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Quantum Coherence in Two Dimensions

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 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 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.Hawking@amtp.cam.ac.uk, J.D.Hayward@amtp.cam.ac.uk1

1. IntroductionThe discovery that black holes emit radiation [1] suggests 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 matrices[2]. This 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 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 possibility, 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 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 curvature 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.The 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 walls[3].

The energy can be distributed in a large number of microscopicconfigurations, but one of two situations 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 large 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 would2

form by thermal fluctuations, and then evaporate again. By CPT one would expect thisprocess to be time symmetric.

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 there’s a separate species of white holes that 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 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 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, Giddings, Harvey and Strominger (CGHS) [4]. Itcontained a metric g and a dilaton φ 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 will 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 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 [5][6]. This singularity could either become naked, that is,visible from future null infinity at late retarded times [7][8][9] or it could be a thunder-boltthat cut offfuture null infinity at a finite retarded time [10][11].

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 one 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 this ambiguity in the measure shouldn’t 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 that3

higher order quantum corrections might prevent the occurence 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 isindependent 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 determine their energy momentum tensor up to constants of integrationwhich can be fixed by boundary conditions.

In the conformal gauge in which the metric isds2 = −e2ρdx+dx−(1)the energy momentum tensor of each of the minimal scalars isT±± = −112 ∂ρ∂x±2−∂2ρ∂x±2 + t±(x±)! (2)T+−= −112∂+∂−ρ(3)where t±(x±) are constants of integration.Consider a situation in which the spacetime is flat, so that the conformal factor is ofthe form ρ = log F(x−) + log G(x+) and the energy momentum is zero before some nullgeodesic γ.

This would be the case if the initial state was the linear dilaton solution. Onthe null geodesic γ one can change the coordinate x−toR x−F 2dx′−so that ρ = 0 onγ.

The range of x−will 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 γ 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 ∂2ρ∂x−2 − ∂ρ∂x−2! (4)Let λ be an ingoing null geodesic at late advanced time.

If the outgoing energy flux crossingλ is non negative,∂2ρ∂x−2 ≥ ∂ρ∂x−2(5)4

To integrate (5) along λ, one needs to know the initial value of ∂ρ/∂x−. Let µ be anoutgoing null geodesic from a point p on γ to a point q on λ.

We shall choose µ to lie inthe asymptotic region, that is, at early retarded times. One can choose the x+ coordinatealong µ so that ρ = 0 on µ.

This fixes the coordinates up to a Poincare transformation.With this choice of coordinates,∂ρ∂x−q = 18Z qpR dx+(6)One would expect the curvature R on µ to be positive and exponentially decreasing if theBondi mass measured at infinity,M ∝e−2φR|x−→−∞(7)on µ is positive. Thus, if one takes the null geodesic µ to be sufficiently far out in theasymptotic region, the integral (6) will be positive.Suppose now that the outgoing energy flux T−−is strictly positive on some interval ofλ around a point r to the future of q.

Then it follows from (5) and (6) that to the futureof r on µρ ≥log(c −b) −log(c −x−)(8)where b is the value of x−at r and c is some finite quantity greater than b. From (8) itfollows that ρ will diverge at some point s on µ 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 λ. It will however be at a finite retarded timex−(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 ¯λ be the portion of λ up to s. Then J−(¯λ), the past of ¯λ, will not include thewhole of the null geodesic, γ, in the initially flat region. It is this kind behavior that givesrise to thermal radiation.

Let ¯h(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 ¯λ.Then ¯h(x−) is not purely positive frequency with respect to the affine parameter on γ(which is proportional to x−) because it is zero in a semi infinite range. Instead, therewill be some wave packet ˆh(x−) which is zero for x−< a and which is such that ¯h + ˆh ispurely positive frequency on γ.

This will mean that the initial vacuum state in each of theminimal scalar fields fi will appear to contain pairs of particles, one in the ¯h mode, andthe other in the ˆh mode. The ¯h mode will appear to contain a particle on the null geodesic¯λ.

But the ˆh will not cross ¯λ, so an observer in the asymptotic region5

will not see this particle. This would mean that the quantum state would appear tobe a mixed state, described by a density matrix obtained by tracing out over the modesfor x−> a.

Thus there will be loss of quantum coherence.In the above, we have implicitly assumed that every outgoing null geodesic that in-tersects ¯λ, also intersects γ. 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 ¯λ, it wouldn’t follow that on ¯λ∂2ρ∂x2−≥ ∂ρ∂x−2Thus the requirement that the 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 metric 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 the previous discussion, the null geodesic γ was at early advanced time, the null geodesicλ was at late advanced time, and the null geodesic µ 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 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, gµν to be conformal to a manifold withboundary and conformal metric ˜M, ˜gµν wheregµν = Ω−2˜gµνΩ= 0on ∂˜MThe curvature scalars of the two metrics are related byR = Ω2 ˜R + 2Ω˜ Ω−2( ˜∇Ω)2(9)where the covariant derivatives are with respect to the conformal metric ˜gµν. The physicalcurvature R will go rapidly to zero in the weak coupling region.

It then follows from (9)6

that the boundary ∂˜M will be null where ∇µΩ̸= 0. The boundary in the weak couplingregion can be divided into future and past weak null infinities I±w .

They will be joined bythe point I0 representing spatial infinity. The conformal factor Ωwill not be smooth atI0.

One can not say anything in general 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 γ, one will have a past strong nullinfinity I−s , but one can not assume that there is necessarily a future strong null infinity.One can take the conformal metric ˜gµν to be flat.

Then one can take ˜M to be theregion in two dimensional Minkowski space bounded by three null geodesics I−s , I−w andI+w (Fig 2). One does not know the form of the boundary on the fourth side, but this doesnot matter for the problem under consideration.The quantity ˜ρ = −log Ωwill differ by a solution of the wave equation from the ρused in the previous section since it will obey different boundary conditions: ˜ρ = ∞on∂˜M while ρ = 0 on γ and λ.

In order to identify the coordinate independent part of ρ and˜ρ we shall introduce a field Z with the couplingZ = −νR(10)˜ Z = −νΩ−2R(11)We shall assume that the physical curvature goes to zero fast enough that Ω−2R is boundedon I+s and I+w . One can then solve the wave equation (3) on the conformal spacetime( ˜M, ˜gµν) with the boundary conditions that Z = 0 on I−s and I−w .

The field Z on Mobtained in this way will correspond to 2νρ where ρ is the conformal factor in the previoussection in the limit that the null geodesic µ is taken to infinity.The energy momentum tensor of the ZTµν = 12(∇µZ∇νZ −12gµν(∇Z)2) + ν(∇µ∇νZ −gµνZ)(12)will correspond to the energy momentum of the radiation in the N minimal scalar fields ifν2 = N/24. Thus the energy out flow across I+w isE = Tµνnµnν = 12 (∇µZnµ)2 + ν∇µ∇νZnµnν(13)= 12dZdt2+ νd2Zdt2 −q dZdt(14)where nµ = dxµ/dt is the tangent vector to I+w , t is a parameter along I+w and nν∇νnµ =q nµ.7

Define a metric ˆgµν = exp(−Zν−1)gµν. This metric is flat and corresponds to the flatbackground metric in section 2 in the limit that the null geodesic µ is taken to infinitelyearly retarded times.

Let t be an affine parameter with respect to the metric ˆg on ingoingnull geodesics. Because ˆg is flat, one can choose t to be constant on each out going nullgeodesic.Near I−s , Z = 0 and the range of t will be (−∞, ∞).

At later advanced times, Z ̸= 0andq = ν−1 dZdt(15)Thus the energy flux across I+w isE = −12dZdt2+ ν d2Zdt2(16)If one replaces Z with 2νρ, (16) becomes the same as (4). If the mass measured on I−w ispositive, R ≥0 near I−w .

If ν > 0, this implies Z ≥0 and dZdt ≥0 near I−w .The argument is now similar to that in section 2. If E is non negative on I+w and isstrictly positive on some interval, then by (16), Z will diverge at a point s on I+w at afinite value of the parameter t. But the range of t on I−s is infinite.

Thus there will be apart of I−s which is not in the past of s which is the future end point of I+w because it isat infinite distance in the natural affine parameter. In other words, the spacetime has aglobal event horizon.Again there is the alternative of a naked singularity.

In claiming that the energymomentum tensor of the Z is equal to the radiation in the N minimal scalars, we haveimplicity assumed that the radiation is uniquely determined by the conservation equations,the trace anomaly and the boundary conditions at infinity. This will not be the case ifthere’s a singularity visible from I+w .

So again the requirement that the radiation haspositive energy implies there is either an event horizon or a singularity. The argumentsabout loss of quantum coherence are then the same as in section 2.8

4. ConclusionsIt is possible that two dimensional black holes are not a good model for the four dimensionalcase.

The fact that the field equations of the CGHS model with back reaction becomesingular at a critical value of the dilaton field, suggests that this may be the case. However,if two dimensional models are any guide to the real world, our results indicate that anyLorentzian description of black hole evaporation must have horizons, or naked singularities,or both.

Of the two possibilities, naked singularities, would seem the worse. Unless one hassome boundary condition at a naked singularity, one can not predict what will happen.There is no obvious candidate for such a boundary condition: the boundary conditionsthat have been proposed seem rather ad hoc.By contrast, in a Euclidean treatment, there is a natural boundary condition, namelythe no boundary condition, which says that there are no singularities and no boundariesin the Euclidean domain, other than asymptotically flat space.

This boundary conditionof no boundaries should mean that is asymptotic Green functions are defined by a pathintegral over all fields and Euclidean metrics that are asymptotically flat. These Greenfunctions can then be used to calculate how ingoing particles evolve to outgoing particles,maybe with loss of quantum coherence.

It is not obvious that this process will have aLorentzian description, but if it does, our results suggest that it will contain horizons.By CPT symmetry, one might expect that there would be past horizons as well as futurehorizons. It is bad enough to lose quantum coherence, but to lose CPT symmetry as wellseems like carelessness.

This leads to a picture in which particles would fall into a holethat was already existing in the vacuum. The hole would grow in size and mass and thenevaporate down to a hole like those in the vacuum.

One might claim that the informationabout the particles that fell in was not lost, that it was still contained in the residualblack hole. But if this residual hole was indistinguishable from holes in the vacuum, theinformation is effectively lost, and the outgoing radiation will be in a mixed state.This picture is similar to that of scattering offan extreme magnetically charged blackhole: the hole grows in mass and then evaporates back to the original zero temperatureblack hole.

One can see that the information is contained in the residual black hole, butthat is just words. The amount of information that can be fed in is infinite, and there isno way the information can be recovered.

Moreover, as the radiation is emitted in a weakfield region, there is no reason to distrust the semi classical calculations that indicate thatit is in a mixed state. It is this effective loss of quantum coherence that is the physicallyimportant result, rather than any semantics about whether the information can be thoughtof as being contained in some remnant.The only difference between the picture being suggested here, and the magnetically9

charged case, is that one would have to imagine that the ground state with zero mass andconserved charge also contained objects with zero temperature future and past horizons.But this is just what there is in the Lorentzian section of a Euclidean wormhole[12].Consider the Euclidean metricds2 =1 + a2x22dx2This corresponds to two asymptotically Euclidean regions connected by a wormhole orthroat of size a. However, the Lorentzian section obtain by x4 →ix4 looks rather different.Its Penrose diagram is shown in figure 3.

It has an outer null infinity Io like flat Minkowskispace but now the light cone of the origin has also been sent to infinity to become an innernull infinity Ii. The two null infinities intersect in two two spheres I+ and I−.

This is thefour dimensional analogue of the Penrose diagram for the linear dilaton solution, whichalso has two infinities. This supports the idea that there is a close connection betweenwormholes and the formation and evaporation of black holes.

Particles and informationfalling into black holes pass into another universe, and particles from that universe enterours in the form of black hole radiation. Further developments of this idea will be publishedelsewhere.This work was supported in part by the U.S. Dept.

of Energy under contract no.DE-AC03-81-ER40050. Part of this work was done while SWH was a Sherman FairchildDistinguished Scholar at Caltech.10

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T_ _> 0cb pq matter wave in after null line Figure 1 : Physical Spacetime Diagramλ µγ ..srλ _

sI-I+ Figure 2 : Conformal Spacetime DiagramwJ-(I+ w)wI-s

I+0I+I-Figure 3 : Conformal Diagram of WormholeI i

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