真实之路 宇宙法则的完全指南 (彭罗斯)30量子态缩小的.pdf

30 Gravity s role in quantum state reduction 30 1Is today s quantum theory here to stay In this chapter I shall put the case to the reader that there are powerful positive reasons over and above the negative ones put forward in the preceding chapter to believe that the laws of present day quantum mech anics are in need of a fundamental though presumably subtle change These reasons come from within accepted physical principles and from observed facts about the universe Yet I Wnd it remarkable how few of today s quantum physicists are prepared to entertain seriously the idea of an actual change in the ground rules of their subject Quantum mechanics despite its extraordinary exception free experimental support and strik ingly conWrmed predictions is a comparatively young subject being only about three quarters of a century old dating this from the establishment of the mathematical theory by Dirac and others based on the schemes of Heisenberg and Schro dinger in the years immediately following 1925 When I say comparatively I am comparing the theory with that of Newton which lasted for nearly three times aslong before it needed serious modiWcationintheformofspecialandthengeneralrelativity andquantum mechanics Even if we are to count Newton s theory as suVering its Wrst modiWcation with the introduction of Maxwellian Welds this still gave it an exception free reign of over a century and three quarters Moreover Newton s theory did not have a measurement paradox While the linearity of quantum theory s U process gives that theory a particular elegance it is that very linearity or unitarity which leads us directly into the measurement paradox 22 2 Is it so unreasonable to believe that this linearity might be an approximation to some more precise but subtle non linearity We have a clear precedent Newton s gravitational theory has the par ticular mathematical elegance that the gravitational forces always add up in a completely linear fashion yet this is supplanted in Einstein s more precise theory by a distinctly subtle type of non linearity in the way that gravitational eVects of diVerent bodies combine together And Einstein s theory is certainly not short on elegance of a quite diVerent kind from 816 that of Newton We also see in Einstein s theory that the modiWcations to Newton s theory that were needed were nothing like the tinkering that I referred to in 29 2 At various times such tinkerings with Newton s theory had indeed been suggested such as a replacement of the power 2 in Newton s inverse square formula GmM r2 see 17 3 by 2 00000016 as suggested by Aspeth Hall in 1894 in order to accommodate those very slight deviations as ascertained in 1843 from the Newtonian predictions of Mercury s motion about the Sun and Hall s suggestion gets good fits also for the other planets as shown by Simon Newcombe 1Einstein s theory subsequently explained these deviations without fuss but the new theory was by no means obtained just by tinkering with the old it involved a completely radical change in perspective This it seems to me is the general kind of change in the structure of quantum mechanics that we must look towards if we are to obtain the in my view needed non linear theory to replace the present day conventional quantum theory Indeed itismyownperspectivethatEinstein sgeneralrelativitywillitself supply some necessary clues as to the modiWcations that are required The 20th century gave us two fundamental revolutions in physical thought and to my way of thinking general relativity has provided as impressive a revolution ashas quantum theory or quantum Weld theory Yet these two geat schemes for the world are based upon principles that lie most uncom fortablywitheachother Theusualperspective withregardtotheproposed marriage between these theories is that one of them namely general rela tivity must submit itself to the will of the other There appears to be the commonviewthattherulesofquantumWeldtheoryareimmutable anditis Einstein s theory that must bend itself appropriately to Wt into the standard quantum mould Few would suggest that the quantum rules must them selves admit to modiWcation in order to ensure an appropriately harmoni ous marriage Indeed the very name quantum gravity that is normally assigned to the proposed union carries the implicit connotation that it is a standardquantum Weld theorythatissought Yet Iwouldclaimthatthere is observational evidence that Nature s view of this union is very diVerent fromthis Icontendthatherdesignforthisunionmustbewhat inoureyes would be a distinctly non standard one and that an objective state reduc tion must be one of its important features 30 2Clues from cosmological time asymmetry What evidence is this Let us Wrst turn to those places where Nature s choice of quantum gravity union most clearly reveals itself I refer to the spacetime singularities of the Big Bang and of black holes and also the Big Crunch if such is to take place In Chapter 27 the extraordinarily special nature of the Big Bang was presented in stark contrast to the seemingly 817 Gravity s role in quantum state reduction 30 2 generic nature of the singularities of collapse Despite the brave sugges tions made in accordance with the Hartle Hawking proposal as discussed in 28 9 I see no escape from a gross time asymmetry being a necessary feature of Nature s quantum gravity union Such a temporal asymmetry would seem to be completely at variance with the implications of any standard quantum Weld theory Let us con sider for example the CPT theorem noted in 25 4 Recall that T stands for time reversal whilst P and C stand respectively for space reversal and for the replacement of particles by their antiparticles If we believe that the CPT theorem applies to our sought for quantum gravity union then we are in trouble If we apply CPT to any allowed generic Wnal singularity of gravitational collapse then we get an initial type singularity as a possibility for the Big Bang or for part of the Big Bang Recall the enormity of the available phase space as described in 27 13 and graph ically illustrated in Fig 27 22 Once such generic initial singularities become allowable then there is nothing to guide the Creator s pin into that absurdly and from the anthropic perspective see Fig 28 13 un necessarily tiny region B that seems to have been the actual starting point of our universe It seems to me to be clear that the mystery of the extraordinarily special nature of the Big Bang cannot be resolved within the standard framework of quantum Weld theory At least this would be the case for any theory for which the word standard entails the validity of the CPT theorem 25 4 Strictly speak ing that theorem is not immediately applicable to a theory that fully respects the curved spacetime basis of Einstein s general relativity One of the premises of the CPT theorem is that the background spacetime is Xat Minkowski space Nevertheless I suspect that most physicists would regard this as an unimportant technicality taking the view that one can re express Einstein s theory if it is desired in the form of a Poincare invariant Weld theory by introducing a Minkowski background as a convenience Personally I have strong reservations about this type of procedure 2yet I would tend to agree that it seems unlikely that the completely time symmetrical classical Einstein theory of general relativity should become so time asymmetrical when submitted to the standard time symmetrical procedures of quantum Weld theory On the other hand we recall that in 25 5 26 5 11 we encountered situations where a symmetry of the classical theory becomes broken when we pass to the quantum theory Might it be that it is this that happens when Einstein s theory becomes brought appropriately within the com pass of standard QFT rules I suppose that this is conceivable but it is hard to see how this could be very much like the type of symmetry breaking that occurs in say electroweak theory where the vacuum state jYi is taken not to share the symmetries of the quantum dynamics 818 30 2CHAPTER 30 If this idea is to work then jYi has to be time asymmetrical I am not sure how one could make sense of this kind of idea It is true that the ket jYi is what would be placed over on the right hand side of all Weld operators in the manner described in 26 11 and could be thought of as representing the initial state of the universe which here means the very particular Big Bang state But in standard QFT the complex conjugate of jYi namely the bra hYj would also feature in the formalism being needed for the formulation of probabilities via expressions like hYjAjYi and it would play a completely symmetrical role to jYi but with time reversed Thus hYj would have to represent the Wnal state of the universe and we have a Wnal state of a similar structure to the initial one in gross contra diction with the entire message of Chapter 27 There are also other features that arise in the process of quantization whereby the quantum theory might not share the symmetries of the classical theory known as anomalies These come about when the classical commutation rules providing the classical symmetry given by Poisson brackets see 14 8 cannot be fully realized by quantum commutators with only a subgroup of the whole classical symmetry group surviving in the quantum theory Anomalies seem normally to be regarded as things to be avoided and we shall see the contortions that theorists sometimes have to perform in order to eliminate such things when we come to consider string theory in the next chapter Yet one might imagine taking a diVer ent view and regard an anomaly as being a good thing in those circum stances when the larger symmetry is something that one does not want to have However in our present case it is a discrete symmetry namely CPT in addition to T CT and PT indeed anything with a T in it that one needs to violate and it is hard to see the relevance of the usual anomaly idea which usually but not always refers just to the continuous symmet ries that can be realized in terms of Poisson brackets However one looks at it it is hard to avoid the conclusion that in those extreme circumstances where quantum eVects and gravitational eVects must both have come together in the spacetime singularities at the Big Bang and in gravitational collapse gravity just behaves diVerently from other Welds Recall the Wnal conclusion in the penultinmate para graph of Chapter 27 concerning this point For whatever reason Nature has imposed a gross temporal asymmetry on the behaviour of gravity in such extreme circumstances 30 3Time asymmetry in quantum state reduction Does this relate to any other clues concerning the possible interrelation between gravity and quantum mechanics I strongly believe so Whereas we perceive no time asymmetry in the U part of quantum theory 27 1 819 Gravity s role in quantum state reduction 30 3 there is an essential time asymmetry in R We can see this very easily in a simple hypothetical quantum experiment Suppose that there is a photon source S which emits single photons from time to time and that whenever it does so this event is recorded 3I shall suppose that the photons have high energy being possibly even X ray photons The photons are aimed at a beam splitter B half silvered mirror angled at 458 to the beam so that if a photon is transmitted through then it activates a detector D at the other side while if it is reXected then it gets absorbed into the ceiling C see Fig 30 1 I am supposing equal amplitudes for these two alternatives so that the detector will register reception of the photon in just one half of the occasions that the source is registered as having emitted the photon This is just a straightforward application of the R procedure There is an amplitude 1 ffi ffi 2 p ignoring possible phase factors for the photon history SBD and an amplitude 1 ffi ffi 2 p for the photon history SBC Application of R s squared modulus rule then gives the correct answer that whenever there is an emission event at S there is a 50 probability of a detection event at D and by inference a 50 probability of the photon reaching C This is simply the correct answer But now let us imagine reading this particular experiment backwards in time I am not proposing that we try to build a backward time source or detector No thephysicalprocessesarenottobealteredinanyway Itisjust that I propose to phrase my questions about them in a reverse time form Rather than asking about the Wnal probabilities let us ask what the initial probabilities are given that there is a detection event at D The relevant amplitudes now refer to the two alternative histories SBD and FBD where C B F DS Fig 30 1A source S randomly emits single high energy photons each such event being recorded aimed at a beam splitter B tilted at 458 to the beam If transmit ted through B the photon activates a detector D route SBD if reXected it is absorbed at the ceiling C route SBC The quantum squared modulus rule correctly predicts probabilities 1 2 1 2 On the other hand given that D registers the photon could have come from S route SBD or from the Xoor F route FBD Used in the reversed time direction the squared modulus rule incorrectly retro dicts probabilities 1 2 1 2 which should be 1 0 820 30 3CHAPTER 30 FstandsforapointontheXoorwiththepropertythatifaphotonweretobe emitted from there it could be reXected at B to be received at D Again the amplitudes are 1 ffiffi 2 p for each of these two histories ignoring phases This must be so because the ratio of the modulus of the amplitudes for going onewayortheotherisjustapropertyofthebeamsplitter Thereisnotime asymmetryhere Now ifweweretoapplythe squared modulusrule toget the probabilities for these two alternatives we would Wnd a probability of 50 for emission at S and 50 by inference for the photon to come from the Xoor F whenever there is a detection event at D This of course is an absurdity There is virtually a zero chance that an X ray photon will jump out of the Xoor aimed at the beam splitter The probabilities are more like 100 that there was an emission event at S and 0 that the photon came from the Xoor F whenever there is a detection event at D The squared modulus rule applied in the past direction has simply given us completely the wrong answer 4 Of course this rule was not designed to be applied into the past but it is instructive to see how completely wrong it would be to do so Sometimes people have objected to this deduction pointing out that I have failed to take into account all sorts of particular circumstances that pertain to my time reversed description such as the fact that Second Law of thermo dynamics only works one way in time or the fact that the temperature of the Xoor is much lower than that of the source etc But the wonderful feature of the quantum mechanical squared modulus law is that we never have to worry about what the particular circumstances might be The miracle is that the quantum probabilities for future predictions arising in the measurement process do not seem to depend at all on considerations of particular temperatures or geometries or anything 5If we know the ampli tudes then we can work out the future probabilities All we need to know are the amplitudes The situation is completely diVerent for the probabil ities for retrodiction Then we do need to know all sorts of detailed things about the circumstances The amplitudes alone are quite insuYcient for computing past probabilities There are however situations in which the quantum probabilities can be computed in a way that is completely symmetrical in time and it is perhaps instructive to have a look at these These occur where the quantum state is measured to be something known both before and after some intermediate quantum measurement To be more explicit imagine a succession of three measurements where the Wrst one projects the state into jci and the third one projects it into jfi there being a yes no measurement between these two described by the projector E 22 6 The probability of yes for the middle measurement is then given by 30 1 30 1 Why Can you derive this formula 821 Gravity s role in quantum state reduction 30 3 jhfjEjcij2 where we assume the normalizations hcjci 1 hfjfi which is cer tainly time symmetrical To set up such a situation one performs the succession of three measurements many many times over and picks out for examination only those cases for which the Wrst measurement yields jci and the third yields jfi The above probability then refers to the fraction of these cases for which the middle measurement yielded yes 6 This has led some people to conclude that there is at root no time asymmetry in quantum measurement 7 However most quantum measurements are not of this kind For the normal forward time use of the squared modulus rule we do not specify a jfi and for the above backward time attempted use we do not specify jci We see that we can perfectly well calculate the quantum probabilities while not specifying jfi but we cannot get away with not specifying jci One might take the view that the reason that the quantum rules work well for the future probabilities has to do with jfi being in some sense random which has to do with the Second Law of thermodynamics Perhaps there is something in this but I Wnd this requirement for jfi rather unclear What does random mean in this context Nevertheless there would certainly seem to be some connection with the Second Law in the measurement question We may take note of the fact that actual measuring devices tend to take advantage of this law in some part of their operation That there is some connection between R and the Second Law is indeed part of my own perspective on the matter And since we have seen that the Second Law is intimately tied up with the missing quantum gravity union we must expect an intimate relation between R and this anticipated union also Before coming to this issue more explicitly it is worth point