CHAPTER 11 THE UNIFICATION OF PHYSICS

As was explained in the first chapter, it would be verydifficult to construct a complete unified theory of everything inthe universe all at one go. So instead we have made progressby finding partial theories that describe a limited range ofhappenings and by neglecting other effects or approximatingthem by certain numbers. (Chemistry, for example, allows us tocalculate the interactions of atoms, without knowing the internalstructure of an atom’s nucleus.) Ultimately, however, one wouldhope to find a complete, consistent, unified theory that wouldinclude all these partial theories as approximations, and that didnot need to be adjusted to fit the facts by picking the valuesof certain arbitrary numbers in the theory. The quest for sucha theory is known as “the unification of physics.” Einstein spentmost of his later years unsuccessfully searching for a unifiedtheory, but the time was not ripe: there were partial theoriesfor gravity and the electromagnetic force, but very little wasknown about the nuclear forces. Moreover, Einstein refused tobelieve in the reality of quantum mechanics, despite theimportant role he had played in its development. Yet it seemsthat the uncertainty principle is a fundamental feature of theuniverse we live in. A successful unified theory must, therefore,necessarily incorporate this principle.
As I shall describe, the prospects for finding such a theoryseem to be much better now because we know so much moreabout the universe. But we must beware of overconfidence -we have had false dawns before! At the beginning of thiscentury, for example, it was thought that everything could beexplained in terms of the properties of continuous matter, suchas elasticity and heat conduction. The discovery of atomicstructure and the uncertainty principle put an emphatic end tothat. Then again, in 1928, physicist and Nobel Prize winnerMax Born told a group of visitors to Gottingen University,“Physics, as we know it, will be over in six months.” Hisconfidence was based on the recent discovery by Dirac of theequation that governed the electron. It was thought that asimilar equation would govern the proton, which was the onlyother particle known at the time, and that would be the end oftheoretical physics. However, the discovery of the neutron andof nuclear forces knocked that one on the head too. Havingsaid this, I still believe there are grounds for cautious optimismthat we may now be near the end of the search for theultimate laws of nature.
In previous chapters I have described general relativity, thepartial theory of gravity, and the partial theories that governthe weak, the strong, and the electromagnetic forces. The lastthree may be combined in so-called grand unified theories, orGUTs, which are not very satisfactory because they do notinclude gravity and because they contain a number ofquantities, like the relative masses of different particles, thatcannot be predicted from the theory but have to be chosen tofit observations. The main difficulty in finding a theory thatunifies gravity with the other forces is that general relativity is a“classical” theory; that is, it does not incorporate the uncertaintyprinciple of quantum mechanics. On the other hand, the otherpartial theories depend on quantum mechanics in an essentialway. A necessary first step, therefore, is to combine generalrelativity with the uncertainty principle. As we have seen, thiscan produce some remark-able consequences, such as blackholes not being black, and the universe not having anysingularities but being completely self-contained and without aboundary. The trouble is, as explained in Chapter 7, that theuncertainty principle means that even “empty” space is filledwith pairs of virtual particles and antiparticles. These pairswould have an infinite amount of energy and, therefore, byEinstein’s famous equation E = mc2, they would have aninfinite amount of mass. Their gravitational attraction would thuscurve up the universe to infinitely small size.
Rather similar, seemingly absurd infinities occur in the otherpartial theories, but in all these cases the infinities can becanceled out by a process called renormalization. This involvescanceling the infinities by introducing other infinities. Althoughthis technique is rather dubious mathematically, it does seem towork in practice, and has been used with these theories tomake predictions that agree with observations to anextraordinary degree of accuracy. Renormalization, however,does have a serious drawback from the point of view of tryingto find a complete theory, because it means that the actualvalues of the masses and the strengths of the forces cannot bepredicted from the theory, but have to be chosen to fit theobservations.
In attempting to incorporate the uncertainty principle intogeneral relativity, one has only two quantities that can beadjusted: the strength of gravity and the value of thecosmological constant. But adjusting these is not sufficient toremove all the infinities. One therefore has a theory that seemsto predict that certain quantities, such as the curvature ofspace-time, are really infinite, yet these quantities can beobserved and measured to be perfectly finite! This problem incombining general relativity and the uncertainty principle hadbeen suspected for some time, but was finally confirmed bydetailed calculations in 1972. Four years later, a possiblesolution, called “supergravity,” was suggested. The idea was tocombine the spin-2 particle called the graviton, which carriesthe gravitational force, with certain other particles of spin 3/2,1, ?, and 0. In a sense, all these particles could then beregarded as different aspects of the same “superparticle,” thusunifying the matter particles with spin ? and 3/2 with theforce-carrying particles of spin 0, 1, and 2. The virtualparticle/antiparticle pairs of spin ? and 3/2 would havenegative energy, and so would tend to cancel out the positiveenergy of the spin 2, 1, and 0 virtual pairs. This would causemany of the possible infinities to cancel out, but it wassuspected that some infinities might still remain. However, thecalculations required to find out whether or not there were anyinfinities left uncanceled were so long and difficult that no onewas prepared to undertake them. Even with a computer it wasreckoned it would take at least four years, and the chanceswere very high that one would make at least one mistake,probably more. So one would know one had the right answeronly if someone else repeated the calculation and got the sameanswer, and that did not seem very likely!
Despite these problems, and the fact that the particles in thesuper-gravity theories did not seem to match the observedparticles, most scientists believed that supergravity was probablythe right answer to the problem of the unification of physics. Itseemed the best way of unifying gravity with the other forces.
However, in 1984 there was a remarkable change of opinion infavor of what are called string theories. In these theories thebasic objects are not particles, which occupy a single point ofspace, but things that have a length but no other dimension,like an infinitely thin piece of string. These strings may haveends (the so-called open strings) or they may be joined upwith themselves in closed loops (closed strings) (Fig. 11.1 andFig. 11.2). A particle occupies one point of space at each instantof time. Thus its history can be represented by a line inspace-time (the “world-line”). A string, on the other hand,occupies a line in space at each moment of time. So its historyin space-time is a two-dimensional surface called theworld-sheet. (Any point on such a world-sheet can bedescribed by two numbers, one specifying the time and theother the position of the point on the string.) The world-sheetof an open string is a strip: its edges represent the pathsthrough space-time of the ends of the string (Fig. 11.1). Theworld-sheet of a closed string is a cylinder or tube (Fig. 11.2):
a slice through the tube is a circle, which represents theposition of the string at one particular time.
Two pieces of string can join together to form a singlestring; in the case of open strings they simply join at the ends(Fig. 11.3), while in the case of closed strings it is like the twolegs joining on a pair of trousers (Fig. 11.4). Similarly, a singlepiece of string can divide into two strings. In string theories,what were previously thought of as particles are now picturedas waves traveling down the string, like waves on a vibratingkite string. The emission or absorption of one particle byanother corresponds to the dividing or joining together ofstrings. For example, the gravitational force of the sun on theearth was pictured in particle theories as being caused by theemission of a graviton by a particle in the sun and itsabsorption by a particle in the earth (Fig. 11.5). In stringtheory, this process corresponds to an H-shaped tube or pipe(Fig. 11.6) (string theory is rather like plumbing, in a way). Thetwo vertical sides of the H correspond to the particles in thesun and the earth, and the horizontal crossbar corresponds tothe graviton that travels between them.
String theory has a curious history. It was originally inventedin the late 1960s in an attempt to find a theory to describe thestrong force. The idea was that particles like the proton andthe neutron could be regarded as waves on a string. Thestrong forces between the particles would correspond to piecesof string that went between other bits of string, as in a spider’sweb. For this theory to give the observed value of the strongforce between particles, the strings had to be like rubber bandswith a pull of about ten tons.
In 1974 Joel Scherk from Paris and John Schwarz from theCalifornia Institute of Technology published a paper in whichthey showed that string theory could describe the gravitationalforce, but only if the tension in the string were very muchhigher, about a thousand million million million million millionmillion tons (1 with thirty-nine zeros after it). The predictions ofthe string theory would be just the same as those of generalrelativity on normal length scales, but they would differ at verysmall distances, less than a thousand million million millionmillion millionth of a centimeter (a centimeter divided by 1 withthirty-three zeros after it). Their work did not receive muchattention, however, because at just about that time most peopleabandoned the original string theory of the strong force infavor of the theory based on quarks and gluons, which seemedto fit much better with observations. Scherk died in tragiccircumstances (he suffered from diabetes and went into a comawhen no one was around to give him an injection of insulin).
So Schwarz was left alone as almost the only supporter ofstring theory, but now with the much higher pro-posed valueof the string tension.
In 1984 interest in strings suddenly revived, apparently fortwo reasons. One was that people were not really makingmuch progress toward showing that supergravity was finite orthat it could explain the kinds of particles that we observe. Theother was the publication of a paper by John Schwarz andMike Green of Queen Mary College, London, that showed thatstring theory might be able to explain the existence of particlesthat have a built-in left-handedness, like some of the particlesthat we observe. Whatever the reasons, a large number ofpeople soon began to work on string theory and a newversion was developed, the so-called heterotic string, whichseemed as if it might be able to explain the types of particlesthat we observe.
String theories also lead to infinities, but it is thought theywill all cancel out in versions like the heterotic string (thoughthis is not yet known for certain). String theories, however,have a bigger problem: they seem to be consistent only ifspace-time has either ten or twenty-six dimensions, instead ofthe usual four! Of course, extra space-time dimensions are acommonplace of science fiction indeed, they provide an idealway of overcoming the normal restriction of general relativitythat one cannot travel faster than light or back in time (seeChapter 10). The idea is to take a shortcut through the extradimensions. One can picture this in the following way. Imaginethat the space we live in has only two dimensions and iscurved like the surface of an anchor ring or torus (Fig. 11.7). Ifyou were on one side of the inside edge of the ring and youwanted to get to a point on the other side, you would have togo round the inner edge of the ring. However, if you wereable to travel in the third dimension, you could cut straightacross.
Why don’t we notice all these extra dimensions, if they arereally there? Why do we see only three space dimensions andone time dimension? The suggestion is that the otherdimensions are curved up into a space of very small size,something like a million million million million millionth of aninch. This is so small that we just don’t notice it: we see onlyone time dimension and three space dimensions, in whichspace-time is fairly flat. It is like the surface of a straw. If youlook at it closely, you see it is two-dimensional (the position ofa point on the straw is described by two numbers, the lengthalong the straw and the distance round the circular direction).
But if you look at it from a distance, you don’t see thethickness of the straw and it looks one-dimensional (theposition of a point is specified only by the length along thestraw). So it is with space-time: on a very small scale it isten-dimensional and highly curved, but on bigger scales youdon’t see the curvature or the extra dimensions. If this pictureis correct, it spells bad news for would-be space travelers: theextra dimensions would be far too small to allow a spaceshipthrough. However, it raises another major problem. Why shouldsome, but not all, of the dimensions be curled up into a smallball? Presumably, in the very early universe all the dimensionswould have been very curved. Why did one time dimensionand three space dimensions flatten out, while the otherdimensions remain tightly curled up?
One possible answer is the anthropic principle. Two spacedimensions do not seem to be enough to allow for thedevelopment of complicated beings like us. For example,two-dimensional animals living on a one-dimensional earth wouldhave to climb over each other in order to get past each other.
If a two-dimensional creature ate something it could not digestcompletely, it would have to bring up the remains the sameway it swallowed them, because if there were a passage rightthrough its body, it would divide the creature into two separatehalves: our two-dimensional being would fall apart (Fig. 11.8).
Similarly, it is difficult to see how there could be any circulationof the blood in a two-dimensional creature.
There would also be problems with more than three spacedimensions. The gravitational force between two bodies woulddecrease more rapidly with distance than it does in threedimensions. (In three dimensions, the gravitational force dropsto 1/4 if one doubles the distance. In four dimensions it woulddrop to 1/5, in five dimensions to 1/6, and so on.) Thesignificance of this is that the orbits of planets, like the earth,around the sun would be unstable: the least disturbance froma circular orbit (such as would be caused by the gravitationalattraction of other planets) would result in the earth spiralingaway from or into the sun. We would either freeze or beburned up. In fact, the same behavior of gravity with distancein more than three space dimensions means that the sunwould not be able to exist in a stable state with pressurebalancing gravity. It would either fall apart or it would collapseto form a black hole. In either case, it would not be of muchuse as a source of heat and light for life on earth. On asmaller scale, the electrical forces that cause the electrons toorbit round the nucleus in an atom would behave in the sameway as gravitational forces. Thus the electrons would eitherescape from the atom altogether or would spiral into thenucleus. In either case, one could not have atoms as we knowthem.
It seems clear then that life, at least as we know it, canexist only in regions of space-time in which one time dimensionand three space dimensions are not curled up small. Thiswould mean that one could appeal to the weak anthropicprinciple, provided one could show that string theory does atleast allow there to be such regions of the universe - and itseems that indeed string theory does. There may well be otherregions of the universe, or other universes (whatever that maymean), in which all the dimensions are curled up small or inwhich more than four dimensions are nearly flat, but therewould be no intelligent beings in such regions to observe thedifferent number of effective dimensions.
Another problem is that there are at least four differentstring theories (open strings and three different closed stringtheories) and millions of ways in which the extra dimensionspredicted by string theory could be curled up. Why should justone string theory and one kind of curling up be picked out?
For a time there seemed no answer, and progress got boggeddown. Then, from about 1994, people started discovering whatare called dualities: different string theories and different waysof curling up the extra dimensions could lead to the sameresults in four dimensions. Moreover, as well as particles, whichoccupy a single point of space, and strings, which are lines,there were found to be other objects called p-branes, whichoccupied two-dimensional or higher-dimensional volumes inspace. (A particle can be regarded as a 0-brane and a stringas a 1-brane but there were also p-branes for p=2 to p=9.)What this seems to indicate is that there is a sort ofdemocracy among supergravity, string, and p-brane theories:
they seem to fit together but none can be said to be morefundamental than the others. They appear to be differentapproximations to some fundamental theory that are valid indifferent situations.
People have searched for this underlying theory, but withoutany success so far. However, I believe there may not be anysingle formulation of the fundamental theory any more than, asGodel showed, one could formulate arithmetic in terms of asingle set of axioms. Instead it may be like maps - you can’tuse a single map to describe the surface of the earth or ananchor ring: you need at least two maps in the case of theearth and four for the anchor ring to cover every point. Eachmap is valid only in a limited region, but different maps willhave a region of overlap. The collection of maps provides acomplete description of the surface. Similarly, in physics it maybe necessary to use different formulations in different situations,but two different formulations would agree in situations wherethey can both be applied. The whole collection of differentformulations could be regarded as a complete unified theory,though one that could not be expressed in terms of a singleset of postulates.
But can there really be such a unified theory? Or are weperhaps just chasing a mirage? There seem to be threepossibilities:
1. There really is a complete unified theory (or a collection ofoverlapping formulations), which we will someday discover if weare smart enough.
2. There is no ultimate theory of the universe, just aninfinite sequence of theories that describe the universe moreand more accurately.
3. There is no theory of the universe: events cannot bepredicted beyond a certain extent but occur in a random andarbitrary manner.
Some would argue for the third possibility on the groundsthat if there were a complete set of laws, that would infringeGod’s freedom to change his mind and intervene in the world.
It’s a bit like the old paradox: can God make a stone soheavy that he can’t lift it? But the idea that God might want tochange his mind is an example of the fallacy, pointed out bySt. Augustine, of imagining God as a being existing in time:
time is a property only of the universe that God created.
Presumably, he knew what he intended when he set it up!
With the advent of quantum mechanics, we have come torecognize that events cannot be predicted with completeaccuracy but that there is always a degree of uncertainty. Ifone likes, one could ascribe this randomness to the interventionof God, but it would be a very strange kind of intervention:
there is no evidence that it is directed toward any purpose.
Indeed, if it were, it would by definition not be random. Inmodern times, we have effectively removed the third possibilityabove by redefining the goal of science: our aim is to formulatea set of laws that enables us to predict events only up to thelimit set by the uncertainty principle.
The second possibility, that there is an infinite sequence ofmore and more refined theories, is in agreement with all ourexperience so far. On many occasions we have increased thesensitivity of our measurements or made a new class ofobservations, only to discover new phenomena that were notpredicted by the existing theory, and to account for these wehave had to develop a more advanced theory. It wouldtherefore not be very surprising if the present generation ofgrand unified theories was wrong in claiming that nothingessentially new will happen between the electroweak unificationenergy of about 100 GeV and the grand unification energy ofabout a thousand million million GeV. We might indeed expectto find several new layers of structure more basic than thequarks and electrons that we now regard as “elementary”
particles.
However, it seems that gravity may provide a limit to thissequence of “boxes within boxes.” If one had a particle with anenergy above what is called the Planck energy, ten millionmillion million GeV (1 followed by nineteen zeros), its masswould be so concentrated that it would cut itself off from therest of the universe and form a little black hole. Thus it doesseem that the sequence of more and more refined theoriesshould have some limit as we go to higher and higherenergies, so that there should be some ultimate theory of theuniverse. Of course, the Planck energy is a very long way fromthe energies of around a hundred GeV, which are the mostthat we can produce in the laboratory at the present time. Weshall not bridge that gap with particle accelerators in theforeseeable future! The very early stages of the universe,however, are an arena where such energies must haveoccurred. I think that there is a good chance that the study ofthe early universe and the requirements of mathematicalconsistency will lead us to a complete unified theory within thelifetime of some of us who are around today, always presumingwe don’t blow ourselves up first.
What would it mean if we actually did discover the ultimatetheory of the universe? As was explained in Chapter 1, wecould never be quite sure that we had indeed found thecorrect theory, since theories can’t be proved. But if the theorywas mathematically consistent and always gave predictions thatagreed with observations, we could be reasonably confident thatit was the right one. It would bring to an end a long andglorious chapter in the history of humanity’s intellectual struggleto understand the universe. But it would also revolutionize theordinary person’s understanding of the laws that govern theuniverse. In Newton’s time it was possible for an educatedperson to have a grasp of the whole of human knowledge, atleast in outline. But since then, the pace of the development ofscience has made this impossible. Because theories are alwaysbeing changed to account for new observations, they are neverproperly digested or simplified so that ordinary people canunderstand them. You have to be a specialist, and even thenyou can only hope to have a proper grasp of a smallproportion of the scientific theories. Further, the rate ofprogress is so rapid that what one learns at school oruniversity is always a bit out of date. Only a few people cankeep up with the rapidly advancing frontier of knowledge, andthey have to devote their whole time to it and specialize in asmall area. The rest of the population has little idea of theadvances that are being made or the excitement they aregenerating. Seventy years ago, if Eddington is to be believed,only two people understood the general theory of relativity.
Nowadays tens of thousands of university graduates do, andmany millions of people are at least familiar with the idea. If acomplete unified theory was discovered, it would only be amatter of time before it was digested and simplified in thesame way and taught in schools, at least in outline. We wouldthen all be able to have some understanding of the laws thatgovern the universe and are responsible for our existence.
Even if we do discover a complete unified theory, it wouldnot mean that we would be able to predict events in general,for two reasons. The first is the limitation that the uncertaintyprinciple of quantum mechanics sets on our powers ofprediction. There is nothing we can do to get around that. Inpractice, however, this first limitation is less restrictive than thesecond one. It arises from the fact that we could not solve theequations of the theory exactly, except in very simple situations.
(We cannot even solve exactly for the motion of three bodiesin Newton’s theory of gravity, and the difficulty increases withthe number of bodies and the complexity of the theory.) Wealready know the laws that govern the behavior of matterunder all but the most extreme conditions. In particular, weknow the basic laws that underlie all of chemistry and biology.
Yet we have certainly not reduced these subjects to the statusof solved problems: we have, as yet, had little success inpredicting human behavior from mathematical equations! Soeven if we do find a complete set of basic laws, there will stillbe in the years ahead the intellectually challenging task ofdeveloping better approximation methods, so that we can makeuseful predictions of the probable outcomes in complicated andrealistic situations. A complete, consistent, unified theory is onlythe first step: our goal is a complete understanding of theevents around us, and of our own existence.