真实之路 宇宙法则的完全指南 (彭罗斯)24狄拉克电子和.pdf

24 Dirac s electron and antiparticles 24 1Tension between quantum theory and relativity The considerations of 23 10 only just begin to touch upon some of the profound issues about the relation between the principles of quantum mechanics and those of relativity Indeed in presenting the detailed way in which quantum theory operates in the preceding three chapters I have taken a very non relativistic standpoint appearing to ignore those import ant lessons that Einstein and Minkowski have taught us as described in Chapter 17 about the interdependence of time and space In fact this is quite usual in quantum theory The standard approach adopts a picture of reality in which time is treated diVerently from space As remarked early in Chapter 22 there is a single external time coordinate but there are many spatial ones each particle requiring its own set This asymmetry is usually regarded as a temporary feature of non relativistic quantum theory which would be merely an approximation to some more complete fully relativistic scheme In this chapter and in the following two we shall begin to witness the profound issues that arise when we try seriously to bring the principles of quantum theory together with those of special relativity The more ambitious union with Einstein s general relativity where gravitation and spacetime curvature are also brought into the picture requires something considerably more and there is as yet no consensus on the most promising lines of pursuit I shall address some of these lines in Chapters 28 and 30 33 It is a particular feature of combining quantum theory with special relativity that the resulting theory becomes not just a theory of quantum particles but a theory of quantum Welds The reason for this can be boiled down to the fact that the bringing in of relativity implies that individual particles are no longer conserved but can be created and destroyed in conjunction with their antiparticles This comment needs some explan ation Why is there this need for antiparticles in a relativistic quantum theory Why does the presence of antiparticles lead us from a quantum theory of particles to a quantum theory of Welds This chapter is largely aimed at the answers to these two questions but particularly the Wrst and 609 with particular reference to Dirac s wonderful insights into the mathemat ical description of electrons Quantum Weld theory itself will be discussed in Chapter 26 and we shall be glimpsing some of the pervasive tension between special relativity and quantum theory that has guided the subject of particle physics into more and more elaborate mathematical schemes We shall Wnd ourselves enticed into a long and fascinating journey When the tension can be resolved appropriately as with the standard model of particle physics discussed in Chapter 25 the resulting theory is found to have a very remarkable agreement with observational fact Yet in many respects this tension has remained and has never been fully resolved Strictly speaking quantum Weld theory at least in most of the fully relevant non trivial instances of this theory that we know is mathematically inconsistent and various tricks are needed to provide meaningful calculational operations It is a very delicate matter of judge ment to know whether these tricks are merely stop gap procedures that enable us to edge forward within a mathematical framework that may perhaps be fundamentally Xawed at a deep level or whether these tricks reXect profound truths that actually have a genuine signiWcance to Nature herself Most of the recent attempts to move forward in fundamental physics indeed take many of these tricks to be fundamental We shall be seeing several examples of such ingenious schemes in this and later chapters Some of these appear to be genuinely unravelling some of Nature s secrets On the other hand it might well turn out that Nature is a good deal less in sympathy with some of the others 24 2Why do antiparticles imply quantum fields The theoretical anticipation of antiparticles in a relativistic quantum theory appears to have unravelled one of Nature s true secrets now well supported by observation We shall be seeing something of the theoretical reasons for antiparticles later in this chapter and most speciWcally in 24 8 For the moment instead of addressing that issue let us restrict attention to the second of the two questions raised above namely why does the presence of antiparticles leads us away from a quantum theory of particles and into a quantum theory of Welds Let us for now just accept that there is an antiparticle to each type of particle and try to come to terms with the consequences of this remarkable fact The key property of an antiparticle at least the antiparticle of a massive particle is that the particle and antiparticle can come together and annihilate one another their combined mass being converted into energy in accordance with Einstein s E mc2 conversely if suYcient 24 2CHAPTER 24 610 energy is introduced into a system localized in a suitably small region then there arises the strong possibility that this energy might serve to create some particle together with its antiparticle Thus with this potential for the production of antiparticles there is always the possibility of more and more particles coming into the picture each particle appearing to gether with its antiparticle Thus our relativistic theory certainly cannot just be a theory of single particles nor of any Wxed number of particles whatever In quantum theory as we shall be seeing in Chapters 25 and 26 particularly if there is the potential for something to happen e g the production of numerous particle antiparticle pairs then this potential possibility actually makes its contribution to the quantum state In at tempting to come up with a theory of relativistic particles therefore one is driven to provide a theory in which there is a potential for the creation of an unlimited number of particles This takes us outside the framework of Chapters 21 24 but we shall see in Chapter 26 how the quantum theory of Welds enables us to accommo date such behaviour Indeed according to a common viewpoint the primary entities in such a theory are taken to be the quantum Welds the particles themselves arising merely as Weld excitations Yet we shall Wnd that this is not the only way to look at quantum Weld theory In the Feynman graph approach which we shall address in Chapters 25 and 26 there is a strong particle like perspective on the basic processes that go to make up the quantum Weld theory where indeed an unlimited number of particles can be created or destroyed It is instructive to elaborate a little more on the reasons underlying particle creation as a feature of a sensible relativistic quantum theory I am still assuming for the moment that antiparticles exist Essentially the reason to expect particle creation comes down to Einstein s famous E mc2 Energy is basically interchangeable with mass c2being merely a conversion constant between the units of energy and mass that are being used When enough energy is available then a particle s mass can be created out of that energy However having the means to produce the particle s mass is not in itself suYcient for the conjuring up the particle itself There are likely tobevariousconserved additive quantumnumbers suchas electric charge or other things e g baryon number which are not supposed to be able to change in a physical process Simply to conjure a charged particle out of pure energy for example would represent a violation of charge conservation and the same would apply to other conserved quantities such as baryon number etc However with the assumption that for every kind of particle there is a corresponding antiparticle for which every additive quantum number is reversed in sign a particle together with its antiparticle can be created out of pure 611 Dirac s electron and antiparticles 24 2 AntiparticleParticle Energy Time energy see Fig 24 1 All the additive quantum numbers will be conserved in this process The rest mass of the antiparticle rest mass being non additive is on the other hand the same as that of the original particle We need suYcient energy at least twice that of the rest mass energy of the particle itself to create both the particle and its antiparticle in this process Conversely if a particle of a given type encounters another particle which is of its antipar ticle s type then it is possible for them to annihilate one another with the productionofenergy Again theenergyhastobeatleasttwicetherestmass energy of the individual particle In either the creation or the annihilation process the energy can be more than this value because the particle and antiparticle are likely to be in relative motion and there will be an energy residinginthismotion thekineticenergy thataddsintothetotal Inany event we see that the presence of antiparticles indeed forces us away from the quantum theory of individual particles as described in Chapters 21 23 24 3Energy positivity in quantum mechanics Let us now return to the road that ultimately leads us to the requirement of antiparticles in a relativistic quantum theory We shall need to examine the framework of quantum theory from a somewhat deeper perspective than before First let us recall the basic form of the Schro dinger equation i h c t Hc Suppose that we require our quantum system to have a deWnite value E for its energy so that c is an eigenstate of energy with eigenvalue E that is since H is the operator deWning the total energy of the system we require Hc Ec Fig 24 1A particle and its antiparticle can be created out of energy All the particle s conserved additive quantum numbers are reversed in sign for the antiparticle to ensure conservation of these quantities in the creation process 612 24 3CHAPTER 24 According to the quantum mechanical R process 22 1 5 such a state c would result from our having performed a measurement on a system asking it the question what is your energy where we have received the speciWc answer E Schro dinger s equation then tells us i h c t Ec The solutions of this equation have the form 24 1 c C e iEt h where C is independent of t i e a complex function of spatial variables only Now it is important that the energy value E be a positive number Negative energy states are bad news in quantum mechanics for various reasons their presence leading to catastrophic instabilities1 24 2 When the energy E is indeed positive the coeYcient iE h of t in the exponent in e iEt h is a negative multiple of i Recall from 9 5 and see Note 9 3 that functions c t of this nature or linear combinations of such functions are said a little confusingly to be of positive frequency Recall also that in 9 3 we addressed the splitting of a function f x of a real variable x into its positive and negative frequency parts in an apparently completely diVerent way namely in terms of the geometry of the Riemann sphere 2There we treated this as just an elegant piece of pure mathematics The real line could be thought as wrapped once around the equator of the Riemann sphere and the positive frequency part of the function f was understood as that part which extended holomorphically see 7 1 into the southern hemisphere the negative frequency part extending likewise into the northern hemisphere But now we have come to a remarkable physical reason for the great importance of this notion Any self respecting wavefunction though it need not itself be an eigenstate of energy ought to be expressible as a linear combination of eigenstates of energy and each energy eigenvalue ought to be positive Thus the time dependence of any decent wavefunction ought indeed to have this crucial positive frequency property It seems to me that this remarkable relation between an essential physical requirement on the one hand and an elegant mathematical property on the other is a 24 1 Check that this is indeed a solution 24 2 Explain why adding a constant K to the Hamiltonian simply has the eVect that all solutions of the Schro dinger equation are multiplied by the same factor Find this factor Does this substantially aVect the quantum dynamics Suppose we are concerned with the gravitational eVect of a quantum system Why can we not simply renormalize the energy in this way under these circumstances 613 Dirac s electron and antiparticles 24 3 wonderful instance of the deep subtle and indeed mysterious relationship between sophisticated mathematical ideas and the inner workings of our actual universe In non relativistic quantum mechanics this requirement of positive fre quency tends to come about automatically as a natural feature of the theory provided that the Hamiltonian comes from a reasonable physical problem where classical energies are positive For example in the case of a single free non relativistic spinless particle of positive mass m we have the Hamiltonian H p2 2m recall 20 2 21 2 The expression p2 and hence the Hamiltonian H itself is what is called positive deWnite 3 13 8 9 Classically this comes about because p2is a sum of squares and this cannot be negative p2 p p p1 2 p2 2 p3 2 Quantum mechanically we must make the replacement of p by i h where x1 x2 x3 and now the positive deWnite assertion refers totheeigenvaluesoftheoperator H2 fornormalizablestates i e elements of an appropriate Hilbert space H and again these cannot be negative esentially for the same reason as in the classical case 24 3 24 4Difficulties with the relativistic energy formula Now let us consider a relativistic quantum particle In this case the Hamiltonian is obtained from the relativistic expression for the energy which is not p2 2m but c2m 2 c2p2 1 2 This expression comes directly from the equation c2m 2 E2 c2p2of 18 7 where m is now the rest mass of the particle The reader who worries that this expression does not look much like p2 2m should refer back to Exercise 18 20 That told us from a power series expansion of c2m 2 c2p2 1 2 that our relativistic expression incorporates Einstein s famous E mc2as a Wrst term This term is the energy contribution coming from the particle s rest mass and it is additional to the kinetic energy of the particle s motion The second term indeed gives us the Newtonian kinetic energy Hamiltonian p2 2m The reader may thereby be reassured about our choice of relativistic Hamiltonian Nevertheless it would be decidedly awkward and not very 24 3 Schro dinger s equation here is c t i h 2m r2c ConWrming Wrst that for an energy eigenstate with energy E we have r2c Ac where A 2m h 2E use Green s theorem R c cr2c d3x R c c cd3x to show that A must be positive for a normalizable state Con versely it is in fact true that for positive A there are many solutions of r2c Ac which tail oV suitably towards inWnity so that the norm kck remains Wnite4and we can normalize to kck 1 if we wish Show how to derive Green s theorem from the fundamental theorem of exterior calculus 614 24 4CHAPTER 24 illuminating to attempt to use this actual power series expression for our Hamiltonian particularly because the classical series does not even con verge when p2 m2 Yet we shall Wnd that the square root half power in the exact expression c2m 2 c2p2 1 2carries its own profound diYculties in relation to preserving the positive frequency requirement Let us try to understand something of the importance of this To avoid cluttering our expressions unnecessarily I shall return to units for which the speed of light is unity c 1 so that our relativistic Hamiltonian including the rest energy is now H m2 p2 1 2 We must bear in mind that the p2 in quantum mechanics is really the second order partial diVerential operator h2H2 so we shall need some considerable mathematical sophistication if we are actually to assign a consistent meaning to the expression m2 h2H2 1 2 which is the square root of a partial diVerential operator To appreciate the diYculty think of trying to assign a meaning to a thing like p 1 d2 dx2 for example 24 4 There is a more serious diYculty with this square root expression because it contains an implicit sign ambiguity In classical physics such things might not worry us because the quantities under consideration are ordinary real valued functions and we can imagine that we could keep the positive values separate from the negative ones However in quantum mechanics this is not so easy Part of the reason for this is that quan tum wavefunctions are complex and the two square roots of a complex number expression do not tend to separate neatly into positive and negative in a globally consistent way 5 4 This should be considered in relation to the fact that quantum mechanics deals with operators acting on complex functions and things like square roots can lead to essential ambiguities that are not simply resolved by just saying take the positive root There is another way of expressing this diYculty In quantum mechan ics one has to consider that the various possible things that might happen in a physical situation can all contribute to the quantum state and therefore all these alternatives have an inXuence on whatever it is that does happen When there is something like a square root involved each of the two roots has to be considered as a possibility so even an unphysical negative energy has to be considered as a physical possibility As soon as there is the potential for such a negative energy state then there is opened 24 4 Make some suggestions either using Fourier transforms 9 4 or a power series or contour integrals or otherwise 615 Dirac s electron and antiparticles 24 4 up the likelihood of a spontaneous transition from positive to negative energy which can lead to a catastrophic instability In the case of a non relativistic free particle we do not have this problem of the possibility of negative energy because the positive deWnite quantity p2 2m does not have this awkward square root However the relativistic expression m2 p2 1 2is more problematic in that we do not