0, lbet uscˆtak0eca s+impleˆbeˆbxam0pble of câncên0tacng+led. .st.ate. | !1 | ! 1 ⊗ | ! 1 1| ! 1 ⊗ 1| ! 1 2 1 1| ! 1 ⊗ 1 1| ! 1 5.7 Enta!nglement and the idea of ‘mixed states’ " 2 5=.7 CEn0tabnglem0 cen+t aγn1dbthe i1dec a+ofγ‘m2ixb ed s2tacte+s’. . . (5.65) Consider two| e!le1ct⊗ro|ns!, 1kept |at!t1w⊗o d| iff!e1rent lo|ca!t1io⊗ns|, a!n1d let each of them have a ‘spin up’ stateCaonndsiade‘rsptwinodeolewcntr’osntsa,tkee.pTt haetntwthoisdisffyesrteenmt lcoacnatihoanvse, a‘fnadctloerteedacshtaotfest’hoemf thheavfeoram‘spin up’ where n b means th#at we have n quanta of type b1 in the state etc. $ | This! 1 statestate a isnd entangleda ‘spin dow nbetw’ stateeen. Th theen t hbis andsyst ecm spacescan have ‘factored states’ of the form The important feature of this state is that the b1 and c1 excitations are ‘entangled’. To ψ = ψ 1 ψ 2 (5.66) understand this in more detail, let us take a si|m!pψle =|ex!aψm⊗p|le!ψof an entangled state. (5.66) | ! | !1 ⊗ | !2 ConsiderExamples aar esystem of two spins Examples are 5.7 Entanglement and the idea of ‘mixed states’ ψ = 1 2 | ! ψ =| ↑! ⊗1 | ↓! 2 Consider two electrons, kept at two |di!ffere1nt| ↑l!oc⊗at|io↓n!s, and l1et each of them have a ‘spin up’ Factored states ψ = (1 1 + 1) 1( 2 + 2) (5.67) state and a ‘spin down’ state. Th|en! tψhis =sy√stem| ↑(!can1 +|h↓a!ve 1⊗‘)fa√ctor↑e(!d 2st+a|t↓e!s’2)of the form (5.67) | ! 2√2 | ↑! | ↓! ⊗ √2 2 ↑! | ↓! etc. But we can also have ‘entangled states which cannot be written as a product of the type etc. But we can also have ‘eψntan=gleψd s1tatesψw2hich cannot be written as a product o(f5t.6h6e)type (5.66()5, .f6o6r),efxoarmexpalemple | ! | ! ⊗ | ! Examples are 1 1 ψ ψ= = ( ( 1 1 2 +2 + 1 1 2)2) ((55..6688)) Entangled states | !| ! √2√2| ↑!| ↑!⊗ ⊗| ↓|!↓! | ↓|!↓!⊗⊗| |↑!↑! ψ = SuppSouspepwoeseawske:a|wsk!h:awt hisatt|his↑e!ts1hte⊗atset|a↓ote!f2eolfeecltercotnro1n?1?FoFrorstsattaetsesofoftytyppee((55.6.666)) wwee ccan annsswweerr tthhisis questqiounes:tiowne: igwneoirgenotrhee tshtea1tsetaotef eolfecetlercotnro2n a2nadn1djujsutstgigvieveththeeaannswsweerr ψψ 11.. BBuut forr ssttaatteess ooff ψ = ( 1 + 1) ( 2 + 2) || !! (5.67) | ! √2 | ↑! | ↓! ⊗ √2 ↑! | ↓! 27 etc. But we can also have ‘entangled states which ca2n7not be written as a product of the type (5.66), for example 1 ψ = ( 1 2 + 1 2) (5.68) | ! √2 | ↑! ⊗ | ↓! | ↓! ⊗ | ↑! Suppose we ask: what is the state of electron 1? For states of type (5.66) we can answer this question: we ignore the state of electron 2 and just give the answer ψ . But for states of | !1

27 wavelengths. Consider a wavemode with wavelength shorter than the one shown in fig.13. Then this mode will evolve for a longer time before it suffers a nonlinear deformation. This situation in depicted in fig.14. On the initial slice we have drawn two wavemodes of different wavelengths. The one with the longer wavelength becomes distorted first, and creates the quanta labeled b1 and c1 on the late time slice. The wavemode with shorter wavelength evolves for a longer time before becoming distorted, and creates the quanta labeled b2, c2.

Figure 14: On the initial spacelike slice we have depicted two fourier modes: the longer wavelength mode is drawn with a solid line and the shorter wavelength mode is drawn with a dotted line. The mode with longer wavelength distorts to a nonuniform shape ﬁrst, and creates an entangled pairs b1, c1. The mode with shorter wavelength evolves for some more time before suﬀering the same distortion, and then it creates entangled pairs b2, c2.

The state of the first pair b1, c1 is of the form γˆb†cˆ† ψ = Ce 1 1 0 wavelengths. Consider a wavemode wi(t5h.6w1a)velength shorter than the one shown in fig.13. | !1 | ! ˆ† Then this mode will evolve for a longer time before it suffers a nonlinear deformation. Here b1 is an operator that creates a quantum in the locaTlihzisedsitwuaatvioenpianckdepticdteedpinctfiegd.14a.s Obn1 tihne initial slice we have drawn two wavemodes of † different wavelengths. The one with the longer wavelength becomes distorted first, and creates fig.14, and similarly cˆ1 creates the quaThentum totalof th wae wvaefunctionvepacket l aisb eale tensord c1. B precaoductuse w eofh atheve wavefuntions the quanta labeled b1 and c1 on the late time slice. The wavemode with shorter wavelength broken up wavemodes into localized wavofep theseackets ,pairswe ca .....n define a sort of local vacuum 0 b1 in evolves for a longer time before becom|ing! distorted, and creates the quanta labeled b2, c2. the region occupied by this mode b1. If we are in this vacuum state then there are no quanta ˆ† in this region, if we act with b1 once then we have one quantum with this wavepacket, if we act ˆ†ˆ† with b1b1 then we have two quanta of this type, and so on. Doing the same for the modes on S− we can write the state (5.61) as

γˆb†cˆ† ψ = Ce 1 1 0 b 0 c (5.62) A similar state is produced by|th!e1 wavemode |w!hi1c|h !st1arted oﬀ with a shorter wavelength on the initial slice. We get particle pairs described by 26 γˆb†cˆ† ψ = Ce 2 2 0 b 0 c (5.63) | !2 | ! 2 | ! 2

The pairs bk, ck for different k lie in regions that do not overlap, so the overall state on the late time slice is the direct product of the states ψ k | ! ψ = ψ ψ ψ . . . (5.64) | ! | !1 ⊗ | !2 ⊗ | !3 ⊗ We have presented a simplified discussion of the created pairs; more technical details can be found in [2, 5, 6, 7]. For a more accurate description we should use a large number of oscillations in making each wavepacket (we have used just a few), and then we will have to consider many wavenumbers in each of the intervals on S± over which the wavepackets extend. But the above approximate description has all the essence of what we need to understand the entanglement of quanta. Figure 14: On the initial spacelike slice we have depicted two fourier modes: the longer wavelength mode is drawn with a solid line and the shorter wavelength mode is drawn with a dotted line. The mode with longer wavelength distorts to a nonuniform shape first, and creates an entangled pairs b1, c1. The mode with shorter 5.6 The entangled nature of ψ wavelength evolves for some more time before suffering the same distortion, and then it creates entangled pairs | ! b2, c2. Consider the state ψ | !1 The state of the first pair b1, c1 is of the form 2 † † γ † † † † γˆb†cˆ† ψ = C 0 b 0 c + γˆb 0 b cˆ 0 c + ˆb ˆb 0 b cˆ cˆ 0 c + . . . ψ 1 = Ce 1 1 0 (5.61) | !1 | ! 1 ⊗ | ! 1 1| ! 1 ⊗ 1| ! 1 2 1 1| ! 1 ⊗ 1 1| ! 1 | ! | ! ! ˆ† " 2 Here b1 is an operator that creates a quantum in the localized wavepacket depicted as b1 in = C 0 b1 0 c1 + γ 1 b1 1 c1 + γ 2 b1 2 c1 + . . . † (5.65) | ! ⊗ | ! | ! ⊗ | ! | !fig.1⊗4,| a!nd similarly cˆ1 creates the quantum of the wavepacket labeled c1. Because we have # broken up wavem$odes into localized wavepackets, we can define a sort of local vacuum 0 b1 in where n b1 means that we have n quanta of type b1 in the state etc. | ! | ! the region occupied by this mode b1. If we are in this vacuum state then there are no quanta The important feature of this state is that the b1 and c1 excitations are ‘enˆt†angled’. To in this region, if we act with b1 once then we have one quantum with this wavepacket, if we act understand this in more detail, let us take a simple example ô†fˆ†an entangled state. with b1b1 then we have two quanta of this type, and so on. Doing the same for the modes on S− we can write the state (5.61) as

5.7 Entanglement and the idea of ‘mixed states’ γˆb†cˆ† ψ = Ce 1 1 0 b 0 c (5.62) | !1 | ! 1 | ! 1 Consider two electrons, kept at two diﬀerent locations, and let each of them have a ‘spin up’ state and a ‘spin down’ state. Then this system can have ‘factored states’ of the form 26

ψ = ψ ψ (5.66) | ! | !1 ⊗ | !2 Examples are

ψ = | ! | ↑!1 ⊗ | ↓!2 1 1 ψ = ( 1 + 1) ( 2 + 2) (5.67) | ! √2 | ↑! | ↓! ⊗ √2 ↑! | ↓! etc. But we can also have ‘entangled states which cannot be written as a product of the type (5.66), for example 1 ψ = ( 1 2 + 1 2) (5.68) | ! √2 | ↑! ⊗ | ↓! | ↓! ⊗ | ↑! Suppose we ask: what is the state of electron 1? For states of type (5.66) we can answer this question: we ignore the state of electron 2 and just give the answer ψ . But for states of | !1

27 iωt Ψ = ψ(x)e− (201) 1 L = ∂ φ∂µφ (202) 2 µ τ (203) 1 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) | ! √2 | ! ⊗ | ! | ! ⊗ | ! GMm GM E = mc2 E = mc2 E 0 r (205) The problem of information− rloss (loss of∼ unitarity)∼ c2

Ψ = [ 0 0 + 1 1 ] | ! | !b1 | !c1 | !b1 | !c1 [ 0 0 + 1 1 ] ⊗ | !b2 | !c2 | !b2 | !c2 . . . [ 0 0 + 1 1 ] (206) ⊗ | !b1 | !c1 | !b1 | !c1

Entangled pairs

11 c b

(a) The b quanta are entangled with the c quanta

(b) Thus there is no state as such for the b quanta alone, but there is a state for the b and c quanta together

(d) There is not supposed to be any such state in quantum mechanics !! iωt Ψ = ψ(x)e− (201) 1 L = ∂ φ∂µφ (202) 2 µ τ (203) 1 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) | ! √2 | ! ⊗ | ! | ! ⊗ | ! GMm GM E = mc2 E = mc2 E 0 r (205) 2iωt − r ∼ Ψ = ψ∼(x)ec− (201)

Suppose you have an entangled state 1 µ Ψ = [ 0 1 0 1 + 1 1 1 L1 ]= ∂ φ∂ φ (202) | ! | !b | !c | !b | !c 2 µ [ 0 0 + 1 1 ] ⊗ | !b2 | !c2 | !b2 | !c2 τ (203) . . . 1 ψ = (1.1 0 0 + 0.9 1 1 ) (204) + [ 0 1 0 1 1 + 1 1 1+ 1 ]b1 c1 b1 c(1206) ⊗ | !b || !!c √| 2!b | !|c ! ⊗ | ! | ! ⊗ | ! iθ iθ e e− GMm (2G07M) E = mc2 E = mc2 E 0 r (205) − r ∼ ∼ c2 Suppose the left atom vanishes What is the state of the right atom ? Ψ = [ 0 0 + 1 1 ] | ! | !b1 | !c1 | !b1 | !c1 [ 0 0 + 1 1 ] ⊗ | !b2 | !c2 | !b2 | !c2 In fact, there is no . . . state, quantum ?? ?? + [ 0 1 0 1++ 1 1 1 1 ] (206) theory is ⊗ | !b | !c | !b | !c violated eiθ (207)

11 Essential problem # 1: The state of the Hawking radiation is entangled with what is left in the black hole, and when the black hole vanishes, the radiation is not in any ‘pure’ quantum state

Can small changes to this state make it ‘non-entangled’ ?

NO, that is not possible ......

That is the essential strength of the information paradox type (5.68) we cannot do this, and only the state of the entire system makes sense. Suppose we nevertheless want to ignore electron 2 in some way. Then we can make a ‘density matrix’

ρ = ψ ψ (5.69) | !" | For the two electron system we get 1 ρ = 2 | ↑!1 ⊗ | ↓!2 1"↑ | ⊗ 2"↓ | 1 + 2 | ↑!1 ⊗ | ↓!2 1"↓ | ⊗ 2"↑ | 1 + 2 | ↓!1 ⊗ | ↑!2 1"↑ | ⊗ 2"↓ | 1 + (5.70) 2 | ↓!1 ⊗ | ↑!2 1"↓ | ⊗ 2"↑ | ber is We can now ‘trace over’ the states of system 2, whichdf3okr the 1above case means that the bra Γ = σ(k) 3 ω (6.77) and ket states of system 2 must be the same in the term(2sπt)haetTwe k1eep. Then we get a ‘reduced density matrix’ describing system 1 − The semiclassical radiation from the hole is ‘thermal’ in this sense. But the essential problem that we have is not created1 by this ‘therma1lity’, but by the entangled nature of the state. ρ = + (5.71) Whether we have the en1tan2gle| d↑!s1ta1t"e↑ (|5.65) 2(w|h↓i!c1h c1a"↓n|be shown to be ‘thermal’ in the above sense) or the entangled state (5.74), which is very different from ‘thermal’, we face the same In general we get a density matrix of the form ρ = Cmn m n . The probability to problem. There is order unity entropy of1 entanmg,lnement |fro!m1 1t"he| state created by each pair of find system 1 in state k is given by the coefficient Ckk. These probabilities must add up to ˆ† † ! unity,ospoewraetohrasve(btkrρ, cˆ=k)1, .aTndheseontrhoeprye tihsatanreseunlttasnfgrolemmiegnntoreinntgrosypsyte(m5.726i)s gfoivrenthbeyradiation which is order Sbek. It is this entanglement that will eventually lead to information loss. By contrast, if a piece of coal burns away complSet=ely tro ρralndρiation, then this radiation is in a (p5u.7r2e)state, even − though it looks much more ‘thermal’ than a state which has the form (5.74) for each of the For theˆ†den†sity matrix (5.71) we can compute S easily since it is a diagonal density matrix (bk, cˆk). Thus ‘thermality’ is not reall1y the1 issue1; th1e issue is the entangled nature of the state created S = [ ln + ln ] = ln 2 (5.73) in the process of black hole ev−ap2orat2ion. 2 2 If the state ψ in (5.69) is ‘factorized’ as in (5.66) then when we make ρ and compute S then | ! 1 we get6.S2= 0C. aRnousgmhlyalslpqeaukainngt, uSmgivgersatvheitlyogeofffetchtesnuemnbceordoef tienrfmosrmin aatsiuomn liinke t(h5.e68r).adiation? The eCntornopsiydeisr tthhues daemriveastuiorne ooff hHoaww‘keinntganrgaldedia’ttihoen sdyisstceumsse1dainndth2earaeb.ove sections. We have used a classical metric and a quantum field φ on this ‘curved space’, but gravity itself has not been 5.8 tErenaterdopays qoufanthtiezeHd;atwhiksiins gcarllaeditahteiosnemiclassical approximation. Thus the semiclassical computation of radiation does not use the physics of quantum gravity anywhere. Since spacetime Let us now return to the black hole. The state (5.65) is not factorized between the b1 and c1 excitactuiornvsa.tuTrhee wnuams bloewr γinis tohrederreugnioitnys, swohtehreefitrhste fwewavteemrmosdeins tdheefosurmmewdilalnbde ocfreraelteevdanpcaer.ticles, this To expwloauinldthseeesmigntioficOurbaencae stategoofotdh eisape ofnptr aothisnxgilme dessentialatnioantu.rBe uofformt tohneesctatne swtiellwwilolnfodrercoinf vtehneiesnmcealrlecpolarrceections that the stwatoeu(l5d.6a5r)isbeyfrtohme sqimuapnletrumstagtreavity effects could change the state of the radiation to a pure state. There are two aspects to this question: 1 ψ 1 = ( 0 b1 0 c1 + 1 b1 1 c1 ) (5.74) (a) The first poin|t t!o no√te2is|th! at⊗a|s!mall |ch!an⊗ge|in! the state of the quantum field will not succeed in making the state of the b quanta a pure state. Focusing again on a given set (b1, c1) The quanta of type b1 lie on the part S+ of the spacelike surface which is outside the hori- we see that their state is a mixed one like (5.65). To get no entanglement of the b quanta with zon, while the quantaAo factorf typeedc1 statelie on wtouldhe pa bert Sof− thewh icformh is inside the horizon. Due to 1the the c1 quanta we would need a state like 28 ψ 1 = (C0 0 b + C1 1 b + . . .i)ωt (D0 0 c + D1 1 c + . . .) (6.78) | " | " 1 Ψ =| "ψ1(x)e− ⊗ | " 1 | " 1 (201)

But the state (6.78) is not a small pert1urbatioµn on a state like (5.65). The two states are L = ∂µφ∂ φ (202) completely diffeTherent ,essentialso we n epointed an iso thatrder2 au nsmallity c hchangeange i nint houre s tstateate o fwillea ch set (bk, ck) before the state can bnotecom make peu rite .a factorThus edif qstateuanτt u: m gravity is to help us, then it must(2co0m3)pletely change the evolution of the wavemodes that we have been drawing in the above sections. 1 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) (b) The second p|oin!t is t√ha2t even| i!f w⊗e h|ad! a state|lik!e (⊗6.7|8!), and thus the radiation quanta bk formed a pure state by themselves, it would not solve the information problem. Consider the Penrose diagisra almostm in fi gas.1 6entangled(a). Ther easa there n initialot two statebut twhree ehadkinds of matter involved in the problem. There is the matter that fell in to make the hole, marked Q. Then there are the Hawking radiation quanta bk (we have labeled them B) and their entangled partners, the ck (labeled C in the figure).

11 Thus a small change in the evolution of the wavemode will NOT solve the information problem

We need a change of ORDER UNITY

How does the information come out when we burn a piece of coal ?

We will find TWO important differences between the coal and the black hole 6.3 What is the difference between Hawking radiation and radiation from a burning piece of coal? Suppose a piece of coal burns away completely, leaving behind only the radiation it emitted. This time we know that subtle correlations in the emitted quanta encode the entire information about the state of the coal. But because these correlations are subtle, we cannot see them First difference: easily. How does this radiation differ from the Hawking radiation emitted by the black hole? Consider the first photonSupposeemitted anby atomthe c ‘a’oa lemits. Th isa pphotonhoton ‘p’can be in a mixed state with the matter left behind in the coal. Let us assume that an atom emits this photon, and that after the emission the spin of theTheato mspinsan doft thesehe sp icann o bef t hentangled,e photon soar thise co looksrrelat likede in an entangled wavefunction as follows the situation in the black hole 1 ψ 1 = ( a p + a p) (6.79) | ! √2 | ↑! ⊗ | ↓! | ↓! ⊗ | ↑! where a stands for the spin up state of the atom, p stands for the spin down state of | ↑! Now consider the next emitted| ↓photon! the emitted photon, etc.. Thus far, the situation looks just like the case of entangled b, c in the black hole. But the crucial difference is that when later photons are emitted from the coal, they can bounce off the atom left behind in the coal, and tThishus tphotonhe spin cans o finteractthese later photons can carry the information left behind in this atom. If thiswithatom thed rfirstifts oatomut it andself (as a piece of ash) then it can also carry the information of its spin. Thusitsa tstatethe willend beth eaffqecteduanta bcyo thellec ting at infinity are entangled only with themselves, and form a purspine st aoft etheca rfirstryin atomg all the information in the initial piece of coal. This gives subtle correlations between emitted photons ... Contrast this with the state of the radiated quanta bk in the black hole case, shown in fig.14. The quanta of type b1 are correlated with the quanta of type c1, which are located at a certain region on the part S− of the spacelike slice. But this place where c1 is located is not involved any further in the process of radiation from the black hole. For example, consider a later pair , say (b10, c10), and look at the region where this mode is suffering its nonuniform deformation. This region is not causally connected to the location where the earlier quanta c1 is located, so c1 cannot have any influence on the later emitted quantum b10. In the case of the coal the atom left behind after the first emission was in causal contact with later quanta leaving the coal. The black hole is different because each pair (bk, ck) is created at a point on a spacelike surface, and then this surface stretches so that the bk, ck quanta are moved away in different directions. New quanta are again created in the middle (i.e. at the horizon), these are again moved away by stretching, and so on. Thus all the created quanta bk, ck are located along different points of a very long spatial slice, with no overlap in their locations. Since the quanta are prevented from influencing each other by being spread out along this very long spatial slice, we should ask the basic question: how did we get this very long spatial slice when the black hole only had a given size GM? Recall from fig.4 that the spacelike ∼ slice inside the horizon was of the form r = constant, and it could be made arbitrarily long while remaining in the region r < 2GM. This possibility is unique to the black hole geometry, since it needs the light cones to ‘turn over’ and make the r = constant direction spacelike. This does not happen for the coal, and so later quanta can (and do) carry the information left in entangled pairs from earlier quanta.

33 wavelengths. Consider a wavemode with wavelength shorter than the one shown in fig.13. Then this mode will evolve for a longer time before it suffers a nonlinear deformation. This situation in depicted in fig.14. On the initial slice we have drawn two wavemodes of different wavelengths. The one with the longer wavelength becomes distorted first, and creates the quanta labeled b1 and c1 on the late time slice. The wavemode with shorter wavelength evolves for a longer time before becoming distorted, and creates the quanta labeled b2, c2.

wavelengths. Consider a wavemode with wavelength shorter than the one shown in fig.13. Then this mode will evolve for a longer time before it suffers a nonlinear deformation. Figure 14: On the initial spacelike slice we have depicted two fouriTerhims soidtuesa:tiothneinlodnegpeirctweadvienlefingg.t1h4.mOodnethise initial slice we have drawn two wavemodes of drawn with a solid line and the shorter waveByleng tcontrast,h mode is dInra wthedniffwe riblacktehntawdaov etholetleendgtlih neachse.. TThhe eparonmeowticlediethwti htpairhe lloonng giseerr wpravoducedelength be cinom aes distorted first, and creates wavelength distorts to a nonuniform shape firplacest, and thatcreat edoess an e ntnothtaenqgu linterfeadntpaaliarberseble1d, cwith1b1. aTnhd ethecm1 oo dnearlieretwheithlatse hpairsotirmteer slice. The wavemode with shorter wavelength wavelength evolves for some more time before suffering the same deivsotlovretsiofno,r aanldontgheerntimt cerbeaeftoerseebnetcaonmgliendg pdaisitrosrted, and creates the quanta labeled b2, c2. b2, c2.

The state of the first pair b1, c1 Ais opairf th eis fmadeorm at the γˆb†cˆ† horizon,ψ = Ce 1 1 0 (5.61) | !1 | ! ˆ† Here b1 is an operator that createsThena qu athentu surfacem in the localized wavepacket depicted as b1 in † fig.14, and similarly cˆ1 creates the strquetches,antum omof tvhese wpairavepacket labeled c1. Because we have broken up wavemodes into localized wavepackets, we can define a sort of local vacuum 0 b in apart, | ! 1 the region occupied by this mode b1. If we are in this vacuum state then there are no quanta ˆ† in this region, if we act with b1 onceThenthen aw enehwav epairon eisq uantum with this wavepacket, if we act ˆ†ˆ† with b1b1 then we have two quantamadeof thi satt ythepe, horizonand so o ...n. Doing the same for the modes on S− we can write the state (5.61) as

A similar state is produced by the wavemγoˆb†dcˆe† which started oﬀ with a shorter wavelength ψ 1 = Ce 1 1 0 b 0 c (5.62) on the initial slice. We get particle| p!airs described| b!y1 | ! 1

γˆb†cˆ† ψ = Ce262 2 0 b 0 c (5.63) | !2 | ! 2 | ! 2

The pairs bk, ck for different k lie in regions that do not overlap, so the overall state on the late time slice is the direct product of the states ψ k | ! Figure 14: On the initial spacelike slice we have depicted two fourier modes: the longer wavelength mode is drawn with a solid line and the shorter wavelength mode is drawn with a dotted line. The mode with longer ψ = ψ 1 ψ 2 ψ 3 . . . (5.64) | ! | ! ⊗ | ! ⊗ | ! w⊗avelength distorts to a nonuniform shape first, and creates an entangled pairs b1, c1. The mode with shorter wavelength evolves for some more time before suffering the same distortion, and then it creates entangled pairs We have presented a simplified discussion of the creatbe2d, c2p. airs; more technical details can be found in [2, 5, 6, 7]. For a more accurate description we should use a large number of oscillations The state of the first pair b , c is of the form in making each wavepacket (we have used just a few), and then we will have to con1sid1er many γˆb†cˆ† wavenumbers in each of the intervals on S over which the wavepackets extend. But the abψov1e= Ce 1 1 0 (5.61) ± | ! | ! approximate description has all the essence of what we needˆ†to understand the entanglement Here b1 is an operator that creates a quantum in the localized wavepacket depicted as b1 in of quanta. † fig.14, and similarly cˆ1 creates the quantum of the wavepacket labeled c1. Because we have broken up wavemodes into localized wavepackets, we can define a sort of local vacuum 0 b in | ! 1 5.6 The entangled nature of ψ the region occupied by this mode b1. If we are in this vacuum state then there are no quanta ˆ† | ! in this region, if we act with b1 once then we have one quantum with this wavepacket, if we act Consider the state ψ ˆ†ˆ† | !1 with b1b1 then we have two quanta of this type, and so on. Doing the same for the modes on S we can write the state (5.61) as −2 † † γ † † † † ˆ ˆ ˆ γˆb†cˆ† ψ 1 = C 0 b1 0 c1 + γb1 0 b1 cˆ1 0 c1 + b1b1 0 b1 cˆ1cˆ1 0 c1 + . . . ψ = Ce 1 1 0 0 (5.62) | ! | ! ⊗ | ! | ! ⊗ | ! 2 | ! ⊗ | ! 1 b1 c1 ! " | ! | ! | ! 2 = C 0 b 0 c + γ 1 b 1 c + γ 2 b 2 c + . . . (5.65) | ! 1 ⊗ | ! 1 | ! 1 ⊗ | ! 1 | ! 1 ⊗ | ! 1 26 where n b means th#at we have n quanta of type b in the state etc. $ | ! 1 1 The important feature of this state is that the b1 and c1 excitations are ‘entangled’. To understand this in more detail, let us take a simple example of an entangled state.

5.7 Entanglement and the idea of ‘mixed states’ Consider two electrons, kept at two diﬀerent locations, and let each of them have a ‘spin up’ state and a ‘spin down’ state. Then this system can have ‘factored states’ of the form

ψ = ψ ψ (5.66) | ! | !1 ⊗ | !2 Examples are

27 Second difference

As the coal radiates, there is less stuff left in the coal

The atom left behind can ﬂoat out as ash, so all the information goes out ﬁnally

Nothing can be left behind, because if there is no mass, there is no matter possible iωt Ψ = ψ(x)e− (201) 1 L = ∂ φ∂µφ (202) 2 µ τ (203)

1 wavelengths. Consider a wavemode with wavelength shorter than the one shown in fig.13. ψ 1 = (1.1 0 b1 0 c1iω+t 0.9 1Thebn1this mode 1will ecvo1lv)e for a longer time before it suffers a nonlin(ear2d0efor4ma)tion. Ψ = ψ(x)e− This situation in depicted in fig.14. On the initial slice we ha(ve2dr0aw1n t)wo wavemodes of In the black| ! hole√, 2the quanta| ! ⊗ ‘c’| falling! in ha| dv!iffeere nnegativt ⊗wavele|ngth!s. Tehe oenergne with the loynger wavelength becomes distorted first, and creates the quanta labeled b1 and c1 on the late time slice. The wavemode with shorter wavelength 1 µ evolves for a longer time before becoming distorted, and creates the quanta labeled b2, c2. 2 L2 = G∂Mµφm∂ φ GM (202) iωt E = miωtc E = mc 2 E 0 r 2 (205) Ψ = ψ(x)eΨ− = ψ(x)e− − r(201) ∼(201) ∼ c τ (203) 1 1 µ µ L = ∂ φ∂L φ= ∂µφ∂ φ (202) (202) µ 2 1 2 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) τ τ | ! √2 | ! ⊗ | !(203) | !(20⊗3) | !

1 1 2 2 GMm GM ψ 1 = ψ(1.11=0 b1 (10.1c10+b10.9E10=bc11 m+c01.9c11)Eb1 = m1cc1 ) (204) E (0204) r (205) | ! √2| ! | !√2⊗ | ! | ! ⊗|| !! ⊗ | ! | ! ⊗ | ! − r ∼ ∼ c2 GMm GMm GM GM E = mc2 EE==mmc2c2 E = mc2 E 0 rEwhen0 r (205) (205) 2 2 Figure 14: On the initial spacelike slice we have depicted two fourier modes: the longer wavelength mode is − r − ∼r ∼∼ c ∼ c drawn with a solid line and the shorter wavelength mode is drawn with a dotted line. The mode with longer wavelength distorts to a nonuniform shape ﬁrst, and creates an entangled pairs b1, c1. The mode with shorter wavelength evolves for some more time before suﬀering the same distortion, and then it creates entangled pairs b2, c2.

The state of the first pair b1, c1 is of the form γˆb†cˆ† ψ = Ce 1 1 0 (5.61) | !1 | ! ˆ† Where is the mass inside the black holeH er?e b1 is an operator that creates a quantum in the localized wavepacket depicted as b1 in † fig.14, and similarly cˆ1 creates the quantum of the wavepacket labeled c1. Because we have broken up wavemodes into localized wavepackets, we can define a sort of local vacuum 0 b in | ! 1 the region occupied by this mode b1. If we are in this vacuum state then there are no quanta ˆ† in this region, if we act with b1 once then we have one quantum with this wavepacket, if we act ˆ†ˆ† with b1b1 then we have two quanta of this type, and so on. Doing the same for the modes on S− we can write the state (5.61) as

γˆb†cˆ† ψ = Ce 1 1 0 b 0 c (5.62) | !1 | ! 1 | ! 1

11 11 11 The problem is that not only do the quanta B have to form a pure state, they have to carry the information of the matter Q. This is because in quantum mechanics the evolution of states is one-to-one and onto, and so different states of the initial matter Q have to give different states of the final radiation B. In fig.16(b) we have drawn the slices as shown in fig.5, with Q, B, C indicated. We see that the quanta Q reach small r first, and exist on each slice. The way we have drawn our slices keeps Q always in a region of low curvature; to achieve this we have evolved the small r region very little as we move from slice to slice. As the evolution proceeds the bk and ck quanta start appearing out of the vacuum modes. But these vacuum modes were localized in the region between the b and c quanta, far away from where Q sits on the slice. So how can the matter Q transfer its information to the bk? This is the essence of the information problem. Note that all the evolution depicted in fig.16(b) has been in a low curvature region, with slices that are smooth and carrying matter that is always low density. Thus it would appear that the situation is like the low curvature physics encountered in the solar system, and no unexpected quantum effects can occur. The only unusual thing is that through the course of The black hole vanishes because the the the evolution the slices stretch by a large amount, as discussed in section 4. In conventional relativity the total stretcenerghing fryo mof iQnit icancelsal to fin athel sl icenerge doesy nofot Cm, anottter ; quantum gravity effects wilnegativl not coem energe in ays longbecauseas the r atherte oef cwashan gnothinge is sma leftll. Tinh ithers facet may not be true in string thequantaory; for a discussion see [8]. Hawking radiation (positive energy)

Q C

Even as the black hole becomes tiny, it contains all the information of Q and all the details of C

Infalling matter (positive mass)

Figure 16: (a) The infalling matter Q and the entangled pairs C, B shown on the spacelike slices in the Penrose diagram. (b) Q, C, B sit at diﬀerent locations on the spacelike slices. To catch all three of these on the slices while staying in a low curvature region we have evolved the small r side less and the large r side more, something that we are certainly allowed to do in classical gravity.

32 The problem is that not only do the quanta B have to form a pure state, they have to carry the information of the matter Q. This is because in quantum mechanics the evolution of states is one-to-one and onto, and so different states of the initial matter Q have to give different states of the final radiation B. In fig.16(b) we have drawn the slices as shown in fig.5, with Q, B, C indicated. We see that the quanta Q reach small r first, and exist on each slice. The way we have drawn our slices keeps Q always in a region of low curvature; to achieve this we have evolved the small r region very little as we move from slice to slice. As the evolution proceeds the bk and ck quanta start appearing out of the vacuum modes. But these vacuum modes were localized in the region between the b and c quanta, far away from where Q sits on the slice. So how can the matter Q transfer its information to the bk? This is the essence of the information problem. Note that all the evolution depicted in fig.16(b) has been in a low curvature region, with slices that are smooth and carrying matter that is always low density. Thus it would appear that the situation is like the low curvature physics encountered in the solar system, and no unexpected quantum effects can occur. The only unusual thing is that through the course of the the evolution the slices stretch by a large amount, as discussed in section 4. In conventional relativity the total stretching from initial to final slice does not matter; quantum gravity effects will not come in as long as the rate of change is small. This fact may not be true in string theory; for a discussion see [8].

The nature of the evolution on spacelike slices looks like this

B and C are entangled Q does nothing but provide the mass needed to make the geometry

Figure 16: (a) The infalling matter Q and the entangled pairs C, B shown on the spacelike slices in the Penrose diagram. (b) Q, C, B sit at diﬀerent loWhatcations ownet hneede spac eislik feorsli cBes .toT obeca tach ‘non-entangled’all three of these o nstatethe s lices while staying in a low curvature region we have evolved the small r side less and the large r side more, something that we are certainly allowed to do in ccarlassircyingal gra vtheity. information of Q

32 Essential problem # 1: The state of the Hawking radiation is entangled with what is left in the black hole, and when the black hole vanishes, the radiation is not in any ‘pure’ quantum state

Essential problem #2: Even if we manage to dis-entangle b, c, we would still need the state of b to carry the information of Q. But Q is nowhere near the place where b, c are being created, so how can this happen? What are the possibilities allowed by this paradox ? What happens if we accept this situation, and have these b quanta entangled with the c quanta ... ?

(A) We can accept that quantum mechanics is violated ... (Hawking 1974). This is bad because quantum mechanics works so well in all other contexts

(B) REMNANTS: After the hole becomes very small (planck size), quantum gravity effects can stop the process of pair creation

The black hole does not evaporate away, the information in Q stays in the hole, and the b quanta stay entangled with the c quanta in the hole

Q C Problem with remnants: Since the starting hole can be as big as we want, there will have to by an inﬁnite number of remnants with planck mass

It is unusual for a theory to have an inﬁnite number of states for a bounded energy and volume

Loop amplitude will diverge because of infinitely many flavors of remnants iωt Ψ = ψ(x)e− (201) 1 L = ∂ φ∂µφ (202) 2 µ τ (203) 1 ψ 1 = (1.1 0 b1 0 c1 + 0.9 1 b1 1 c1 ) (204) | ! √2 | ! ⊗ | ! | ! ⊗ | ! √ GMm S = 2πG√Mn5(√n1 + n¯1)(√np + n¯p) (57) E = mc2 E = mc2 E 0 r (205) − r ∼ ∼ c2 E ! = 2π√n5( ) (58) √m1mp Ψ = [ 0 0 + 1 1 ] b1 c1 b1 c1 S = 2πd√nn1n5npnkk (59) | ! √N |n !√|n +! 1 | ! | ! √N √n + 1 (n + 1) n (175) (C) − W[ 0e canb2 0 findc2 + some1 b2 mechanism1≈c2 ] to change dthet ∝ ⊗ | ! | ! | ! | ! S = 2π√n1n5nkk(√np + n¯p) (60) evolution of field modes1 by order unity near the . . . ω = [ l 2 m m + m n] = ωgravity (176) horizon R R − − − ψ φ R E ! [ 0 b1 0 c1 + 1 b1 1 c1 ] = 2π√n1n5( (206)) (61) ⊗ | ! | ! |m! =|n! + n + 1, n = n n √mpmkk (177) L R L − R How icanθ this happen?iθ e e− λ m Sn=+ m2π√mn=1n05,(√nNp += 0 n¯p)((2√07n)kk + √n¯kk) (178) (62) | − ψ φ | From c , ! , G λ =the0 ,onlmy ψlength= l ,scalen = is0 , N l!=p 0 (208) (179) (63) − ∼ gravity 1 ωI = ωI n 6 l (180) (64) ∼ p 0 ψ M < 0 ψM 0 T 4 S1 (181) (65) | % | % 9,1 →| % ≈4,1 × × E/(2mkk) = 0.5 (66)

E/(2mkk) = 1.2 (67) 2 4 g α! √n1n5np 1 L [ ] 3 R (68) z ∼ V R ∼ s So it would seem that quantum gravity effects∆ Smust stay (69) near the singularity eS (70) eS+∆S (71)

S = 2π n n n (1 f) + 2π n n n f(√n + √n¯ ) (72) 1 5 p − 1 5 p k k " 1 ∆E! 1 nk = n¯k = = (73) 2 mk 2Dmk 2 4 n1n5npg α! 1 √ 3 D [ ] RS (74) ∼ V Ry ∼

∆S = S 2π√n1n5np = 1 (75) − A S = (76) 4G 2 G5 D mk 2 (77) ∼ G4 ∼ G5 1 1 D G 3 (n n n ) 6 R (78) 11 ∼ 5 1 5 p ∼ S

α N lp (79) eS (80)

5 10 S = 2π√n5(√n1 + √n¯1)(√np + n¯p) (57) E ! = 2π√n5( ) (58) √m1mp

S = 2π√n1n5npnkk (59)

S = 2π√n1n5nkk(√np + n¯p) (60) E ! = 2π√n1n5( ) (61) √mpmkk

S = 2π√n1n5(√np + n¯p)(√nkk + √n¯kk) (62)

l!p (63) ∼ 1 n 6 l (64) ∼ p M M T 4 S1 (65) 9,1 → 4,1 × × E/(2mkk) = 0.5 (66)

E/(2mkk) = 1.2 (67) 2 4 g α! √n1n5np 1 3 Lz [ ] Rs (68) ∼ V R A ∼ Sbek = = 2π√n1n5np (15) ∆S 4G (69) S Semicro = 2π√n1n5np (70) (16) eS+∆S1 ∂M (71) = T = ( ) (17) T −L zz ∂L S S = 2π n n n (1 f) + 2π n! n n f(√n + √n¯ ) (72) 1 5 p − 1 5 p k k " =1 ∆πE√! ( 1 E ) 0 (18) nk = n¯kT= ∈ =N ∈" ≈ (73) 2 mk 2Dm#k∈$√ 2 4 # √n1n5npg α! 1 $ D [ L ] 3 R (74) S = 2π√n5(√n1 + √n¯1)(√np + n¯p) = conSst. (57) (19) ∼ V Ry T M ∼ E ! = 2π√n∆5(S = S )2π√n1n5npS˜1= 1 (58) (75) (20) √m1m−p A dn √ √ dn T 4 √ S1 √ (21) √N n √n + 1 SN=√N2nπ√√nn1nn+5+n1p1nkSk = (n +N1) n + 1ndn (5(91)75)(n + 1)(76) n (175) −√N n √n + ≈1 − √Nd4tG∝√≈n×+ 1 dt(n∝+ 1) n (175) 2 − S = 2π√n1n5nkk(√n≈pS+= 2n¯π1p√) n1n5(√np + n¯pd)t ∝(60) (22) 1 G5 gDravity gravity mωR = [ l 2 mψm + mφn] = ω (77) (176) ωR = [ l 2 mψm +kmφnG] 2= ωRG (17R6) R − − − 1 E ∼! 4R∼− −5 − E % gravity = 2π√n1n5( 1 ) √ (61) ωR = [ mlm 2= 2πm1Nψm(2 + mφ)n] = ωR (23) (176) m = nL + nR + 1√, np 3=kknmL =nnR6L + nR + 1, n = nL (1n7R7) (177) RD −G−(n1n−5np) R&S 2mp − (78) ∼ 5 − ∼ S = 2λπ√nm1nψ5n(√+nmp +φm n¯=p)0(√, nkkNλ+=√m0n¯ψkkn) + mφm = 0, N(6(=21)708) (178) But a black hole is made of a large mnumber= n Lof +quanta|nR− + N 1,= ,son 1wnne5 m=ust|nL nR (24) (177) | − | α ask if the relevant length scales are l! p or N lp − (63) (79) λ = 0, mψ = ∼l, n = 0λ, = 0N, =m0ψ =4 l, 1 n =˜10, N(17=9)0 (179) 1 M∼9,1 M3,1 T − S S (25) −λ mψn +S mφm = 0, N = 0 (178) n 6 glravity e→ × × gra×vity (64) (80) ωI |= ω−p |ωI = ωI (180) (180) ∼ S =I 2π√n1n5(√np + n¯p)(√nkk + √n¯kk) (26) 4 1 M9,1 λ M=4,10, T mψS = l, n = 0, N =(650) (179) 0 →ψ × <×0 ψ50 0 ψ E < 0 ψ (1801) (181) | % | % | |%=≈%−2π√N%| % | % ≈ (27) E/(2mkk) = 0.5 grmavimty (66) ωI = ωI p kk (180) E/(2mkk) = 1.2 (67) N1 (28) 2 4 0 ψ 0 ψ 0 (181) g α! √n1n5np 1 L [ | % ] 3 R| %N N1 & | % ≈ (68) (29) z ∼ V R ∼ s − ∆S E1 (69) (30) S e E E1 (70) (31) − In this case the black hole would Sbe+∆ Sreplaced by a horizon - sized e E1 (71) quantum ‘fuzzball’. S = 2π N1 (2 ) (32) S = 2π n n n (1 f) + 2π n n n f(√n + √n¯ &) 2mp (72) 1 5 p − 1 5 p %k k String theory computations" suggest that such is the case ...(E E ) 1 ∆E! S =12π (N N ) 1 (33) nk = n¯k = = 1 − (73) 2 mk 2Dmk − mpmkk 2 4 % n1n5npg α! 1 E √ 3 D [ ] RS # = (74) (34) ∼ V Ry ∼ 2mkk 1 ∆S = S 2π√n1n5np = 1 (75) − (35) A √# S = (76) 4G 2 3 G5 D mk 2 (77) ∼ G4 ∼ G5 1 1 D G 3 (n n n ) 6 R (78) ∼ 5 1 5 p ∼ S

α N lp (79) eS (80)

10 10

10 Some misconceptions, and summary (since the remnant can result from an arbitrarily large black hole). Allowing the theory to have inﬁnitely many states within a bounded spatial region and within a bounded energy range is unnatural, and creates many problems for the theory. It would therefore seem best if somehow we could get the black hole to disappear and yet have the quanta bk be left in a pure state. Let us now discuss what would be needed for this to be possible.

6 Common misconceptions about information loss

We will ﬁnd it helpful to start by considering several common misconceptions about how information can come out of the black hole. Some common misconceptions (in my opinion) 6.1 Is the emitted radiation exactly thermal? A common argument about Hawking radiation is the following. The above discussed computa- (A) ‘To see if the informationtion scomesgive ‘the routmal r awdieat ishouldon’, but th seeere c oiful dthebe corrections (from the gravitational backreac- spectrum departs from thermality’tion of the created pairs, for example) which generate small deviations from ‘thermality’, and these deviations can encode the information that should escape from the hole.

This is incorrect. Black hole radiation has ‘grey body factors anyway, so it is not planckian

Figure 15: The planck distribution; small deviations from this distributions are indicated by the dotted curve.

The problem here is the word ‘thermal’. What is thermal radiation? One might think that The problem is the entangled‘ tnaturhermal’em ofean sthethe sstatepectrum. Wofer acandiatio makn shoueld itb evperlanyk ian; this spectrum is depicted by entangled and still make the 1-pointthe solid c ufunctionrve in fig.15 .exactlSmall dyev iplanckianations from this spectrum are shown by the dotted curve in fig.15. Can such a change in spectrum bring out information from the black hole? We will now see that the shape of the spectrum itself does not have much to do with whether the information comes out. For one thing, the spectrum of the semiclassical radiation from the black hole is not of the planck shape; the spectrum is modified by greybody factors. This is a general feature of radiation from any warm body – there is a modification to the spectrum if the emitted wavelength is comparable to the size of the body. For black holes, this wavelength is GM, ∼ which is the same order as the black hole size r 2GM. Thus the spectrum is not planckian ∼ anyway. A more correct definition of ‘thermal’ radiation is that if the body has an absorption cross section σ(k) for quanta of a certain wavenumber, then the emission rate for the same wavenum-

30 (B) The `Transplankian problem’: Hawking radiation arises from modes that were very high frequency in the past. Can we trust their evolution? Maybe they carry information after all ...

This is incorrect. We can b argue in the following steps:

(A) Look at the mode when it is still << M, but when we know the physics, say ~ fermi c (B) Spacetime is vacuum at the horizon. We assume the vacuum is unique. Then Figure 13: A fourier mode on the initial spacelike surface is evolved to later spacelike surfaces. In the initial part of the evolution the wavelength increaseswabut vthelengthere is no signiﬁc a~nt dfisermitortion of the general shape of the we know the physics of mode. At this stage the initial vacuum state is still a vacuum state. Further evolution leads to a distorted fermi scale modes waveform, which results in particle creation.

this is the crucial result for the physics of information, so we will now spend some time in understanding it.

25 (C) There are two possibilities:

(i) The mode is not in the vacuum state. Then there will be particles giving nuclear mass density at the horizon, which contradicts the traditional black hole picture.

(ii) The mode is in the vacuum state. In that case it will evolve as noted before into entangled pairs SUMMARY ‘Small’ quantum gravity effects CANNOT encode the information in subtle correlations in the Hawking radiation quanta.

We need to alter this evolution completely b

But this is evolution of wavelength ~M modes on smooth vacuum spacetime

So how can we change c it so much ??

this is the crucial result for the physics of information, so we will now spend some time in understanding it.

25 m = n = n + R + 1, n = n n Exppo[ωCF T t] (223) L L − R I gtt = 0 gtt > 0 (224) 1 S = Rd4x CF T (225) m = nL = n + R + 1, 16nπ=G nL nR Exppo[ωI t] (223) ! − CF T m = nL = n + R + 1, n = nL nR Exppo[ωI t] (223) gtt −=10 gtt >1 0 (224) R 2 2 (226) gtt = 0 gtt∼>L0 ∼ (GM) (224) 1 4 S =4 R2d x (225) 1 d 1x6π4 G(GM) (227) S = Rd∼CxF T! (225) m = nL = n + R + 1, n = nL nR16πEGxppo[1ωI t]2 1 (223) − ! S GM (228) R ∼2 2 (226) gtt = 0 gtt > 0 1 ∼ 1L ∼ (GM) (224) 1 1 1 iES t 1 iEAt ψ = R ψS +2 ψA 2 e− ψS + e− ψA (226) (229) 1 | # 2| ∼ #L 2∼| (dG#4Mx→) (2GM)2 | # 2 | # (227) S = Rd4x ∼ (225) Mathematical16πG statementd4x of inf(GormationMS)2 paradoGMx:2 (227) (228) ! ∼ψ1 ∼ψ2 ψn (230) In the classical black hole| solution,# 2 the| state# at the| horizon# is 1 11 S 1GM 1 iES t 1 iEAt (228) R the vacuumψ = 0ψ + ψ ψ 0 ψ e− 0 ψ0 ψ+ e−(1226)ψ (2(3212)9) 2 | #S2→ ∼| # A % | # ≈ %S | # ≈ A ∼1 L | ∼# (1G2M| ) # 2|1 # → 2 1 | # 2 | # This state evolves to give entanglediES pairs,t we get informationiEAt loss ψ = ψS + ψA e− ψS + e− ψA (229) | # d24x| #(GM2|)2 # → 2 | # 2 | #(227) We∼ need quantum effectsψ to1 change thisψ2 state at theψ horizonn (230) 2 | # | # | # S GMψ 0 ψ ψ ψ < 0 ψ > 0 (228) (230) (231) ∼ | 1# | |# 2→# | # | n# | ≈ 1 1 1 iES t 1 iEAt ψ = ψ + ψ 0such thate−ψ ψ 0 +ψ e− 0 ψ0 ψ 1 (229) (231) | # 2| S# 2| A#| →# →2 | # | S%#| #2≈ |% A| ## ≈ The paradox is that all such attempts failed, quantum corrections ψ alwaψys turned out ψto give (230) | 1# | 2# | n# 0 ψ 0 ψ 0 0 ψ 1 (231) | # → | # % | # ≈ % | # ≈