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1、arXiv:gr-qc/9502017v1 8 Feb 1995DAMTP-R/95/4gr-qc/9502017January, 1995Quantum Coherence and Closed Timelike CurvesS. W. HawkingDepartment of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeSilver StreetCambridge CB3 9EWUKAbstractVarious calculations of the S matrix have shown that

2、it seems to be non unitary forinteracting fields when there are closed timelike curves. It is argued that this is becausethere is loss of quantum coherence caused by the fact that part of the quantum statecirculates on the closed timelike curves and is not measured at infinity. A prescription isgive

3、n for calculating the superscattering matrix $ on space times whose parameters canbe analytically continued to obtain a Euclidean metric. It is illustrated by a discussion ofa spacetime in with two disks in flat space are identified. If the disks have an imaginarytime separation, this corresponds to

4、 a heat bath. An external field interacting with theheat bath will lose quantum coherence. One can then analytically continue to an almostreal separation of the disks. This will give closed timelike curves but one will still get loss1of quantum coherence.21. IntroductionThis paper is about what sens

5、e, if any, can be made of quantum field theory on a spacetimebackground that contains closed timelike curves. The development of causality violationin a bounded region is classically forbidden in the sense that it can not occur if the weakenergy condition holds 1. However, quantum field theory in cu

6、rved spacetime has manyexamples like the Casimir effect where the expectation value of the energy momentumtensor fails to obey the weak energy condition. It has therefore been suggested 2 that anadvanced civilization might be able to create a wormhole in spacetime which could be usedto travel into t

7、he past. This has led to a lot of interest in the problem of the formulationand behavior of quantum field theory in spacetimes with closed timelike curves.In general, it seems that divergences in the energy momentum tensor occur when onehas closed or self intersecting null geodesics 3. These diverge

8、nces may create spacetimesingularities which prevent one from traveling through to the region of closed timelikecurves 1.However, Kim and Thorne 3 have suggested that quantum gravitationaleffects may smear out the divergences and lead to a non singular spacetime. It is thereforeof interest to consid

9、er the properties of quantum field theory in spacetimes with closedtimelike curves.In particular, a number of authors have studied what I shall call, confined causalityviolating spacetimes. In these, there are well behaved initial and final regions and thecausality violations are restricted to a reg

10、ion in the middle. One can make this definitionmore precise but I shall refrain from doing so in order that I can discuss as wide a class ofexamples as possible. On such spacetimes, one might hope to calculate an S matrix whichwould relate the quantum state in the final region to the state in the in

11、itial region. Someauthors have claimed 4, 5 that this S matrix will be unitary for free fields but will benon unitary if there are interactions. To try to make sense of such non unitarity, Hartle6 has suggested that the usual amplitudes should be normalised by a factor that dependson the initial sta

12、te. This would restore conservation of probability but at the heavy priceof making quantum mechanics non linear.In principle, one would be able detect thenon linearity produced by a wormhole, which an advanced civilization might create in thedistant future. There would be a paradox, if this informat

13、ion were to cause the advancedcivilization to change its mind about creating the wormhole. But such paradoxes occuranyway with closed timelike curves.Anderson 7 has suggested that one should evolve the quantum state only with theunitary part of the S matrix U = (SS)1/2S. The trouble with this propos

14、al is that itdoesnt obey the usual composition law 8: the unitary part of S1S2is not the productof the unitary part of S1and S2separately. A third proposal is to extend the S to be a3unitary transformation on a larger Hilbert space 8. The trouble with this idea is that thelarger Hilbert space may ha

15、ve to have an indefinite metric.The message of this paper is that there is no need to propose non linear modificationsof quantum theory, or indefinate metrics on Hilbert space. The reason that the S matrix,calculated according to the usual rules, is non unitary, is that there is loss of quantumcoher

16、ence when there are closed timelike curves. This means that the probability to gofrom an initial state to a final state is given by a superscattering operator, $, rather thanby SS. Thus it does not matter that the object that one might think was the S matrix,is not unitary.A proposal has been made b

17、y Deutsch 9 and Politzer 10 for calculating the evolutionin the presence of closed timelike curves. This approach is based on finding a consistentsolution for the density matrix. This solution will involve loss of quantum coherence ingeneral. However, it will also depend in a non linear way on the i

18、nitial state 11, whichmeans that one loses the superposition principle.If one simply requires that the quantum theory is linear, the most general relationbetween the initial and final situations is not an S matrix but a superscattering operator,$, that maps initial density matrices, to final ones. I

19、n what follows it will be helpful to useindex notation. I shall represent a vector in a Hilbert space, by a quantity with an upperindex.A HThe corresponding vector in the complex conjugate Hilbert space, will carry a lower index.A HThe S matrix is a linear map from the initial Hilbert space to the f

20、inal Hilbert space, soit can be written as a two index tensor.+A= SABBHowever, the most general description of the quantum state of a system is not a vectorin a Hilbert space, but a density matrix. This can be regarded as a Hermitian two indextensor on Hilbert space. Then the most general linear evo

21、lution is given by a four indextensor that maps initial density matrices to final ones.A+B= $ABCDCDMany familiar quantum systems obey the Axiom of Asymptotic completeness 12.This requires that the Hilbert space of the interaction region in the middle is isomorphic4to the initial and final Hilbert sp

22、aces. In other words, there are unitary maps between theinteraction region Hilbert space and the initial and final Hilbert spaces. If this is the case,there will evidently be a unitary map from the initial Hilbert space to the final Hilbertspace. The superscattering operator $ will factorize into th

23、e product of an S matrix andits adjoint.$ABCD= SACSBDIn this situation, a density matrix corresponding to a pure quantum state will be carriedinto a pure quantum state. There will be no loss of quantum coherence.However, there are quantum systems that do not obey the Axiom of AsymptoticCompleteness.

24、 An example is provided by a particle interacting with a heat bath. A heatbath is not in a single quantum state. Rather it can be in any quantum state |ni withprobability expEnTIn other words, it is in a mixed quantum state . A particle thatis initially in a pure quantum state, which interacts with

25、the heat bath, will end up ina mixed quantum state. This loss of quantum coherence is to be expected: informationabout the original quantum state of the particle is lost into the heat bath. However, Iwill give examples of systems with closed timelike curves that are very similar to particlesinteract

26、ing with a heat bath.The only difference is that the temperature of the heatbath is imaginary. This corresponds to a spacetime that is identified periodically in realLorentzian time, rather than periodically in imaginary time, as in a normal heat bath.It should therefore come as no surprise that one

27、 gets loss of quantum coherence in thesecases. More generally, whenever one has confined causality violations, one has part of thequantum state that is circulating on closed timelike curves. When one makes measurementsat infinity, one does not see this part of the state. One will therefore have to d

28、escribe thestate at infinity by a mixed state, obtained by tracing out over the part of the state thatone cant see.In this paper I shall show that the usual rules of quantum theory seem to lead to lossof quantum coherence when there are closed timelike curves. I should emphasize that thisloss is not

29、 an optional feature that one can choose whether or not to have in the theory.Rather, like radiation from black holes, it is an inevitable consequence of the standardassumptions of quantum field theory in curved spacetime. The only way to protect thepurity of quantum states is either to abandon one

30、or more of these standard assumptions,or subscribe to the Chronology Protection Conjecture 1. This says that quantum effectsbecome so large when closed timelike curves are about to appear that they either preventthe curves appearing or they bring the spacetime to an end at a singularity. In either c

31、asethe laws of physics conspire to prevent causality violations.52. Euclidean ApproachIt is clear how to define quantum field theory on a curved spacetime background that isglobally hyperbolic. That is to say, it can be covered by a family of Cauchy surfaces. Inthis case, one can choose the commutat

32、or of two free field operators to be the half advancedminus half retarded Green function, which is well defined.(x),(y) = iD(x,y)However, it is much less clear how to proceed if the spacetime is not globally hyperbolic,and in particular, if it contains closed timelike curves. In this case, it is not

33、 clear howto generalize the half advanced minus half retarded Green function and field operators atpoints that are locally spatially separated may not commute because the points can bejoined by a timelike curve that goes round a large loop and returns to the same neighbour-hood.The approach I shall

34、adopt is to analytically continue the parameters of the spacetimewith closed timelike curves to get a metric with a real Euclidean section. On this section,all the field operators commute and the Green functions are well defined. One then analyt-ically continues back, both in the parameters of the m

35、etric, and in the points themselvesin a certain order, to get Green functions in the original Lorentzian spacetime. One canthen calculate the superscattering operator from the Green fun ctions according to certainrules which involve displacing the points slightly into the complex. This is equivalent

36、 tothe usual i prescription.One can illustrate this with a simple causality violating spacetime which has beendiscussed by Politzer 8. Two flat spacelike three dimensional disks of radius R are locatedin Minkowski space at t = 12T and t =12T and the same spatial coordinates. One makesthe following i

37、dentifications:1 The lower surface of the bottom disk is identified with the upper surface of the topdisk.2 The upper surface of the bottom disk is identified with the lower surface of the topdisk.The resultant spacetime is geodesically incomplete at the edges of the disks. However,one can impose bo

38、undary conditions there that make the wave equation well behaved, atany rate locally. The first identification is not very significant and is imposed just to avoidfree surfaces. But the second introduces closed timelike curves in the region between thedisks.In order to define the Green functions, I

39、shall first take the separation between thedisks T to be pure imaginary. One then has a Euclidean spacetime with time coordinate6 = it. On this one can define Green functions in the normal way as n point expectationvalues.G(x1,x2,.,xn) = h(x1)(x2).(xn)i=Zd(x1)(x2).(xn)eIIn the case of free fields, t

40、he n point function will be built out of all combinationsof two point functions. But if there are interactions, one will have the usual Feynmandiagram expansion in terms of the two point function on the background.On the Euclidean spacetime, all points are spacelike separated from each other. Thisme

41、ans that the field operators at different points commute with each other. Thus the npoint Green function does not depend on the order in which the n points are taken, as isobvious from the representation of the Green function by a path integral. On the otherhand, the n point Wightman functions in Lo

42、rentzian spacetime certainly do depend onthe order of the points, because the field operators do not commute at timelike separatedpoints. The way this comes about is that one analytically continues the n point expecta-tion values from the Euclidean to the Lorentzian regime, keeping a small imaginary

43、 timedisplacement between each field point 12. The displacement is in the positive imaginarytime direction between each point in the Lorentzian expectation value reading from left toright. That is, for the expectation valueh(x1)(x2).(xn)ione requires that Im(x01) Im(x02) . Im(x0n). The purpose of th

44、e displacement is toevaluate the analytically continued expectation values on the right side of the singularitiesthat occur on the complex light cone. It is equivalent to the usual i prescription, forintegrating round the singularities in the propagator.73. The Superscattering Operator $I now come t

45、o the rules for calculating the superscattering matrix from the Lorentzianexpectation values 12. One makes the usual assumption, that the field in the initial andfinal regions can be expanded in terms of annihilation and creation operators. =Xifiai+ fiaiThe annihilation operators aiare multiplied by

46、 positive frequency wave functions fiandthe creation operators aare multiplied by negative frequency wave functionsfi. Onecan invert this relation and express the annihilation and creation operators as integralsover spacelike surfaces of the field operator with negative or positive frequency wavefun

47、ctions.ai=Zfi(x)(x)d(x)One is interested in the expectation value of certain operators Q in a given initialstate with density matrix |ih|. This will be given byhIQIiwhere I is a string of creation operators that create the given initial state:|i = I|0iIn particular, one is interested in the probabil

48、ity of the final state containing a set ofparticles created by the string F of creation operators. This would correspond to takingQ = FF. Thus the superscattering matrix element between the initial state |ih|and the final state |+ih+| is determined by hIFFIi.One can express the annihilation and crea

49、tion operators in terms of the field operators.In this way, the superscattering matrix can be calculated from an integral of expectationvalues with initial and final wave functions. In order to get the right operator ordering inthe expectation value, these integrals over the initial and final surfac

50、es have to be slightlydisplaced in the imaginary time direction. The rule is, the initial creation operators havethe greatest displacement in the positive imaginary time direction. They are followed by thefinal annihilation operators, the final creation operators, and then the initial annihilationop