1994 二维黑洞的超散射矩阵-精品文档资料整理.pdf

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1、arXiv:hep-th/9401109v1 21 Jan 1994The Superscattering MatrixFor Two Dimensional Black HolesS. W. HawkingDepartment of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeSilver StreetCambridge CB3 9EWUKCalifornia Institute of TechnologyPasadena CA 91125USANovember 1993AbstractA consist

2、ent Euclidean semi classical calculation is given for the superscattering op-erator $ in the RST model for states with a constant flux of energy. The $ operator isCPT invariant. There is no loss of quantum coherence when the energy flux is less than acritical rate and complete loss when the energy f

3、lux is critical.11. IntroductionIn classical general relativity a lot of information is lost in the formation of a black hole.A black hole will settle down rapidly to a stationary state. The classical no hair theoremssay that this state is characterized just by the values of conserved charges, like

4、mass,angular momentum, and electric charge, that are coupled to gauge fields. In other words,a black hole of a given mass, angular momentum, and electric charge, can be formed bythe collapse of a very large number of different objects.This loss of information was not a worry in the purely classical

5、theory, because onecould say the information was still inside the black hole, even if one couldnt get at it. Butthe situation changed when it was discovered that according to quantum theory, black holesshould radiate and slowly evaporate. This made the question about the information in ablack hole m

6、uch more pressing. If the black hole evaporated and disappeared completely,what would happen to the information? There were three possible ways the informationmight be preserved:1 The information might come out again at the end of the evaporation. The problemwas that information requires energy to c

7、arry it and there wouldnt be much energyleft in the final stages.2 The information might come out continuously during the evaporation. The difficultyhere was that the information would be carried by the in-falling matter, far beyondthe apparent horizon. So, if it were to appear outside the horizon,

8、it would violatecausality. If one could send information faster than light like this, one could also sendit back in time.3 The black hole might not evaporate completely, but might leave some long lived rem-nant that could be said still to contain the information. But this would violate CPT,if black

9、holes could form, but never disappear completely.In this paper I want to take seriously the idea that information is lost. As I showedsome time ago1, this would imply a loss of quantum coherence. One could form a blackhole from matter in a pure quantum state, but it would decay into radiation in a m

10、ixedquantum state described by a density matrix. In ordinary quantum field theory the evo-lution is described by an S matrix. This can be thought of as a two index tensor on theHilbert space that maps initial states to final states:fA= SABiBHowever when one goes to quantum gravity, the possibility o

11、f forming real or virtual blackholes means that the evolution is given by what I call a super scattering operator, $. This2can be thought of as a four index tensor on Hilbert space that maps initial density matricesto final ones:fAB= $ABCDiCDIn general, the $ operator will not be the product of an S

12、 matrix with its complex conjugate:SABCD6= SACSBDThis proposal of a non unitary evolution was greeted with outrage by most particle physi-cists. I was accused of violating quantum mechanics. That is not the case. One gets lossof quantum coherence and a mixed state whenever there is part of a system

13、you dontmeasure. All I have done, is point out you cant measure the part of the quantum statethat is inside the black hole.The argument about whether quantum coherence is lost in black hole evaporation hasdragged on inconclusively over the years because general relativity is not renormalizable.This

14、has meant we have not been able to calculate what happens in the final stages ofblack hole evaporation. Maybe the answer lies in supergravity or superstring theory, butwe dont know how to use them to do the calculation. However, in the last two years,there has been a revival of interest in two dimen

15、sional black holes. These have the greatadvantage of being renormalizable, so it should be possible to use them to decide the issue.Those who felt a mission to defend quantum purity, hoped that these two dimensionalmodels would show that information and quantum coherence were preserved. However,they

16、 have been disappointed with the results so far: all calculations to date have shownloss of information and quantum coherence.This has left the unitary S matrix campmumbling something about the breakdown of the large N approximation, but with no realargument.Most of the calculations that have been d

17、one on two dimensional black holes, have beencarried out in Lorentzian spacetime. They have assumed no horizons or singularities in thepast. One can show2 that this implies that there must be horizons or naked singularitiesin the future. Thus the calculations are manifestly not CPT invariant. This i

18、s reflected inthe fact that the outgoing energy flux they predict is always below a certain critical level.However, the ingoing energy flux can have any value. Thus the super scattering operatorgiven by these calculations will not be CPT invariant. To lose quantum coherence is badenough, but to lose

19、 CPT as well, sounds like carelessness.I shall therefore outline a different approach to calculating the super scattering oper-ator which is guaranteed to give CPT invariant results. It is the Euclidean path integralmethod, coupled with the no boundary condition3.In the cosmological case, the no3bou

20、ndary condition literally meant that spacetime had no boundary. In other words, thequantum state is defined by a path integral over all compact positive definite metrics. Butin the particle scattering case, the appropiate quantum state is defined by a path integralover all positive definite metrics

21、that have one of more asymptotically Euclidean regionsbut no other boundary. One can show that the asymptotic Green functions in this quan-tum state are CPT invariant4. It then follows that the super scattering operator is CPTinvariant, in the sense of detailed balance5: the probability to go from t

22、he initial purestate A to a final state B, is the same as the probability to go from the CPT conjugate ofB to the CPT conjugate of A.2. Two dimensional black holesThe starting point for two dimensional black holes is the CGHS model6. This has ametric, g, and dilaton field, , coupled to N minimal sca

23、lar fields, fi.L =12ge2(R + 4()2+ 42) 12NXi=1(fi)2#The quantum effective action of the minimal scalars in the curved metric, g, can be eval-uated exactly and added to the action of the classical CGHS model. One can then definenew field variables, and , in which the theory is a conformally invariant

24、quantum fieldtheory and the action looks rather like that of the Liouville model:S =1Zd2x+ + + + 2exp?2( )?+12NXi=1+fifi#In making the theory, a conformal field theory, one has to modify the action. One can eitherchange the kinetic term, as Russo, Susskind, and Thorlacius do7, or the non derivativet

25、erm, as Bilal and Callan8, and de Alwis9 do. Both modifications end up with thesame Liouville like theory, but the relation between the Liouville fields, and , and thephysical fields, g and , is different. The RST version has the great advantage that itadmits the linear dilaton as a solution. The ot

26、her, ABC, version does not seem to haveany natural ground state. I shall therefore work with the RST version.In the RST model the Liouville fields are related to the physical fields by =2 +e2 = 2 +e24where the metric is in the conformal gaugeds2= e2dx+dxThe Liouville model looks to be almost linear,

27、 but this is deceptive. Both in the ABCand RST versions, the field, , is bounded below as a function of the physical field . Ittakes its minimum value, c, when = c. There are two possible attitudes one can taketo this. One, propounded by de Alwis9, is that one should take the quantum theory tobe def

28、ined by a path integral over the full range of and from minus infinity to plusinfinity. The other attitude, put forward by RST7, is that the path integral should berestricted to the range of that corresponds to real physical fields. I shall adopt this latterapproach. The physical fields are what are

29、 important. The Liouville fields, and , canbe regarded just as mathematical constructions that help to solve the field equations, butare of no physical significance of themselves.The field equations for the Liouville model are simple.+( ) = 0+( + ) = 22exp?2( )?The first equation corresponds to the

30、freedom to choose coordinates. One can thereforetake = 0at least locally. It is then easy to write down the general solution in this coordinate gauge: = 2x+x+ F(x+) + G(x)To make this solution Euclidean, introduce a complex coordinate z in the plane withz = x+, z = xThis very simple theory becomes h

31、ighly non trivial if one imposes the requirementthat the physical fields be non singular. One way to get a non singular solution is if omegais always above its minimum value, c. The only solutions like this that are also staticand with only an asymptotically flat region can be written down in the =

32、gauge: =2z z + M +25 is a function only of r in polar coordinates, with a minimum at the origin.Thesesolutions represent Euclidean black holes with the horizon at the origin. The polar angle, corresponds to imaginary time. Because this is periodic, the black hole is in thermalequilibrium. There is a

33、 steady flow of energy in at infinity and a similar flow out.Suppose now that one has a field configuration in which actually reaches the criticalvalue on some curve, C. Then will drop below the critical value in a neighbourhood ofC and the physical fields will be singular unless the gradient of van

34、ishes on C. Thus thenecessary condition for the physical fields to be non singular is|C= 0I shall refer to this as the RST boundary condition. With this boundary condition, nosignal can propagate through the curve C. One can therefore cut offthe spacetime beyondC and consider only the region between

35、 C and asymptotically flat infinity.One canregard the curve C as being like the axis of symmetry in the dimensional reduction ofa spherically symmetric spacetime to two dimensions. On the basis of this analogy, theappropiate boundary condition on the minimal scalars would be that their normal gradie

36、ntshould vanish on C.If C is time like, the normal direction will be space like. The field equation+ = 2exp?2( )?will then imply that the second derivative of in the normal direction will be positive.This means that cin a neighbourhood of C and the physical fields can be nonsingular. On the other ha

37、nd, if C is space like, the normal direction will be time like andthe second derivative of in the normal direction will be negative. Thus will go belowcand the physical fields will be singular. Thus the RST boundary condition can not beimposed in Lorentzian spacetime if the boundary is space like.By

38、 contrast, in the Euclidean regime the normal direction to C is always space like.This means one can always satisfy the boundary condition.Thus the Liouville modelwith the RST condition and the no boundary condition, is a well defined quantum theory.One should therefore be able to calculate the supe

39、rscattering operator and see whetherit involves loss of quantum coherence. I shall give evidence that it does, in at least somesituations.63. The quantum RST modelIf one just had the Liouville model without any restriction on the range of , the theorywould be linear and the semi classical approximat

40、ion would be exact. This means thatthe quantum theory would be determined completely by solutions of the quantum effectiveaction, which will be the same as the original Liouville action. However, if the range of is restricted, and the RST boundary condition imposed, the theory becomes effectivelynon

41、 linear. This means that the solutions of the Liouville theory, are no longer the wholestory. But one can still hope that they will give a first approximation.As well as the field equations obtained by varying and , there are constraint equa-tions obtained by varying the components of the metric tha

42、t are zero in the conformalgauge. In terms of the physical fields, the constraint equations can be written in a formlike the Einstein equations:he2+4i(4 22) =12NXi=1fifi ( 2+ t)The term on the left that is the analogue of the Einstein tensor, is the trace free part ofthe second covariant derivative

43、of . The first term on the right is the classical energymomentum tensor of the scalar fields. The remaining terms on the right can be regardedas the quantum induced energy momentum tensor. It is non local because it contains thefunctions, t(x), that have to be determined by boundary conditions. Give

44、n a solutionof the field equations, one can always find functions t(x) that will satisfy the constraintequations. I shall therefore regard the left hand side of the constraint equations as thedefinition of the total energy momentum tensor, classical plus quantum, of the scalar fields.In other words,

45、 given a solution of the and field equations, one can read offfrom theconstraint equations what the energy momentum flow is.In the = gauge there are a two parameter family of solutions which have a Killingvector in the direction of the polar angle, . The two parameters are the coefficient ofthe logr

46、 term and the constant term in . Imposing the RST boundary condition that = 0 where = cputs one relation between the two parameters. One therefore hasa one parameter family of static Euclidean solutions. =2z z P log(2z z) P?log?P?+ 1?4hlog?4?+ 1iThe linear dilaton is the member of this family with P

47、 =4. In it, has a minimum ofcon a circle around the origin. The total energy momentum tensor of the scalar fields7calculated in the manner I described, is zero. Thus this solution represents the vacuumwith no energy flux at infinity. However for 0 P ceverywhere. There is a oneparameter family of bla

48、ck hole solutions, but all have the same energy flux at infinity.There are no regular static Euclidean solutions with an energy flux greater than this value.That is what one might expect because if one sent in energy at such a rate, one wouldcreate a black hole that would grow faster than it could r

49、adiate itself away. So one couldntget a static solution.4. The superscattering operatorOne would expect these Euclidean solutions to give the dominant contributions to thesuperscattering operator for incoming states with a steady flux of energy. Consider firstthe solutions, like the linear dilaton,

50、that have a boundary at cIn these, the region insidethe boundary circle is not regarded as part of the spacetime. This means theres no reasonto identify the polar angle, , with any particular period. It is therefore natural to take to run the full range from minus infinity to infinity.It is straight