5416833 the Problem of Physic

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A n O P U S b o o kTHE P R O B L E M S OF P H Y S I C S


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OPUS General EditorsKeith ThomasAlan Ryan

Walter Bodmer

OPUS books provide concise, original, and authorita t iveintroductions to a wide range of subjects in the humani t iesand sciences. They are written by experts for the generalreader as well as for students.

The Problems of ScienceThis group of OPUS books describes th e current state ofkey scientific subjec ts, with special emphasis on the questionsnow at the forefront of research.

The Problems of Physics A. J. LeggettThe Problems of Biology John Maynard SmithThe Problems of Chemistry W. Graham RichardsThe Problems of Evolution Mark RidleyThe Problems of Mathematics Ian Stewart


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The Problems ofPhysicsA. J. LEGGETT

O x f o rd N e w Y o r kO X F O R D U N I V E R S I T Y P R E S S


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by Oxford University Press Inc., NewYorkA.J. Leggettl987

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British Library Cataloguing in Publication DataLeggett,A.J.

The problems of physics.(OPUS).1. Physics I. Title 530 QC21.2Library of Congress Cataloging in Publication Data

Leggett,A.J.The problems of physics.

(OPUS)1. Physics. 2. Particles (Nuclear physics)

3. Condensed matter. 4. Cosmology.I. Title. H. Series.

QC21.2.L44 1987 530 87-12278Printed in Great Britainon acid-free paper by

Biddies Ltd., King's LynnISBN 0-19-921124-8(Pbk.) 978-0-19-921124-1 (Pbk.)

1 3 5 7 9 1 0 8 6 4 2


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To my parents.To Haruko and Asako.


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PrefaceThis book is intended as an introduction to some of the majorproblems studied by contemporary physicists . I have assumedthat the reader is familiar with those aspects of the atomicpicture of matter which can reasonably be said to be par t of ourcontem porary cul ture but , a par t f rom this, does not necessarilyhave any technical bac kgroun d. W here argum ents about specif icpoints of physics are given in the text, I hav e tried to ma ke themself-contained.It would be quite impossible in a book of this length to giveeven a supe rficial discussion of all the diverse aspects and areasof the subject we call 'physics'. I have had, per force , to ignoretotally not only the 'applied', or technological, aspects of thesubject, but also a whole range of fascinating que stions con cerningits organization and sociology. Even within physics regardedpurely as an acade m ic discipline, I have not attempted completecoverage; huge subfields such as atomic, molecular , and evennuclear physics are completely unrepresented in this book, andothers such as astrophysics and biophysics are mentioned onlybriefly. Rather , after an introduction which at tempts to reviewhow we got to wh ere we are, I have concentrated on f our m ajor'fron tier ' areas of cu rren t research w hich I believe are rea sonab lyrepresentative of the subject: particle physics, cosmology,condensed-matter physics, and ' foundationaP quest ions, thefrontiers co rrespo nding , one m ight say, to the very sma ll, the verylarge, the very complex, and the very unclear . I do not believethat inclusion of other subfields such as geophysics or nuclearphysics would int roduce many qualitatively new features whichare not already exemplif ied in these four areas.The focus of this book is the cur ren t problems of physics, notthe answers which physics has already provided. Thus, I havespent time detailing our current picture only to the extent thatthis is necessary in order to attain a vantage-point from whichthe problems can be view ed. Also, w hile I have tried to explainschematically the basic principles involved in the acquisition ofexperimental information in the various areas, I have not


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viii Prefaceattempted to discuss the practical aspects of experimental design.The reader who wishes for more detailed information, on eitherthe structure of existing theory or the experimental methodscurrently in use, may consu lt the v ariou s books and articles listedunder ' further reading'.Finally, a word to my professional colleagues, in case any ofthem should happ en to pi ck up this book: it is not m eant for them!They will undoubtedly find some points at which they will disagreewith m y presentation; ind eed, I hav e repeatedly and consciouslyhad to resist the temptation to qualify some of my statements w iththe string of technical reservations which I am well aware wouldbe essential in a journal review article, say, but which would m akethe book quite indigestible to a non-specialist readership. Inpart icular, I am acutely conscious that the brie f discussion in thelast chapter of the evidenceor rather lack of it for theapplicability of the quantum-mechanical formalism in toto tocomplex systems grossly oversimplifies a highly complex andtechnical issue, and that, read in isolation, i t may seem to someprofessional physicists misleading and possibly even outrageous.All I can say is that I hav e giv en extended and technically detaileddiscussions of this topic elsewhere; and I would appeal to potentialcrit ics, before they indignantly take pen in hand to tell me howself-evidently i l l - informed and preposterous m y assertions are, toread also these more technical papers.I am grateful to many colleagues at the Univers i ty of Illinoisand elsewhere for discussions wh ich hav e helped m e to clarify m ythoughts on some of the subjects discussed here; in part icular,I thank Gordon Baym, M ike Stone, Jon Thaler, Bill Watson, andBill Wyld for reading parts of the draft manuscript andcomment ing on it. In a more general way, the overall att i tude tothe established canon of physics which is implicit in this boo k hasbeen influenced by conversations over many years with BrianEaslea and Aaron Sloman; to them also I extend m y thanks.Finally, I am grateful to my wife and daughter for their moralsupport, and for putt ing up with the disruption of domestic lifeentailed by the writ ing of this book.

A. J. LEGGETT


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ContentsList of f igures xi

1 Setting the stage 12 W h a t are things made of? 353 The un iverse: i ts s t ru ctu re and evolu t ion 794 Physics on a human scale 1115 Skeletons in the cupboard 1 446 Outlook 173

Notes 181Further reading 184Index 187


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List of figures1.1 A n exper iment to discriminate between part ic le- l ike and wave-l ikebehaviour1 .2 The conventional divis ion of the electromagnetic spectrum(approximate; note that the scale is logarithm ic, that is, each m ark eddivision corresponds to an increase in f requency by a factor of 1000)2 .1 The interact ion between electrons mediated by a vir tual photon2.2 Th e beh aviou r of the cross-section for scattering of a ph oto n b y anatom as a function of photon energy (or frequency)2 .3 The scattering of an electron by a pro ton , viewed as mediated byexchange of a vir tua l photon2.4 The K-capture process , viewed as mediated by exchange of anin termed iate vector boson , W +5 .1 A gas confined to a small volum e which is part of a larger one (top);the same gas after expanding into the whole large volume5.2 A q uan tum -m echan ical system m akin g transi t ions between possiblestates5 .3 Schematic set-up of an experiment to measure the polarization ofphotons emitted in an atomic cascade process


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1Setting the stage

The word 'physics' goes back to Aristotle, who wrote a book w iththat title. But the derivation is from the Greek word physis,meaning 'grow th' or 'nature'; and many of the questions w hichAristotle discussed would today be more naturally classifiedas the subject-matter of philosophy or biology. Indeed, thename did not stick, and for many centuries the subject wenow call 'physics' was called 'natural philosophy'that is ,philosophy of n atu re. In B ritain, some pro fessors o f physics stillhold chairs w ith this title. Let us start by taking a quick look ata few of the milestones in the development of the subject overthe last few hundred years. In the process I will try to point outto the reader some of the essential concepts we will meet in laterchapters.A lthou gh some isolated elements of the world-view enshrinedin modern physics can certainly be traced back to the culturesof China, India, Greece, and other civilizations more thantwo thousand years ago, the origins of the subject as thecoherent, quantitative discipline we now know would probablybe placed by m ost historians in Europe in the late medieval andearly Renaissance period. Just how and why these seminaldevelopments occurred when and where they did, how theyreflected m uch older traditions in the culture, and how some ofthese pre-scientific trad itions may still be colo uring the consciousor unconscious assumptions of physicists todayal l these arefascinating questions, but I have neither the space nor thehistorical expertise to discuss them in this book . In the presentcontext, it is enough to recall that physics as we know it beganwith the systematic and quantitative study of two apparentlydisparate subjects: mechanics and astronomy. In each case thetechnological developments of the period were essential ingredientsin their development. This is obvious in the case of astronomy,where the invention of the telescope around 1600 was a majorlandmark; but in mechanics, the development of accurate clocks


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2 Setting the stageplayed an even more fundamental role , perhaps. For the firstt ime a reproducible and quantitative standard not only oflength but of t ime was available, and one could begin to thinkquantitatively about concepts such as velocity and acceleration,which would eventually form the language in which Newton'sdynamics was formulated. As the technology of the sixteenthand seventeenth centuries developed, more and more of theconcepts of macroscopic physics as we now know i t m a s s ,force, pressure, temperature, and so o n b e g a n to acquire aquantitative meaning, and m any empirical laws relating to themwere foun d, some of which are well kn ow n to every schoolchildtoday: Snell 's law for the refraction of light in a materialsuch as glass, Boyle's law relating the pressure and the volumeof a gas, Hooke's law relating the extension of an elastic springto the force applied to it, and so on. At the same time, in thesister subject of chemistry, scientists began to formulatequantitatively the laws according to which different substancescomb ine, al though at the t im e an d indeed for a couple ofcenturies or so the reaf te ri t was not clear what thesegeneralizations had to do with the mechanical problemsinvestigated by the physicists.The man who is usually regarded as the father of physicsas we know it is of course Isaac Newton (1642-1727). Newtonmade fundamental contr ibutions to many different branches ofphysics, but it is his work on mechanics and astronom y whichhas left an indelible stamp on the subject. W hat he did was, first,to formulate explicitly the basic laws of the mechanics ofmacroscopic bodies which we still believe to be valid today(to the extent that the effects of quantum mechanics andspecial and general relativity can be neglected, which is usuallyan excellent approximation at least for terrestrial bodies); second,to develop w ith others the relevant m athematics (differential andintegral calculus) to the point where the equations of motion couldactually be solved for a n um be r of interesting physical situations;and th ird , to recognize that the principles of mechanics wereequally valid on earth and in the heavens, and specifically, thatthe force which held the planets in their orbits around the sunwas the very same force, gravitation, which was responsible forthe downward pul l on objects on earth. In each case, thegravitational force was proportional to the product of the masses


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Setting th e stage 3of the bodies involved and inversely proportional to the squareof their separation.Newton's laws of motion are so fundamental a bedrock ofmodern physics that it is difficult to imagine what th e subjectwould have been like without them. It is worth giving themexplicitly here:

(1) Every body continues in its state of rest, or of uniform mot ionin a straight line, unless it is compelled to change that state by forcesimpressed upon it.(2) The change of motion is proportional to the motive force impressed,and occurs in the direction of the straight line in which that force isimpressed (that is, it is parallel to the force).(3 ) To every action is always opposed an equal reaction; or, the m utua lactions of two bodies upon each other are always equal, and directedto contrary parts (that is, in opposite directions).

Perhaps even more fundamental than the detailed form of thelaws themselves are the assumptions implicit in them about th ekinds of questions we want to ask. Consider in particular thesecond law , w hich in modern notation reads: the acceleration ofa body is equal to the force acting on it divided by its mass.Acceleration is the rate of change of velocity with time, andvelocity is the change of position with time; so, if the force ona body as well as that body's mass are known, we can calculatethe 'rate of change of the rate of change' of the position withtime. In mathematical terms, the equation of motion is a second-order differential equation with respect to time; so, to obtain adefinite solution, we need tw o additional pieces of information:for example, th e position of the body at some initial time andits velocity at that time. As a m atter of fact, these are not theonly items of knowledge which uniquely fix the solution: forexam ple, if we specify the position of the body at both the initialand the final time, the solution is again uniquely determined.However, it is fairly obv ious that in m any practical problemsfor exam ple, in tryin g to calculate the trajecto ry o f a cannon-ballor the motion of the planetsthe tw o pieces of information whichwe are most likely to have are indeed the initial position andvelocity. Consequently, we tend to look upo n the process ofspecifying these initial data and then using N ew ton's second lawto derive the behaviour of the body in question at a later time


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4 Setting the stageas a paradigm of 'explanation' in mechanics; and then, byextrapolation, to assume that all or most explanation in thephysical sciencesall explanation of specific events at leastm ust consist in a procedu re which will permit us, when given theinitial state of a system, to predict or derive its subsequentbehaviour. For example, in cosmology we tend to assume withouteven thinking about it that an 'explanation' of the present state.of the universe m ust refer to its past: the universe is the w ay itis 'because of the way it started, not because of the way it willend or for any other type of reason. It is interesting to reflect thatthis view of 'explanation', which is so clearly rooted in purelyanthropomorphic considerations, has withstood the violentconceptual revolutions of relativity and quantum mechanics,which, one might think, would make these considerations quiteirrelevant much of the t ime.Newton's formulation of the basic equations of classicalmechanics would have made much less impact, of course, in theabsence of his second great contribu tion, the development of themathematics necessary to solve them in interesting situations.However, his third ach ievementthe recognition of the unity ofthe laws of mechanics on earth and in the heavenswas perhaps,in the long run, the most important of all. Today it is difficultfor us to appreciate the colossal concep tual leap involved in thepostulate that the force which made the anecdotal apple fall tothe ground was the very same force which held the planets in theirorbits. This is the first in a long line of great 'unifications' in thehistory of physics, in which apparently totally disparatephenomena of nature were recognized as different manifestationsof the same basic effect. Here, as in some other cases, theperception of un ity involved extrapolation far beyond what onecould measure directly: Newton could devise methods formeasuring the gravitational force directly on earth, but he hadto postulate that this force acted on the scale of the solar system,and verify this by working out the consequences, some of whichwere to become directly testable only centuries later. At first sight,the lesson of the colossal success of Newton's mechanics is thatit pays to be bold in extrapolating the laws of nature from thecomparatively small region in which we can test them directlyto regions vastly distant, not just in space or time bu t , forexample, in the density of matter invo lvedand , as we shall


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Setting th e stage 5see, in some areas of physics that lesson has been very well learnedindeed.For the ce ntury or so after New ton's death, the course of physicsran comparatively smoothly. On the one hand, great progress w asmade in applying his laws of motion to more and morecomplicated problems in mechanics. To a large extent, thisdevelopment could be regarded as lying in the area of appliedmathematics, rather than physics, since the problem was simplyto solve Newton's equations for more and more complicatedsituations; and once the initial state and the forces are specified,this is a purely m athematical operation, even thoug h its 'pay-offlies in the area of physics. (I will return to this point in moregenerality below.) In the course of this mathem atical development,many elegant reformulations of N ew ton's law s w ere devised (forexample, one w ell-kn ow n such principle states that a body willfollow that path between two points which, given certainconditions, takes the shortest time); and it eventually turned outthat some of these alternative formulations were a crucial cluein the development of quan tum mechanics a century or more later.It is an interesting though t that, had mathematicians of the lateeighteenth and early nineteenth centuries had access to moderncomputing facilities, they would have been deprived of much ofthe motivation for developing these elegant reformulations, andthe eventual transition to qua ntum mechanics w ould very likelyhave been even more traumatic than it in fact was.

In a different direction, a great deal of progress w as made, bothexperimentally and theoretically, in this period in the areas ofelectricity and magnetism, optics and the rmodynamicswhat w etoday think of as the subject-matter of 'classical' physics. Froma study of static electrification phenomena and related effects,there gradually emerged the concept of electric charge, regardedas something like a liquid which could occur in two varieties,positive and negative, and of w hich a body could possess a varyingquantity. Eventually it was realized that the force between twocharges is proportional to the product of the charges (beingrepulsive for like charges and attractive for unlike ones) andinversely proportional to the square of the distance betw een them(Coulomb's law). Simultaneously, the properties of electriccurrents ^vere being investigated, and they also were found tointeract, with a force w hich again varied as the inverse squ are of


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6 Setting the stagethe distance, but depended in detail on the relative orientationof the current-carrying wires (the Am pere and Biot-Savart laws).It gradually became apparent that i t w as conceptually simpler toformulate the laws of electricity and magnetism (as well as thelaw of grav ity) in term s of fields, rather than direct interactions.For example, an electric charge was regarded as producing an'electric field' whose strength was proportional to the inversesquare of the distance from the charge; this field then exerted aforce on any other charged body which happened to be around.Similarly, an electric current would produce a 'magnetic field'which would act on other currents (it was , of course, recognizedthat magnetic substances such as iron could also produce and beacted on by such fields, hence the name); and a massive bodywould produce a 'gravitational field'. Originally, these fields werevisualized as some kind of physical d istortion or disturbance ofspace; b ut, as we shall see, they g radu ally came to be thou gh t ofin more and more abstract terms, finally ending up (with theadvent of quantum mechanics) as little mo re tha n the potentialityfor something to happen at the point in question. It wasappreciated early on, of course, that the current w hose m agneticeffects could be detected w as nothing but the flow of electriccharges, and later, that i t was proportional to the electric fieldin the conductor carrying it (Ohm's law). A second crucial linkbetween electricity and magnetism came with Faraday's discoverythat the motion of a m agnet could induce currents in a conductor,or, in term s of fields, that a varying m agnetic field automaticallyproduced an electric field.In optics, a long debate took place betw een those w ho believedthat light w as a stream of particles (the view favoured by Newton)and the school which held it to be a form of wave motionanalogous to waves on water . The proponents of the w ave theorycould point to the phenomenon of interference as evidence; andsince this phenomenon is so fundamental to modern atomic andsubatomic physics, it is as well to take a moment to explain it,using a rather artificial, but very w ell-w orn, example. Supposew e have a source of particles for example, bul le ts posi tionedsome distance behind a screen Sj in which tw o slits are cut(Figure 1.1). A second screen (S2) provides a w ay of catching andregistering the particles. W e m ay assume that the particles bounceoff the walls of the slits in some fairly random w ay which m ay


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Figure 1.1 An experiment to discrimin ate betw een particle-like and wave-like behaviournot be calculable or predictable in detail. However, we can makeone very firm prediction which is quite independe nt of such details.Suppose wecount the number of particles arriving per second oversome small region of S2, first with only slit 1 open, next withonly slit 2 open and finally with both slits open. The source issupposed to be spewing out particles at the same rate th roughout .Moreover , w e assume (this can be checked if necessary) that theact of opening or closing one or other of the slits does notphysically affect the other slit, and that there are not enoughparticles in the apparatus at the same tim e to affect one another.Then, if N I is the number arriving when only sl i t 1 is open, N 2the number when only sl i t 2 is open, and 7V 12 the number whenboth are open, it is immediately obvious that we must haveN i2 =Ni+N2. That is, the total nu m b e r of particles arriving atthe small region of S2 in question must be equal to the sum ofthe numbers of those w hich wo uld have arrived throu gh each slitseparately.Now suppose we have the same set-up as regards the screensand the position of the source, bu t that they are now partiallyimme rsed in a bath of water , and that the source is in fact some

Setting the stage


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8 Setting the stagekind of stirring device which produces not particles, but waveson the water surface. Suppose w e again open only slit 1 andobserve the w aves arriving at a particular region of S 2. How shallwe describe the ' s t rength ' of these waves? W e could perhapsdescribe it by the ' ampli tude ' of the wave tha t is, the height ofthe water above its original level; but this quantity varies in timeand, moreover , can be either positive or negative (in fact , undernormal condit ions we will see a regular sequence of crestsinterspersed with troughs). It is much more convenient to takeas a measure o f the w ave its ' intensity' that is, the average energyit brings in per second and this quanti ty turns out to beproportional to the average of the square of the ampli tude; thus,it is always positive. Suppose now tha t we repeat the sequenceof operations carried out above for the 'particle' case: that is,we first open only slit 1, then only slit 2, and finally both slitssimultaneously. Only this time we record the wave intensity ineach case: let the relevant values be respectively 7 l5 72, and 712.Do we then find the relationship I\2 =I\ + 72? That is, does theintensity of the wave arriving when both sli ts are open equal thesum of the intensities arriving through each slit separately? Ingeneral, it certainly does not. In fact, there will be some pointson the screen w here, althoug h the intensity is positive w ith eitherslit alone open , it is zero (or nearly so) when both are open! Inthe case of a water wave it is very easy to see why this is so, forit is the amplitude (height) of the wave which is additive tha tis, when both slits are open , the height of the wave at any givenpoin t at a given time is the su m of the h eights of the waves arrivingthrough each slit separately. Since the ampli tudes add, theintensities, which are propor t ional to the squares of theamplitudes, cannot simply add. In fact, if Al, A2, and Andenote the ampli tudes under the various condit ions justmentioned, w e have A12 = A{+A2, and therefore

whereas the sum of I{ and 72 is jus t A{2 +A22. So the twoexpressions differ by the term 2A^A 2. In particular, if AIhappens to be equal to A2 that is, if the wav e arriving at thepoint in question through slit 1 has a crest just w here that comingthrough sli t 2 has a t rough , and vice versa then 7j and 72 are


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Setting the stage 9each separately positive, but I\2 is zero. W e say that we are seeingth e effects of ' interference' (in this case 'destructive' interference)between the waves arriving throug h the two slits. The phenomenonof interference is characteristic of wave phenomena, and can bereadily observed with all sorts of waves: water waves, soundwaves, radio waves, and so on. However, it is imp ortant to observetha t the essential characteristic of a 'wave ' in this connection isnot that it is a regular periodic disturbance, but that it ischaracterized by an amplitude which can be either positive ornegative; without this feature there would be no possibility ofdestructive interference, at least. The actual physical nature ofthe am plitude depends on the wave, of course. In the case of waterwaves it is the height above background; for sound waves it isthe deviation of the air density or pressure from its equilibriumvalue; for radio waves it is the value of the electric field; and soon. But in each case there is some q uan tity w hich can h ave eithersign; and in each case, since we wish the intensity of the waveto be a positive quantity, it is taken to be proport ional to thesquare (or average of the square) of this quantity. Actually, ineach case it is possible to define th e intensity simply as the flowof energy; then, since the overall energy of wave motion isconserved (see below), i t - fo l lows that if the interference isdestructive in one region, there must be other regions in whichit is ' cons t ruc t ive 'tha t is, in which 712 is greater than the sumof 7j and 72.

The first half of the nineteenth century saw more and moreexperiments in optics which could apparently only be easilyexplained in terms of interference, and hence pointed to theinterpretation of light as a wave phenomenon. Nevertheless,perhaps in part because of the tow ering prestige of N ewton, thereremained a substantial num ber of em inent scientists who held thatlight was a stream of particles. In the history of their gradualconversion, one episode is particularly amusing. In an at temptto ref ute a theo ry of diff ractio n developed by the w ave theorists,it was pointed o ut by Poisson, one of the advocates of the p articlepicture, that this theory led inevitably to the prediction that inthe centre of the shadow o f an opaque circular object there shouldappear a bright s p o t w h i c h , he claimed, w as patently absurd.The proponents of the wave theory prom ptly went away and didth e experiment, and show ed that such a spot did indeed exist! Such


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10 Setting the stageinstances of being hoisted with one's ow n petard are notunc o m m o n in the history of physics. By around the middle ofthe nineteenth century few people doubted that l ight was indeeda w a v e a conclusion w hich, as w e shall see later, w as som ew hatpremature .A third major area of development in this period w astherm ody nam ics that is , the s tudy of the re la tionships amongm acroscopically observ able properties of bodies such as pressure,volume, temperature, surface tension, magnetization, electricpolarizat ion, and so on. One major theme which ran throughthis development was the concept of conservation of energy.In mechanics the 'kinetic energy ' of a b o d y t h a t is, theenergy resulting from its m o t i o n i s conventionally definedas one-half its mass times th e square of its velocity, and its'potential energy' as the energy which it has as a result of theforces which can act on it; for example, a body moving inthe uniform gravi tat ional field of the earth near i ts surfacehas a potential energy equal to its mass t imes its height abovethe ground t imes the constant of gravitational acceleration(roughly 1 0 metres per second per second). For motion in a moregeneral gravitational field (for example, for the planets movingin the field of the sun, w hich is not u nifo rm ) or w hen the forcesare not gravitational in origin, one can often m a k e a naturalextension of this definition of potential ene rgy. Its m ain pro pe rtyis that it depends only on the position of the body , not on itsvelocity or acceleration. The point of defining kinetic and potentialenergy in this way is that for an isolated system acted on onlyby forces such as gravitation and electricity, it is a consequenceof Newton's laws that al though the two forms of energy can beconverted into one another, their sumthat is, the total energyremains constant. This is known as the ' law of conservation ofenergy' . Physicists love conservation law s, bo th because of theirintrinsic elegance and simplicity and because they often makecalculat ions much simpler . Indeed, Newton's original first lawsimply states the law of conservation of momentum (mass t imesvelocity) for a bo dy sub ject to no forces; while his third law , w hencombined with his second, states in effect that for a system ofbodies interacting only with one another the total m o m e n t um isconserved. W e shall meet many other examples of conservationlaws below.


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Setting the stage 1 1It was natural , therefore, to try to extend the law ofconservation of energy not only to wave phenomena (for whichit w or ks well) but also to the rea lm of thermodynamics. In somecases this was obv iously quite plausible. For example, a mass hungon the end of a light spring and allowed to oscillate will satisfythe principle of energy conservation, at least for a short time,provided that we extend the definition of potential energy toinclude a term associated with the stretching of the spring.However , there are at least two obvious problems. In the firstplace, if the mass is subject to any frictional forces at a l l f o r

example, from the viscosity of the surrounding a i r i t willeventually come to rest, and it is then easy to check that its totalenergy (the sum of kinetic and potential terms) has decreased inthe process. Second, there are clearly cases in which a systemappears to gain energy for no obvious mechanical reason: forexam ple, if I fill a coffee tin w ith w ater, jam the lid on hard, andheat the tin over a flame, then the lid will eventually be blownoff w ith considerable veloc ity that is, w ith a considerable gainin kinetic energy, despite the fact that the re is no loss of poten tialenergy associated with the process. W ith the help of examples suchas these it was eve ntually realized that heat original ly thoughtof as a fluid, 'caloric', somewhat similar to the contemporaryconcept of electric c h a r g e wa s actually nothing but a form ofe n e r g y t h e kinetic and potential energy of the random motionof the molecules which by that tim e w ere becom ing accepted,mainly on the basis of chemical evidence, as the microscopicbuilding blocks of matter. Once it became possible to measureheat quantitatively and the conversion factor between the unitsof heat as conventionally measured and those of mechanicalenergy w as found , it becam e clear that the total e n e r g y t h a t is,heat plus mechanical energyis indeed conserved; thus, forexample, in the case of the oscillating mass on the spring, themechanical energy 'lost' in friction is in fact merely converted intoan equ ivalent am oun t of h eat. This generalized version of the lawof conservation of energy is know n as the first law ofthermodynamics. The better-known second law expresses th e factthat although heat may be converted into mechanical work (orother useful forms of energy) to a certain extent, there are strictl imitations on the efficiency of this process; in particular, thereis a certain fun ction of the state of a sys temits 'entropy' such


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12 Setting the stagethat the total entropy of the un ive r s e by which in this contextis meant the system plu s whatever interacts w ith i t m u s t increaseor, at best, remain constant. Although a body does not containa fixed 'amount of heat' h e a t is only one form of energy, andcan be exchanged with other f o r m s t h e heat added to a bodyis related to its change in entropy, so it is natural to think ofentropy as a measure of the amount of disordered, randommolecular motion. If we do so, then the second law ofthermodynamics expresses the intuitively plausib le fact that it ispossible to convert 'ordered' m o t i o n a s in the oscillations ofthe mass on the spr inginto 'disordered' m o t i o n t h e randommotion of the molecu le sbut not to reverse the process, at leaston a global scale. Thus, if a body is cooled in a refrigerator, itsentropy thereby being reduced, other parts of the universe mustexperience a compensating increase in en t r opy i n this example,the room is heated. As we will see, statistical m echanics later gav ea more quantitative foundation to the interpretation of entropyas a measure of disorder.The second half of the nineteenth century saw two majorunifications of apparently disparate branches of physics, as wellas crucial progress tow ards a third. To take the latter first, it hadlong been recognized that each o f the v arious chem ical elementsand compounds, at least when in gaseous fo rm , emitted light notat random but only w ith certain specific wavelengths characteristicof the element or compound in question. Since the differentwavelengths are ben t by different am ou nts b y a glass prism , theyappear on the photographic plate of a simple spectrometer asisolated vertical lines. The specific set of wav elengths em itted bya substance is known as its (emission) 'spectrum', and theexperimental study of these patterns as 'spectroscopy'. Itspractitioners rapidly learned to identify the chemical com ponentsof unknown substances by the set of spectral lines emitted, atechnique which is now adays qu ite essential, not only to chem icalanalysis but also to astrophysics (see Chap ter 3). It soon becameclear that there were a number of regularities in the spectra ofthe elements. In particular, in the case of hydrogen, which on thebasis of chemical evidence was recognized as the s implest element,there were a number of apparently quite precise numericalrelationships between the various waveleng ths em itted, involvingonly integral numbers. Although the count of these numerical


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Setting the stage 13relationships grew rapidly, their origin remained quite obscureas long as no microscopic picture of the atom w as available. Inaddit ion, the properties of so-called blackbody radiat iontheradiation fo un d w hen a cavity w ith completely absorbing ('black')walls is maintained at a given temperaturewere studied in detail,with results which were puzzling in the extreme.The first major unif icat ion which w as actually achieved inthe latter half of the nineteenth century was of optics withelectromagnetism (and, in the process, of electricity and mag-netism with one another). Newton's mechanics and gravita-t ional theory, and subsequent theories of electric and magneticeffects, had implicitly relied on the notion of ' instantaneous actionat a distance' t h a t is, that the gravitational force, for examp le,exerted b y one astronom ical body w ould be felt instantaneouslyby a second body distant from it , without any need for a finitedelay in transmission. Th e question o f how the force came to betransmitted instantaneously, or indeed at all, was regarded asmeaningless: instantaneous action at a distance w as simply a basicfact of life which did not itself requ ire explanation. Certainly therewere people w ho felt that this approach w as unsatisfactorymetaphysically, and who hankered after a detailed account of theway in which the force w as propagated; but the exponents ofaction at a distance could point out, correctly, that while theconcept might look a little odd and even contrary to ' commonsense' , calculations based on it always seemed to give goodagreement with experiment. (W e will see in a later chapter thata similar situation obtains today with regard to the theory ofmeasurement in quantum mechanics.) At the same t ime, i t hadbeen known since the end of the seventeenth century, fromastronomical observations, that the speed of light w as finite (andindeed the value inferred w as quite close to that accepted today,approximately 3 x 108 metres per second); however, no specialsignificance w as attached to this fact, since there w as no particularreason to believe tha t light played a ny special role in the schem eof physics.In the 1860s, in the course of thinking about the laws ofelectricity and magnetism as then known , the British physicist J. C.Maxwell noticed an odd asymmetry: namely, that a changingmagnetic field could induce an electric field, but , as far as wasknown, not vice versa. M axwell argued fro m consistency that this


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1 4 Setting the stagecould not be the case, and that there was a missing term in theequation describing the observed pheno m enon, and once this wasassumed a remarkable conclusion emerged: electric and magneticfields could propagate through empty space as a kind of wavemotion. Moreover, if one put into the equations the constantsmeasured in laboratory experiments on electricity and magnetism,the velocity of such a wav e could be calculated and turned outto be just about the velocity of light! The obvious conclusion wasthat light is nothing but an electromagnetic wave that is, a wavein which electric and magnetic fields oscillate in directionsperpendicular to one another and to the direction of propagationof the light. The phenomenon of 'polarization' , which had longbeen kn ow n experimentally, could then be easily und erstood : theplane of polarization is simply the plane of the electric field. W enow know, of course, that visible light is j u s t one small part ofthe 'electromagnetic spectrum' , corresponding to waves withwavelengths in the range detectable by the human eye(approximately 4 x l O ~

7 t o 8 x l O ~ 7 metres); as far as is known,electromagnetic waves can propagate with any wavelengths, X ,and the associated frequency, v = c/\ (where c is the speed oflight), from waves in the kilohertz (kHz) frequency region emittedby radio stations to the 'hard 7-rays' with frequencies of the orderof 1028 hertz (Hz) observed in cosmic radiation. The con-ventional division of the electromagnetic spectrum is shown inFigure 1.2.O ne imm ediate consequence of the theory of electromagneticfields as developed by Maxwell is that the velocity of propagationof electromagnetic effects is finite equal, in fact , to the velocity

Figure 1.2 The conventional division of the electromagnetic spectrum(approximate; note that the scale is logari thmic, that is, each markeddivision corresponds to an increase in frequency by a factor of 1000)


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Setting the stage 15of light, c. If , for exam ple, we tu rn on a switch in the laborato ryand thereby set off a current in a circuit, the magnetic fieldproduced by that current at some point a distance R away is notimmediately felt there, but takes a time R/c to materialize; it isonly because the velocity of light is so enormous relative toeveryday speeds that we have the illusion of instantaneous actionat a distance. Thus, a question which within the f ramework ofthe original Newton ian scheme was 'm erely philo so ph ical' H owis action at a distance poss ib le?has now acquired not only aphysical meaning but a very satisfying answer, at least as regardselectromagnetism: namely, that it isn't possible; the fieldspropagate the interaction from each point to its immediateneighbourhood at a finite speed.The second great unification of the late nineteenth century wasof the molecular theory of matter with thermodynamics. It hadlong been accepted tha t sim ple chemical substances w ere bu ilt outof small identical building blocks, molecules, and that themolecules in turn were composed of sub-units, atoms, with eachchemical element having its characteristic type of atom; and thatchemical reactions could be interpreted in terms of the breakupof mo lecules and the rearrangem ent of atoms to fo rm new ones.The precise nature of these atoms and molecules was quiteobscure; however, there were some situations where, plausibly,their detailed structure m ight not matter too much. For example,it was known that in a reasonably dilute gas (such as air at roomtemperature and pressure) the molecules are quite far apartcompared with their typical packing in a liquid or solid; so itseemed reasonable to th ink of them to a first approximation assimply like tiny billiard-balls whose detailed structure wasirrelevant to most of their behaviour in the gas. Even given thismodel, however, how could one do anything useful with it? Ifone thought about the problem within the f ramework ofNew tonian mechanics, then to know the motion of the moleculesof the gas, one would need to know the initial positions andvelocities of each molecule, and moreover, to know them to anincredibly high degree of accuracy. (As any billiards-player knows,even with two or three colliding objects a tiny change in the initialconditions can rapidly lead to a huge change in the subsequentmotion!) The eighteenth-century French m athematician Laplace hadindeed contemplated, as a sort of philosophical thought-experiment,


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16 Setting the stagea being who had the capability of acquiring all this information,not just for a gas but for the whole universe, and had concludedthat to such a being both the past and the future of the universewould be totally determined and predictable. (This view,sometimes kn ow n as 'Laplacean determinism', has some curiousechoes in modern times.) However, whether or not one acceptsthe outcome of the argument, in practice it is totally unrealisticto pose the problem in this way; we can neither acquire thenecessary inform ation nor process i tthat is, solve the Newtonianequations of m o t i o n i n any reasonable time; and no advance,however spectacular, in the computing power available to us inthe remotely foreseeable future is likely to change this state ofaffairs.Into this situation there comes to the rescue the disciplinewe now call 'statistical mechanics'. It starts with th e crucialrecognition that not only is it impossible in practice to know thedetailed behaviour of each individual molecule, but that in anycase there would be no point in doing so. Most of the propertiesof macroscopic bodies which we can actually measuremagnetization, pressure, surface tension, and so o n are in factthe cumulative outcome of the action of huge numbers ofmolecules, and it is a general result of the science of statistics thatsuch averages are very insensitive to details of individualbehaviour. As an example, let us consider the pressure exertedby a gas such as steam on the walls of the vessel containing it .This pressure is nothing but the sum of the forces exerted by theindividual molecules as they collide with, and are reflected by,the walls; the faster the inciden t molecules, the greater the force.Now any one molecule m ay come up to the wall slowly or fast,and therefore exert a small or a large force: but since the measuredpressu re arises from a vast number of such individual events, thefluctuations rapidly even out, and the resulting pressu re is usuallyconstant within the accuracy of our measurements. Thus we donot need to know the detailed behaviour of each molecule: allwe need is statistical information that is , inform ation about theprobability of a rando m ly chosen m olecule having a given velocity.This is what it is the business of statistical m echan ics to pro vide .How does statistical mechanics set about providing thisinform ation? Historically there have been two main approaches,the 'ergodic' and the 'information-theoretic'; while they give the


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Setting the stage 17same answers, at least as regards th e standard applications ofstatistical m echanics, the w ay in which the basic assumptions areformulated look rather different. Both approaches rest on the factthat there are very few simple macroscopic variables of the systemwhich are conserved in the mot ion. For example, for an isolatedcloud of gas the only such conserved quantities are the totalenergy, total mome nt um, and total angular mo men tu m;1 for agas enclosed in a therm ally insulating flask, only th e total energy;and so on. There are indeed known to be other quantities whichm ust be cons erved, bu t they are generally ve ry com plicated, andthe hope is usually that by ignoring them one will not getmisleading results.The ergodic approach proceeds by arguing, crudely speaking,that since the dynamics of a macroscopic system of particles isextremely com plicated, it is plausible to say that whatever stateit starts in, it will in time pass through all other states which arecompatible with the conservation laws (for example, in the caseof the gas in the flask, all states of the same total energy as theinitial one). Since in a norm al experiment on a macroscopic systemwe are in effect averaging the observed quan tities over times w hichon a n atom ic scale are very long, it is argued that w e are in effectaveraging over all accessible states of the system. The mainproblem with th e ergodic approach is that while the basichypothesis that the system will in tim e pass throu gh all accessibles ta tesmay be thought to be intuitively plausible, it has beenactually proved only for an extrem ely small class of systems, noneof which are pa rticu larly realistic from an experimental point ofview.The information-theoretic approach avoids this problem bysimply postulating from th e start that all states which arecompatible with the knowledge we have about the system areequally probable. For example, suppose we have a fairly small(but still macroscopic) subsystem which is in thermal contactthat is, can exchange heat energywith a m uc h larger system (the' environment ' ) , the whole being thermally insulated from theoutside world. (A n example which approximately fulfils theseconditions might be a carton of milk in a refrigerator.) Then w eknow that the energy, E, of the whole must be constant, but thatthat of the subsystem s need not b e, and w e can ask the quest ion:What is the probability, Pn, that the subsystem is in some


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18 Setting the stageparticular state, n, which has energy Enl The answer is inprinciple very simple: it is simply pro po rtiona l to the n um ber ofstates available to the environment which are com patible with thishypothes i sthat is, which have energy EEn. One might thinkthat it would be very difficult to determine wh at this num ber is,but it turns out that for a macroscopic system it is actually rathereasy to get an approxim ate expression w hich is negligibly in e rrorprovided that the number of particles is sufficiently large. Thefinal answer is that Pn is a simple function of the energy Ennamely, a constant times exp (-En/Eo), where E0 is a quanti tyof the dimensions of energy which characterizes the state. (Thisa n s we r t h e so-called Boltzmann dis t r ibu t ionis also obtainedrather more indirectly from the ergodic approach.) If one thencompares the predictions w hich follow from this expression formacroscopically measurable quantities with the formulae ofthermodynamics, it turns out that E0 is nothing but thetemperature, provided that the latter is measured in appropriateunits with an appropriate origin. Nowadays in physics it isconventional to take the origin 'appropriately', that is, to measuretem peratures on the so-called abso lute, or K elvin, scale, on w hichthe absolute zero of temperature corresponds to about -273degrees Celsius; however, it is conventional, though somewhatillogical, to continue to measure temp erature in degrees rather thanin units of energy, so one needs a constant to relate E0 to theconventional temperature in degrees. In fact one has to writeEo = kftT, where the quant i ty & B, k n o w n as Bol tzmann ' sconstant, is numerically about 1.4x 1 0~ 2 3 joules per degree. Itis important to appreciate that & B, unlike the speed of light, c,and Planck's constant, h, which w e will meet below, really hasno fundamental significance, any more than does the constantwe require to convert feet into metres; in fact, had the historyof physics been different, w e might well have done withoutseparate units of temperature altogether, merely measuring'hotness' in units of energy.Arm ed with the Boltzm ann distribution, and the techniques forcounting states which led to it, we can not only make detailedcorrespondences between all the form ulae of thermod ynam ics andthe results wh ich follow from statistical mechanics, but in manycases, if we are prepared to assume some specific microscopicmodel, we can actually calculate the quantities which in


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Setting the stage 19thermodynam ics have to be taken from experimen t. For example,if we take as a model of a gas a set of freely moving billiard-ball-like molecules, we can just ify the classical phenomenological lawsof Boyle and Charles relating pressure, volum e, and tem perature.This kind of demonstration of correspondence between themicroscopic picture and the macroscopically measured quantitieslies at the basis of the bulk of modern work in the physics ofcondensed matter (though I shall argue in Chapter 4 that it is easyto m isinterpret its significance). One qu ite general result emergesfrom a comparison of the thermodynamic and statistical-mechanical form ulae. The mysterious quantity entropy, w hich wasoriginally introduced in a thermodynamic context in connectionwith the second law, turn s out to. be related to the number of stateseffectively available to the system subject to what w e know aboutit. That is, it can be interpreted intuitively either as a measureof the 'disorder' of the system , or, m ore strikingly, as a measureof our ignorance about it. That a qua ntity which at first sight hassuch an anthropomorphic interpretation can actually play a rolein the 'objective' thermodynam ic behaviour of the system is ratherstriking and puz zling , and is related to some even m ore un settlingquestions which I shall take up in Chapter 5.With these major advances in hand , the physics com m unityentered the twentieth century in a mood of high confidence. Allthe pieces of the jigsaw seemed to be com ing together: Ne w tonianmechanics was a complete description of the motion of all possiblemassive bodies, from planets down to atoms; M axwell 's theoryof electromagnetism not only explained all of optics but held outthe prospect of understanding the interactions, presumed to bemainly electrical, between atoms and molecules; and statisticalmechanics would allow the explanation of the properties ofm acroscopic bodies in term s of those of the atoms composing it.True, there were puzzles left: there w as still no understanding ofthe detailed structure of atoms or of the mysterious regularitiesinvolved in their spectra; and wh ile there was a good qualitative,and even quantitative, microscopic theory of the macroscopicbehaviour of gases, no such understanding was yet available forliquids or solids. But these were matters of detail. W ho coulddoubt that the f ramework of physics w as sound, and that thesepuzzles would eventually find their explanation within it? Theprevailing mood of the period was expressed by the British


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20 Setting the stagephysicist Lord Kelvin, w ho , in a lecture in 1900 su rveyin g the stateof physics as it entered the twent ieth century, concluded tha t allwas w ell apart from tw o ' small clouds ' on the horizon (of whichmore be low); in all other respects there was no reason to doubtthat the subject w as f i rmly set, in the right direction. I t may bea good ant idote , in 1987, to some of the more breathless popularaccounts of recent advances in particle physics or cosmology toput oneself in the posit ion of our fo r e runne r s in 1900 and to tryto appreciate how very firm and unshakable the basis of theirenterpr i se must have seemed to t h e m , and how unthinkable th eidea tha t the i r whole conceptual f ramework might be in error .Physicists , alas, do not always make good prophets . In theevent , within l i t t le more than tw o decades, Lord Kelvin 's smallc louds had each blown up into a major hurr icane: the wholeedifice of classical physics was in ruins, and the very questionswhich were being asked would in some cases have seemednonsensical to nineteenth-century physicists. Inde ed, if one exceptsperhaps the years of Newton 's grea t work, the first thir ty yearsof the twen t ie th century w ere by any reasonable account the mos texciting period, and certainly the m ost revolut ionary, in the wholeof the history of physics . One of Lord Kelvin 's c louds was theexper iment of Michelson and Morley , which had failed to showthe expected dependence of the velocity of l ight on the motionof the observer; this l edlogica l ly , even i f not h is tor ica l lytoEin stein 's special theo ry of relativity, and eventual ly his generaltheory . The second cloud concerned the specific heat ofpolyatomic molecules, and this w as resolved only with thedevelopment of quantum mechanics. Both these theories specialand general relativity, an d quantum mechanicsviolen t lychallenged the whole conceptual f r amework in which classicalphysics had been formula ted .I will discuss the special and general theories of relativity inChapter 3 in the context of their cosmological applications, andhere only summarize a few of the features of the special theorywhich are re levant to the topics to be ment ioned in the nextchapter. These are: that the velocity of l ight, c, is i ndependentof the f r a m e of reference in which it is measured , and is afundamenta l cons tant of nature ; that this velocity is the upperlimit of velocity of propagat ion of any kind of physical effect ,and that while bodies of zero m ass (such as light itself) automatically


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Setting the stage 21travel at speed c, no bod y of finite mass can in fact be acceleratedup to this velocity; that energy, E, and mass, m, areinterconvertible according to Einstein's famous relation E=mc2;that for isolated systems, the laws of conservation of energy andmom entum are not two different principles, but the same principleviewed from different frames of reference, and that in any frameof reference, energy, E, and momentum, p, are related by theequation

(the famous E =mc2 being the special case appropriate to aparticle at res tthat is , with p = 0 ) ' , and that 'moving clocksappear to run slow', that is, that a physical phenomenon suchas the decay of a particle which occurs at a given rate when theparticle is at rest appears to an observer with respect to whomthe particle is moving to take place at a reduced rate.To introduce qu an tum mechanics, let us digress for a mom entto a third ma jor advance of the first three decades of the twentiethcentury, which although not in itself as revolutionary from aconceptual standpoint as relativity or quan tum theory neverthelessplays a central role in modern physics: the theory of atomicstructure. A major clue had come in the last decade of thenineteenth century with the discovery that electric charge, or atleast the negative kind of electric charge, was not a continuouslydivisible fluid, but came in discrete units , in fact in the form ofthe particles w e now call 'electrons'. The electron charge, e (about1 . 6 x l O ~ 1 9 coulombs), is a fundamental constant of nature.While it w as clear that electrons were a constituent of atoms, andtherefore that the latter, being electrically neutral, had to containalso some positive charge, the nature and disposition of thispositive charge remained obscure. It was clarified when LordRuther ford , in a famous series of experiments, discovered thatthe positive charge was concentrated in a tiny b l o b w h a t we nowcall the 'nucleus' in the centre of the atom, which containedall but a very small fraction of the atom's mass. Since the nuclearcharge itself appeared to occur only in multiples of e, one wasled to deduce the existence of a positively charged particle withcharge e and mass about tw o thousand times that of thee l ec t ronwhat we now call the 'proton'. The general picture of


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22 Setting the stagethe atom which emerged was of a central nucleus, at most a fewtimes 10 ~ 1 5 metres in diameter, containing protons and (with thevirtue of hindsight!) possibly other thing s, surrounded by a swarmof electrons which orbit the nucleus at a distance of about 10 ~ 1 0metres. The electrons orbiting the small central nucleus behavevery m uch like the planets orbiting the sun, the main difference(apart fro m scale) being that they are kep t in orbit by electrostatic,rather than gravitational, attraction; in fact, the model of the atomwhich emerged from Rutherford's experiments is of ten called the'planetary' model.

B ut planetary m otion involves acceleration, and according toclassical electrodynamics, accelerating charges radiate (theprinciple that is involved in radio transmission), which in the caseof the atom leads to two major problems. First, one would expectthe electrons to be able to radiate light of any freq uenc y, w hereasthe spectroscopic evidence show s that any given atom radiates onlyat certain special freque ncies. Second, by radiating, the electronslose ene rgy, which means that they should rapidly spiral downinto the nucleusthat is, the atom shou ld collapse. T o solve theseproblem s, the D anish physicist Niels Bohr introduced a hypothesiswhich, within the f ramework of classical mechanics, looked totallyarbitrary and unjustif iable: namely, that the electrons wereconstrained to move only in certain particular orbits, and thatwhen they jum pe d between these allowed orbits, the f requency,v, of the emitted radiation was related to the difference in energyof the orbits, A by the relation AE=hv. In this formula hrepresents a constant which had been introduc ed by Planck a fewyears earlier in the theory of blackbody radiation, and is namedafter him ('Planck's constant'); it has the dimensions of energytimes time and is num erically abou t 6.6x 10~ 3 4 joule-seconds. Itis essential to appreciate that h, like the speed of ligh t, c, but un likeBoltzmann's constant, B, is a genuinely funda m ental con stantand not just a conversion factor between different arbitrarilychosen scales:2 while we are, of course, free to choose our scalesof length, time, and mass (and hence energy) so that h and/orc take th e numerical value 1 (a practice which is common inmodern atomic and particle physics), according to Bohrphenomena in m echanics which inv olve values of the product ofenergy and time comparable to h or smaller (such as the motionof electrons in atoms) appear qualitatively different from those


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Setting the stage 23where this combination is large compared to h (as in the motionof the planets of the solar system). The occurrence of such an'intrinsic scale' that is, the fact that, if we measure thecombinat ion Ex t in uni ts of h, the behaviour is quite differentfor Ext>l and for Ext lis totally alien to classicalNe w tonian mechanics, and it is worth emphasizing (with an eyeto some speculations I shall make later) that there is absolutelyno w ay in w hich, simply by inspecting classical mechanics ' f romthe inside ' , one could ever hav e deduced the existence, or indeedthe possibility, of such a scale. In the end i t w as forced on u s byexper iment , and can be understood only in the light of a totallynew theory which rejects the whole classical f r amework quantum mechanics , to which I now turn .Let us proceed, as is common practice in introducing newconcepts in physics, by considering an experiment which from apractical or historical po int of view is rather art ificial , but whichencapsulates in a simple and clear w ay features which in real lifeare inferred from more indirect and complicated experimentsthe 'two-slit ' set-up described abov e and illustrated in Figure 1.1.We saw that if we fired ordinary particles such as billiard-ballsthrough this apparatus, then no matter how complicated thescattering at each slit, the total n u m b er of particles arriving ata given point on the screen S2 w hen both sli ts are open is jus t thesum of the num bers w hich arrive throu gh each slit separately wh enit alone is open; if , on the other han d, we propagate a wave (suchas a surface wave on water) through the apparatus and measurethe intensity arriving at a given point on S2, then in general theintensity detected with both slits open is not the sum of theintensities arriving through each sli t separately, but shows theinterference effects characteristic of wave phenomena. As we saw,historically i t was observations of this type which led to theconclusion that light was a wave phenom enon, rather than astream of particles. N ow suppose we do the experim ent w ith (say)a source such as a hot fi lament which can emit electrons; andsuppose that w e tu rn the source down very low (or insert a filterbetween the source and S j) so that the probabili ty of more thanone electron being in the apparatus at a t ime is negligible, andmoreover use for S2 a type of screen which will detect the arrivalof each electron separately (this is in practice not difficult withsuitable amplifying devices). What will we see? Well , first, we will


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24 Setting the stageconfirm that as regards their arrival at S2 the electrons do indeedbehave like part ic lesthat is , each electron on arrival gives asignal at one specific point on the screen; the signal is not spreadout diffusely over a large area. So it certainly looks as if'something' has arrived at a definite point, and at this stage thereseems no good reason to doubt that the electron is indeed a particlein just th e same sense as are billiard-balls. But wait: as we firemore and more electrons through the apparatus , keeping countof where they arrive, and plot the distribution of their arrivalpoints, w e begin to see that al though the point of arrival of eachelectron appears to be quite rando m , a pattern is eventually builtup; definitely, m ore electrons arrive in some areas of the screenthan in others, and gradually it becomes clear that we are seeingan interference pattern very similar to the pattern we would getfor water waves (or for light). It m ight then occur to us to checkwhether the number of electrons arriving at a part icular pointwhen both slits are open is indeed the sum of those arrivingthrough each slit separately when it alone is open. The answeris no: indeed, there are points at which electrons arrive wh en eitherslit alone is open , but none arrive when both slits are opensimultaneously. In other words , the electrons seem to beexperiencing an interference effect precisely similar to thatassociated with wave phenomena .W e might then ask: if an electron can show wave-like aspects,can light show particle-like aspects? Indeed it can. If we do thesame experiment with light, then, if our detection apparatus isof the fairly crude type norm ally used in, for example, secondaryschool optics experiments, then all we will see is the average lightintensity arrivin g on a given area of S2 over a fair ly long period,and this will show the interference effects typical of a wave. Butit is perfec tly possible to ref ine our detection apparatus so thatit can register very small intensities, and if we do this, we can verifythat the light, does no t in fact arrive co ntin uo usly , but in discretechunks, or packets. That is, we can think of light, at least forsome purposes, as after all consisting of a stream of particles;these particles are called 'photons'. Thus both our traditional'particles' such as electrons and our traditional 'waves' such aslight actually show some aspects of both wave and particlebehav iour. This feature of quantum mechanics is known as 'wave-particle dual i ty ' .


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Setting the stage 25W e can make the picture a good deal more concrete than this.For definiteness let us concentrate on electrons (the case ofphotons, or of other particles such as protons, can be discussedsimilarly). Let us tentatively associate w ith an e lectron in a givenstate a wave, for the moment of unknown nature; and let usassume that this w ave undergoes interferen ce effects jus t like anyother. Then, just as with the water waves discussed earlier, theamplitudes of waves coming by different paths will add that is ,

Ai2=Ai+A2and the intensities, which we again take to beproportional to the average square of the amplitudes, will ingeneral not addthat i s , /1 24=/ i + /2. If w e now interpre t theintensity of the wave as giving a measure of the probability ofdetecting an electron at the point in question, w e will have at leasta qualitative explanation of the observed interference effects. Tomake it quantitative, we will need to calculate the actual behaviourof the wave, and this requires us to make some correspondencebetween the wave and particle aspects of the electron's b ehavio ur.The correspondence which has, in fact, been found to besuccessful is this: if the e lectron, view ed as a particle, is in a statewith definite mom entum p, then the associated wave has adefinitewavelength, X , which is related to p by the equation X = h/p, wherh is Planck's constant. This fundamental equation is known asthe 'de Broglie relation'. From the formulae of special relativityit then follows that the energy, E, of the 'particle' is related tothe frequency, v, of the associated 'wav e' by the relation E=hv(and the same relation ho lds between the frequency of a classicallight wave and the energy of the associated pho tons).With the help of these relations, and the crucial assumptionsmade above about the interpretation of the wave intensity as theprobability of finding a particle at the pdint in question, we cannow explain quantitatively the observed interference pattern.M oreo ver, w e can begin to unde rstand the underlying reasons forthe success of B ohr's postulates about the behaviour of electronsin atoms.Consider, for example, an electron moving around the atomicnucleus in a circular orbit of radius r. Plausibly, the associatedwave must fit back on to itself as we go once around the orbitthat is, there must be an integral number, n, of wavelengths, X ,in the circumference, 2?rr: thus n\= 2irr. But, by the abovecorrespondence, this means thatpr=nh/2ir, and this was precisely


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26 Setting the stagethe condition originally postulated by Bohr for an orbit to be' a l lowed' . Moreover , when the electron ju m ps between levels , itgenerally emits a single photon; by energy conservation, thephoton energy, E, mus t be equal to the difference between theenergies of the electron orbits involved, AE, and so from therelation E=hv above, we have AE-hv as postulated by Bohr .A q uanti tat ive account of the behaviour of the amp li tude of thequantum-mechanical wave is given by the fundamental equat ionof non-relativistic quantum mechanics , Schrodinger 's equat ion;but i t should be emphasized that this amplitude has, i tself, nodirect physical interpretationit is only the intensity which hasa meaning, as a probabil i ty.Viewed f rom the perspective of classical physics, quantummechanics has many bizarre features. In the first place, it enablesone to predict only the probability that an electron or photon willbe detected at a particular point on the screen; just why aparticular electron arrived at the point it did, and not somewhereelse, is a question the theory cannot answer, even in principle.(This featu re alone wo uld have horrified m ost nineteenth-centuryphy sicists!) Second, it does not allow us to consistently attributeto a m icroscopic e nti ty such as an electron a full and completelydefined set of 'particle' properties simultaneously. For example,in classical physics we would think of a particle as possessing atany given t ime both a definite posit ion, x, and a definitemomen tum p. But in quantum mechanics the probabil i ty offinding the particle is associated with a wave intensity, and a wavewhich has a pe rfectly well-defined waveleng th is quite unb oun dedin spatial extent. If we wish the intensity to extend over only afinite region of space, we m ust ma ke up a comb ination of wavesof different wavelengths, and by the relation \ = h/p, there wilthen be a spread, or indeterminacy, in the value of the momentum,p, of the particle. This feature is summarized in the famousHeisenberg indeterminacy principle3

where Ap is an appropriately defined indeterminacy in mo m entumand Ax a similar indeterminacy in position. A similar relation,rather more subtle in interpretation, exists between theindeterminacy, At, in t ime t at which a process occurs (or, to put


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Setting the stage 27it rather crudely, how long i t lasts) and the indeterminacy in theenergy associated with it, AE: AE-At^h/4Tr. Thus , if anintermediate state of the system 'lasts' for a t ime At, its energyis indeterminate to within an amount of the order H/4irAt, so thatwe m ay 'borrow' an energy of this order for it without violatingthe principle of energy conservation; an application of thisprinciple is made in the next chapter.Third, and most alarming of all,qua ntum m echanics requires ,at least prima facie, that in the two-slit experiment the electronshould behave in a quali tat ively different w ay when it is left toi t s e l f t ha t is, when passing through the two-slit apparatusand w hen it is observed (at the fixed screen). In the first situationit behaves l ike a wave, as it must to give the interference effects;while in the second it appears as a particle (that is, gives a singlesignal on the screen; a wave would give a spread-out , diffusesignal). The reader might well ask whether w e can't arrange toobserve it during its passage th rou gh the a ppa r a tusfor example,by placing a detecting device at each of the slits to see which oneit actually went thro ug h? Indeed we can,and if we do, w e alwaysfind that each ind ividu al electron did indeed come thro ug h o neslit or the other, n ot b o t h b u t , then there is no interference effectin the intensity pattern. So in some curious sense th e electron seemsto behave as a wave as long as we don't observe it, but as a particlewhen we do! This is actually a special case of a m uch m ore generaland, to many people, worrying paradox in the founda t ions ofquantum mechanics to which I shall return in Chapter 5.O ver the last sixty years, the form alism of qu an tum m echanics,augmented by the generalizations necessary to accommodatespecial relativity and field theory, has had a success which it isalmost impossible to exaggerate. It is the basis of just abouteverything w e claim to und erstan d in atomic and suba tom icphysics, most things in condensed-matter physics, and to anincreasing extent much of cosmology. For the majority ofpractising physicists toda y it is th e correct description of nature,and they f ind it difficult to conceive that any current or fu tu reproblem of physics will be resolved in other than q u an t u m-mechanical terms. Yet despite all the successes, the re is a persistentand, to their colleagues, sometimes irri tat ing minority who feelthat as a complete theory of the universe quan tum mechan icshas feet of clay, indeed 'carries wi th in it the seeds of its


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28 Setting the stageow n destruction' . Just w hy this should be so will emerge inChapter 5.To conclude this introd uctory cha pter, let m e say a few w ordsabo ut the relationsh ip betw een physics and other disciplines, andabout the basic 'guide-lines' w hich most physicists accept in theirw o r k . A s regards th e distinctions between physics and its sistersubjects such as chem istry or astron om y, these are ma inly ofhistorical significance. The subjects share, to a large extent, acomm on methodology and rely on the same basic law s of n ature,and the fact that stars and complicated organic molecules tendto be studied in different univ ersity departm ents from crystallinesolids or subatomic particles is an accident of the developmentof science. (Indeed, th e fact that there is no firm dividing line isrecognized in the formal existence of cross-border disciplines suchas chemical physics and astrophysics.)With respect to disciplines of a rather different nature, suchas mathematics and philosophy, the relationship is moreinteresting. Historically, the development of physics and that ofm athem atics have been closely intertwined: problem s arising fromphysics have stim ulated imp ortan t developm ents in mathematics(an excellent example is the development of the calculus, w hichw as stimulated by the need to solve Newton's equations ofm otion ); conversely, the prior existence of bran ches of apparentlyabstract m athem atics has been essential to the fo rm ulation of newways of looking at the physical world (for example, in thedevelopment of quantum mechanics). Most physicists today,particularly those whose business is theo ry rather than experiment,move backwards and forw ards across the dividing line b etweenthe two disciplines so frequently that they are barely consciousof its existence.Nevertheless, the distinction is an important one. Suppose aphysicist is faced with a new type of phenomenon which he doesnot understand; for example, he (or a colleague) has measuredthe total magnetization of a piece of iron as a function of its'history' t h a t is, of the operations carried o ut on it in the pastand has fo un d it to b ehave in an unexpected way. As a very crudefirst appro xim ation , w e can separate his attack on the p robleminto two stages. At the first stage, he tries to identify what heth inks are the crucial variables and to formulate what he thinksare the correct relationships between the m th at is, to build a


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Setting the stage 29plausible 'model' of the system. For example, he might decidethat the crucial variables are the density of magnetization at eachpoint of the sample and the local magnetic field and temperatureat that point, and that these variables are related by a particularset of differential equations. In deciding that these are indeed thecorrect variables and the correct equations, he is in effect makingan intelligent guess (see Chapter 4), and at this stage he is genuinelydoing physics. However, it is very rare that the answer to thequestion he really wants to a s k h o w does the magnetizationdepend on history? springs directly out of the equations he haswritten down, so he must now proceed to the second stage andgo through the often complicated and tedious business of solvingthe equations to produce the required information. At this stagehe is not doing physics but mathematics; in fact, in principle hecould at this stage hand his equations over to a puremathematician who had no inkling of what the symbols in themrepresent in the physical world, and the solution would be nonethe worse for that (though the physicist would, of course, haveto tell the mathematician exactly what information he wanted toextract). Of course, this simple scheme is a considerableidealization of real-life practice; it is probably a reasonableapproximation to what goes on in at least some areas of particlephysics and cosmology, but is a much worse description of thetypical situation in condensed-matter physics, where it is quitecommon for the physicist to set up a model, solve the relevantequations, find that the solution predicts results which wheninterpreted physically are clearly ridiculous (for example, that themagnetization increases without limit), realize that he has left someessential physical effect out, go back to stage 1, construct a newmod el and repeat the cycle, sometimes several times. In fact, thereal-life situation is often even more subtle than this, as we willsee in Chapter 4.

Nevertheless, although this effort less (and sometimes un -conscious) switching between what I have called stages 1 and 2is very typical of modern research in physics, it is important tobear in mind that the whole nature of the exercise at the twostages is really quite di f fe rent . At stage 2 one is trying to extract,by the rigorous process of deduction characteristic of math-ematics, the information which is in a sense already implicit inone's original equations; i f one makes a mistake at this stage,


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30 Setting the stageone is not making a mistake about the real world but a logicalerror. At stage 1, on the other h an d, one is trying to 'interface 'one's mathematical description with the real world: if one getsthings wrong at this stage, one has simply made a bad guess abou thow nature actually behaves. It is particularly important to keepthe distinction in mind in the context of the increasingly w ide-spread use of computers at stage 2; despite the appearance inrecent years of a subdiscipline k now n as 'computational physics',computers don't do physicsthey do applied mathemat icsandthe outp ut of a com puter is as good, or as bad, as the model fedinto it by its human programmer as a result of his stage 1considerations.The question of the relationship betw een the disciplines ofphysics and philosophy is an intriguing and controversial one; Iwill touch on it, implicitly, in Chapter 5 and make only a fewremarks here. M ost contem porary w ork ing physicists themselvestend to use the word 'philosophical' (not infrequentlyaccompanied by the word 'merely'!) to refer to questionsconcerning the very basic framework in which physics is done,and the language in which the questions are posed. Such questionsmight include, for example, whether all explanation in physicsmust in the last analysis be of the form 'A happens now becauseB happened in the past'; whether a theory (such as quantummechanics) which refuses in principle to give an account of thereasons for individual events can be adequate; whether theconscious observer should be allowed a special role; and so on.B ecause such questions, of their nature, are not susceptible to anexperimental answer (or at least, not so long as the experimentsare interpreted within the currently reigning conceptualframework) , the word 'philosophical' has become, in the argotof many contemporary physicists, more or less syno nym ous with' irrelevant to the actual practice of physics' . Whether or not thispoint of view is a valid one is a question which will be raised again,by implication, in Chapter 5. It is interesting that, among thosephilosophers who have paid close attention to the conceptualdevelopment of modern physics, views seem to be divided on thispoint: one school feels that the very success of (for example)quantum mechanics shows that any a priori, 'philosophical'objections to the formalism must automatically fail, and thatphilosophy has to accommodate to physics rather than vice versa ;


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Setting the stage 3 1while another school bel ieves that suff ic ient ly s t rong a prioriconceptual objections m u st brand even a successful physical theoryas at best a t rans ient approximat ion to the t r u t h .Let us now t u rn ve ry brie f ly to some features of the genera lconceptual f ramework which most phys ic is ts accept , of tenunconsc ious ly , w hen they th ink abou t th e world. Needless to say,much of th is f ramework is in no way pecul iar to phys ics , but i scharacteris t ic more general ly of those act ivi t ies we would classas ' scientif ic' ; but it is perhaps easiest to analyze it with in thecontext of phys ics , and it is probably no accident that w o r k e r sin subjects such as sociology who wish to make the i r subjects'scientific' tend to look on physics as the paradigm to be imitated.Despite the fact that all the assumptions l is ted below m ay seemso obvious as to be ha rdly wo rth stating, almost eve ry one of themhas been seriously qu est ioned at some t im e or other in the historyof h u m a n t h o u g h t ; and indeed what seems to us common sensehas been very far f rom the intel lectual orthodoxy at var iousper iods of h is tory .Perhaps the most bas ic assumpt ion which under l ies all ofphysics , and indeed al l of science in the sense in w hich the w ordis currently u nders tood, is that the world is in principle susceptibleto unde r s t and ing by human be ings ; that i f w e fail to unde r s t anda given pheno men on, then the f au l t is in us, not in the wor ld ,and that some day someone cleverer than ourselves will show u show to do it. Of cou rse, exact ly wh at is meant by ' unders tanding 'is itself a subt le ques t ion, and I will re tu rn to it by implicat ionin later chapters . H ow ever, it seems clear that unless we had somesuch bel ief , the re w ould be little point in even t ry ing to carry ou tscientific research at all.A rather more specific assump tion is that the w orld exhibits somekind of regulari ty and constancy in space and t ime: that the ' lawsof n a t u r e ' will not arbi t rar i ly change f rom day to d a y , or f romplace to place. For example , a theory which he ld that th e ratioof the frequencies of oscillation of two atomic clocks was quitedifferent a million y ears ago f rom wha t it is today w ould probablyhave di f f icu l ty gainin g acceptance, u nless i t cou ld give a generalformula for the change of the ratio w ith time an d, pre fera bly , m akepredict ions a b o u t the resul ts of experiments in the f u t u r e .A s th i s rem ark sugg es t s, another ingred ient which phy s ici s tst end to requ i re in any th ing they a re willing to regard as an


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32 Setting the stage'explanation' is predictive power. That is , the theory g iven shouldnot only explain relevant facts which are already known; it shouldalso be able to pred ict the resu lts of som e experiments which havenot yet been carried out. If one thinks about it from a purelylogical point of view, this is an odd requirement: the logicalrelation of the theoretical results to the experimental ones cannotdepend on the temporal order in w hich they w ere obtained. Butphysicists are hum an beings, and everyone know s how m uch easierit is psychologicallyand how much less f ru i t fu lto generatea complete explanation for a set of existing experiments than topredict the result of a future one. Ind eed , one of the tell-tale signsof a paper of the type physicists usu ally refer to as 'crackpot ' thou gh it is certainly not peculiar to them ! is that such papers,while apparently able to explain a lot of existing data, rarely ifever venture to forecast the results of any experim ent which hasyet to be carried out. Needless to say, there are cases where thedemand for predictive power cannot reasonably b e m et: in m uc hof cosmology and even astrophysics, the events und er considerationare either by their v ery natu re in the past or so far in the futureas to be beyond experimental reach; and, as we have seen evenin more mundane physics, the demand for exact predictability asregards individu al events has had to be dropped with the adventof quantum mechanics. Still, there are few things w hich m ake theaverage physicist sit up and take notice more than a cleanprediction of a qualitatively new phenomenonfor example, anelementary particle of a new type.It is almost too obvious to be worth stating that physics is anexperimental subject. Yet, it is w orth rem em bering, perhaps, thatit would not have seemed at all obv ious to our m edieval ancestors,most of whom w ould have regarded experimentation as essentiallyirrelevant to an

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5416833 the Problem of Physic