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1、arXiv:hep-th/9305165v2 31 May 1993DAMTP-R93-12CALT-68-1861Quantum Coherence in Two DimensionsS. W. Hawking&J. D. HaywardDepartment of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeSilver StreetCambridge CB3 9EWUK&California Institute of TechnologyPasadenaCalifornia 91125 USAMarch
2、 1993AbstractThe formation and evaporation of two dimensional black holes are discussed. It isshown that if the radiation in minimal scalars has positive energy, there must be a globalevent horizon or a naked singularity. The former would imply loss of quantum coherencewhile the latter would lead to
3、 an even worse breakdown of predictability. CPT invariancewould suggest that there ought to be past horizons as well. A way in which this couldhappen with wormholes is described.S.W.Hawkingamtp.cam.ac.uk, J.D.Haywardamtp.cam.ac.uk11. IntroductionThe discovery that black holes emit radiation 1 sugges
4、ts that they will evaporate andeventually disappear. In this process it seems that information and quantum coherencewill be lost and the evolution from initial to final situation will be described not by an Smatrix acting on states but by a super scattering operator $ acting on density matrices2. Th
5、is proposal of a non unitary evolution evoked howls of protest when it was firstput forward and three possible ways of maintaining the purity of quantum states were putforward:1 The apparent horizon eventually disappears and allows the information that went intothe hole to return.2 The back reaction
6、 to the emission of radiation introduces subtle correlations betweenthe different modes. These allow the information to come out continuously as theblack hole evaporates.3 The black hole does not evaporate completely but leaves some small remnant thatstill contains the information.The first possibil
7、ity, that the information comes out at the end of the evaporation, hasthe difficulty that energy is required to carry the information remaining in the black hole.However, there is very little rest mass energy left in the final stages of the evaporation. Theinformation can therefore be released only
8、very slowly, and one has a long lived remnant,like in possibility three.The second possibility, that the information comes out continuously during the evap-oration, has problems with causality. The particles falling into the hole would carry theirinformation far beyond the horizon before the curvatu
9、re would become strong enough forquantum gravitational effects to be important. Yet the information is supposed to appearoutside the apparent horizon. If one could send information faster than light like that, onecould also send information back in time, with all the difficulties that would cause.Th
10、e third possibility, black hole remnants, has problems with CPT if black holes couldform but never disappear completely. Consider a certain amount of energy placed in a boxwith reflecting walls3. The energy can be distributed in a large number of microscopicconfigurations, but one of two situations
11、will correspond to the great majority: eitherjust thermal radiation, or thermal radiation in equilibrium with a black hole at the sametemperature.Which possibility has more phase space depends on the energy and thevolume of the box.Suppose the energy is sufficiently low and the volume sufficiently l
12、arge that just ther-mal radiation, with no black hole, corresponded to more states. Then for most of thetime there would be no black hole in the box. However, occasionally a black hole would2form by thermal fluctuations, and then evaporate again. By CPT one would expect thisprocess to be time symmet
13、ric. That is, if you took a film, it would look the same runningforwards and backwards. But this is impossible if black holes can form from nothing butleave remnants when they evaporate. One can not even restore CPT, and get a sensiblepicture, by supposing theres a separate species of white holes th
14、at would have existedfrom the beginning of time. The number of white holes would always be going down, andthe number of black hole remnants would be going up, so one could never have a statisticalequilibrium in the box. We shall have more to say about CPT later. It is difficult to see howinformation
15、 and quantum coherence could be preserved in gravitational collapse. However,because General Relativity is non renormalizable, it is not clear what will happen in thefinal stages of black hole evaporation. Thus the question of whether quantum coherenceis lost is still open. For this reason there has
16、 recently been interest in two dimensionaltheories of quantum gravity which show an analogue of black hole radiation and whichhave the great advantage of being renormalizable.The first two dimensional theory that could describe the formation and evaporation ofblack holes was put forward by Callan, G
17、iddings, Harvey and Strominger (CGHS) 4. Itcontained a metric g and a dilaton coupled to N minimal scalar fields fi. In the classicaltheory a black hole can be created by sending a wave of one of the scalar fields. Quantumtheory on this classical black hole background then predicts the black hole wi
18、ll radiateat a steady rate indefinitely. CGHS hoped that the inclusion of the back reaction wouldcause the field configuration that initially resembled a black hole to disappear without asingularity or a global event horizon. Thus they hoped there would be no loss of informationand hence no loss of
19、quantum coherence.However, the most straightforward inclusion of the back reaction in the semi classicalequations did not realize this hope. There was necessarily a singularity where the dilatonhad a certain critical value 56. This singularity could either become naked, that is,visible from future n
20、ull infinity at late retarded times 789 or it could be a thunder-boltthat cut offfuture null infinity at a finite retarded time 1011. In either case part ofthe information about the initial quantum state would be lost on the singularity, whichwould be space like for at least part of its length, so o
21、ne might expect loss of quantumcoherence.The back reaction used in these calculations is based on the obvious andunambiguous measure for the path integral over the minimal scalars and the ghosts but itis not so clear what measure to use for the dilaton and the conformal factor. In the largeN limit t
22、his ambiguity in the measure shouldnt matter but the main hope of would-bedefenders of quantum purity was that the large quantum fluctuations when the dilatonwas near its critical value would cause the large N approximation to break down and that3higher order quantum corrections might prevent the oc
23、curence of singularities and preservequantum coherence. However, in this paper it will be shown that if the emission in scalarhas positive energy, then there must be either naked singularities or event horizons orboth. This argument depends only on the known measure for the minimal scalars, and isin
24、dependent of any corrections to the equations of motion that may arise from the measureon the dilaton and conformal factor or from higher order quantum effects.2. The conservation equationsThe argument is based on the fact that the conservation equations and the trace anomalyof the scalar fields det
25、ermine their energy momentum tensor up to constants of integrationwhich can be fixed by boundary conditions. In the conformal gauge in which the metric isds2= e2dx+dx(1)the energy momentum tensor of each of the minimal scalars isT= 112 ?x?22x2+ t(x)!(2)T+= 112+(3)where t(x) are constants of integrat
26、ion.Consider a situation in which the spacetime is flat, so that the conformal factor is ofthe form = logF(x) + logG(x+) and the energy momentum is zero before some nullgeodesic . This would be the case if the initial state was the linear dilaton solution. Onthe null geodesic one can change the coor
27、dinate xtoRxF2dxso that = 0 on. The range of xwill be (,). From the assumption that the energy momentumtensor is zero initially, it then follows that t(x) = 0 for all x.Suppose now that a wave with positive energy is sent in from the asymptotic regionof weak coupling at an advanced time x+later than
28、 and creates some black hole likeobject which radiates energy in the N minimal scalar fields. By equation (2), the outgoingenergy flux in the minimal scalar fields will beE =N12 2x2?x?2!(4)Let be an ingoing null geodesic at late advanced time. If the outgoing energy flux crossing is non negative,2x2
29、?x?2(5)4To integrate (5) along , one needs to know the initial value of /x. Let be anoutgoing null geodesic from a point p on to a point q on . We shall choose to lie inthe asymptotic region, that is, at early retarded times. One can choose the x+coordinatealong so that = 0 on . This fixes the coord
30、inates up to a Poincare transformation.With this choice of coordinates,x?q=18ZqpRdx+(6)One would expect the curvature R on to be positive and exponentially decreasing if theBondi mass measured at infinity,M e2R|x(7)on is positive. Thus, if one takes the null geodesic to be sufficiently far out in th
31、easymptotic region, the integral (6) will be positive.Suppose now that the outgoing energy flux Tis strictly positive on some interval of around a point r to the future of q. Then it follows from (5) and (6) that to the futureof r on log(c b) log(c x)(8)where b is the value of xat r and c is some fi
32、nite quantity greater than b. From (8) itfollows that will diverge at some point s on where x= a c. The point s may or notbe singular in the sense of the curvature R being unbounded but it will necessarily be atan infinite affine parameter distance along . It will however be at a finite retarded tim
33、ex(Fig 1). This means that the original hope of CGHS, that the black hole would evap-orate without global horizons or singularities, can not be realized in any two dimensionalquantum theory in which the energy emission is positive.Let be the portion of up to s. Then J(), the past of, will not includ
34、e thewhole of the null geodesic, , in the initially flat region. It is this kind behavior that givesrise to thermal radiation. Leth(x) be a wave packet that is zero for x a and is purelypositive frequency with respect to the affine parameter on the late time null geodesic.Thenh(x) is not purely posi
35、tive frequency with respect to the affine parameter on (which is proportional to x) because it is zero in a semi infinite range. Instead, therewill be some wave packeth(x) which is zero for x a. Thus there will be loss of quantum coherence.In the above, we have implicitly assumed that every outgoing
36、 null geodesic that in-tersects, also intersects . This allows us to deduce that the constant of integrationt(x) = 0 on each outgoing null geodesic. However, if there was a singularity that wasnaked in the sense that it was visible from, it wouldnt follow that on2x2?x?2Thus the requirement that the
37、radiated energy is positive implies either that there is anhorizon and associated loss of quantum coherence, or there is a naked singularity. In ouropinion, this would be much worse.The discussion so far has been in terms of a semi classical metric. However it shouldalso apply to each individual met
38、ric in a path integral over all metrics and dilaton fieldbecause our conclusions depend only on the asymptotic form of the metric in the far futureand past. Thus we would expect loss of quantum coherence, or naked singularities, or both,in the full quantum theory.3. Conformal Treatment of InfinityIn
39、 the previous discussion, the null geodesic was at early advanced time, the null geodesic was at late advanced time, and the null geodesic that connected them was at earlyretarded time. To make the arguments about the positive mass and energy of the emittedradiation, one wants to take the limit that
40、 these three null geodesics are at infinitely earlyor late advanced or retarded times. A precise and elegant way of doing this is to use theconcept of conformal infinity that was introduced by Penrose in the four dimensional case.One takes the spacetime manifold and metric M,gto be conformal to a ma
41、nifold withboundary and conformal metricM, gwhereg= 2 g = 0on MThe curvature scalars of the two metrics are related byR = 2R + 2 2()2(9)where the covariant derivatives are with respect to the conformal metric g. The physicalcurvature R will go rapidly to zero in the weak coupling region. It then fol
42、lows from (9)6that the boundary M will be null where 6= 0. The boundary in the weak couplingregion can be divided into future and past weak null infinities Iw. They will be joined bythe point I0representing spatial infinity. The conformal factor will not be smooth atI0. One can not say anything in g
43、eneral about the part of the M that lies in the strongcoupling region because one does not know how R will behave there. However, in the casethat spacetime is flat before some ingoing null geodesic , one will have a past strong nullinfinity Is, but one can not assume that there is necessarily a futu
44、re strong null infinity.One can take the conformal metric gto be flat. Then one can takeM to be theregion in two dimensional Minkowski space bounded by three null geodesics Is, IwandI+w(Fig 2). One does not know the form of the boundary on the fourth side, but this doesnot matter for the problem und
45、er consideration.The quantity = log will differ by a solution of the wave equation from the used in the previous section since it will obey different boundary conditions: = onM while = 0 on and . In order to identify the coordinate independent part of and we shall introduce a field Z with the coupli
46、ngZ = R(10)Z = 2R(11)We shall assume that the physical curvature goes to zero fast enough that 2R is boundedon I+sand I+w. One can then solve the wave equation (3) on the conformal spacetime(M, g) with the boundary conditions that Z = 0 on Isand Iw. The field Z on Mobtained in this way will correspo
47、nd to 2 where is the conformal factor in the previoussection in the limit that the null geodesic is taken to infinity.The energy momentum tensor of the ZT=12(ZZ 12g(Z)2) + (Z gZ)(12)will correspond to the energy momentum of the radiation in the N minimal scalar fields if2= N/24. Thus the energy out
48、flow across I+wisE = Tnn=12(Zn)2+ Znn(13)=12?dZdt?2+ ?d2Zdt2 qdZdt?(14)where n= dx/dt is the tangent vector to I+w, t is a parameter along I+wand nn=q n.7Define a metric g= exp(Z1)g. This metric is flat and corresponds to the flatbackground metric in section 2 in the limit that the null geodesic is
49、taken to infinitelyearly retarded times. Let t be an affine parameter with respect to the metric g on ingoingnull geodesics. Because g is flat, one can choose t to be constant on each out going nullgeodesic.Near Is, Z = 0 and the range of t will be (,). At later advanced times, Z 6= 0andq = 1dZdt(15
50、)Thus the energy flux across I+wisE = 12?dZdt?2+ d2Zdt2(16)If one replaces Z with 2, (16) becomes the same as (4). If the mass measured on Iwispositive, R 0 near Iw. If 0, this implies Z 0 anddZdt 0 near Iw.The argument is now similar to that in section 2. If E is non negative on I+wand isstrictly p