Why String Theory? Page 9
It is not true that progress is never made by changing the rules of the game. The discovery of quantum mechanics is the perfect example of when this was both necessary and correct. The laws of quantum mechanics really are different from the laws of classical mechanics, and there is no way to massage the latter into the former. Quantum mechanics is, as far as we know, fundamentally right, and classical mechanics is – we do know – fundamentally wrong. The laws of classical mechanics are not the basic laws that describe this world, and there is no way to gloss this statement to make it otherwise.
This, however, is an exception and not the rule. The clash between human puzzlement and the laws of nature is generally like a clash between Chelsea and Bradford City. The result may not be guaranteed,21 but there is only one side to bet on. Even if you are Einstein and Feynman rolled into one, nature is still smarter than you are. Almost every claim of new physics, whether the repeated identifications of the nature of dark matter or the discovery of faster-than-light neutrinos, fails. Your inability to solve a problem with the tools available is not a sign that the problem is insoluble and requires a change in the laws. Even the fact that an observation appears completely in contradiction to the known laws of physics is a very poor reason to believe that this contradiction exists.
There is perhaps a human aspect to this as well. Physicists working on understanding the fundamental laws of nature do not generally regard themselves as intellectually deficient. Excessive humility is not observed to be a common weakness within the subject. It is always hard to spend a lot of time attacking a problem, thinking hard about an idea and getting nowhere. It can be easier to believe that the inability to make progress is due to a need to reformulate physics, rather than the fault lying in a personal failure to be the smartest cookie in the biscuit tin.
A previous age would have called this pride. A modern one may assign it to the nature of psychological stimulation in early childhood. The moral is the same: there are few true revolutions. It is much easier to propose new laws than it is to understand existing ones.
With this caveat in mind, we now move on to discuss the reasons why our current theories of physics appear inadequate.
1In the classical astronomy of Ptolemy, who lived in the second century AD, the stars were fixed in a celestial sphere that lay far beyond the region of the planets. This sphere rotated daily, producing the movement of the stars through the sky when viewed from the fixed earth.
2It is often a mark of the greatest breakthroughs that they are both complex and simple: complex viewed from the old perspective and simple from the new.
3Max Planck’s life was one that passed through many ages. Born in 1858 into a prosperous and intellectual German family, Planck had a long and successful career, winning the Nobel Prize in 1918. However, before he died in 1947 he would also live through the death of one son in the first world war, the death of two daughters in childbirth, the collapse of Germany into Nazism, the destruction of all his scientific records and correspondence in an Allied bombing raid and finally the execution of a second son for participation in the attempted assassination of Hitler.
4Technically, there can be more than just one source of infinities: however, this method can be used as long as the number of sources of infinity is finite.
5Even the greats fossilise. Paul Dirac, although one of the founders of quantum mechanics and one of the pioneers of quantum field theory, could never truly accept quantum field theory due to the infinities that arise. Already at retirement age when Wilson did his pioneering work, he was never able to internalise Wilson’s conceptual insights.
6A quantitative estimate of the correction to the electron mass is
which is numerically small compared to the original electron mass, even when the ratio within the logarithm is 1021.
7Coleman was famous, among other things, for the introductory lectures on quantum field theory he gave at Harvard for many years during the 1970s and 1980s. These lectures were filmed and are now available online. To the modern viewer one of the most striking features of these lectures is the cigarette attached to one of Coleman’s hands in the same way that the chalk is attached to the other.
8My recently retired Oxford colleague and fellow of the Royal Society, Graham Ross, launched a distinguished career in the 1960s by the simple expedient of studying and learning quantum field theory when almost no one else thought it worthwhile.
9Weinberg, who alongside his research has written with grace and learning on a variety of topics, is one of the leading scientific intellectuals of the last hundred years. In addition to research that produced a Nobel Prize, he is the author of many books, both for the general reader and for professional scientists. Mostly right and always interesting, when he speaks the wise attend.
10As we shall discuss further below, for the case of discrete symmetries, symmetries-that-are-almost-but-not-quite-symmetries have perhaps been most useful.
11The mathematics of these rotations involves one subtlety. The time part of these rotations effectively involves an extra minus sign compared to purely spatial ones. As i2 = –1, the mathematical fiction of ‘imaginary time’ renders the mathematics identical to that of spatial rotations, at the cost of sounding entirely baffling.
12For full technical accuracy, the statements here hold for what is called the left-handed electron.
13But how can this be a symmetry if the electron has charge and a neutrino has no charge?! This is why the statements in the paragraph only refer to the behaviour under the gauge symmetry of the weak interaction, and not to electromagnetic interactions.
14Mathematically, this is called the dimension of the representation of the symmetry.
15To see that this does exchange left and right, do the following. Curl your right hand in a ball, and then extend your thumb, forefinger and middle finger to define a set of axes. Do the same with your left hand. Your fingers differ only by the exchange of x with ‘minus x’, but brief contortional attempts will convince you that there is no way to rotate the two axes into another without a broken finger.
16We are in no position to sneer at those who thought otherwise. It is less than a century since professional astronomers were all underestimating the size of the visible universe by a factor of ten million.
17‘Measurement began our might’: any error in calibration or in object identification can throw this method into disarray. In the 1950s extragalactic distances suddenly doubled as the German astronomer Walter Baade realised that Cepheid variable stars came in two kinds, and the brighter and dimmer kinds were being confused in distance calibrations.
18This fact – recession velocity grows linearly with distance – is a relationship discovered by the American astronomer Edwin Hubble in 1927 and is known as Hubble’s law.
19The originally derisive description ‘Big Bang’ was coined by the British and Yorkshire astrophysicist Fred Hoyle, who regarded it as an ugly and theistic way for the universe to start. Blunt, brilliant and stubborn, Hoyle got himself into many arguments over the years. His suspicions would not have been allayed by the fact that the first proposer of the idea of a big bang was the Belgian priest-astronomer Georges Lemaître.
20 For afficionados of buzzwords, the gravity theory is type IIB string theory on AdS5 × S5 and the field theory is maximally supersymmetric Yang-Mills theory at large N.
21As the 2015 fourth round FA cup scoreline Chelsea 2 Bradford City 4 shows.
CHAPTER 4
The Truth Is Out There
The previous chapter was a whirlwind tour of the great advances made in our understanding of nature. These advances are associated with the icons of physics – names such as Albert Einstein, Marie Curie, Paul Dirac or Richard Feynman. For all their achievements though, we should not be beholden to great names. However much Einstein contributed, we now know far more, and understand it far more deeply, than Einstein did even at his intellectual peak. One important aspect of this understanding is that our understanding is incomplete. The the
oretical jigsaw outlined in the previous chapter has several large and crucial pieces missing. We do not know for certain what these pieces look like, or where to find them, but we do know where they fit in.
The purpose of this chapter is to explain why something conceptually new is needed. I aim to describe some of these missing jigsaw pieces – of which the first and most prominent is a quantum theory of gravity.
Some scientific theories are not expansionist. They are important in their own sphere but do not extend beyond that. The theory of fracture propagation in metals is important to anyone who boards an aeroplane, but it does not affect all of science. These Switzerlands of the scientific world do what they do well, and are content with that.
This is not the case with quantum mechanics. This is instead a jealous theory that brooks no rivals. It is not content merely with the successful explanation of known phenomena, but is instead continually seeking lebensraum. The claim that the world is described by quantum mechanics is not simply a limited statement applicable to atoms at short distances. It is instead a statement about every last bit of the world – as it was in the beginning, is now, and ever shall be. In this sense, quantum mechanics is a totalitarian theory. It refuses to coexist with alternative accounts of reality but demands for itself a universal applicability to the fundamentals of nature. In this it has been successful, and since its discovery it has spread through the population of scientific theories like a successful mutation.
The quantum field theories that so successfully describe the strong, weak and electromagnetic interactions involve a quantum mechanical treatment of the particles and the forces. The motion and interaction of the elementary particles is described with the full technology of quantum mechanics. However, these particles move against a spacetime backdrop that is taken straight from Einstein’s theory. This spacetime is pristine, relativistic and classical, being entirely innocent of quantum theory.
This may have been good enough if spacetime were itself a fixed entity. Unfortunately, it is not. We saw in the previous chapter that Einstein’s second great insight, general relativity, made spacetime itself dynamical. The geometry of spacetime warps and shimmers in response to the matter that passes through it. Spacetime geometry is no more intrinsically special and fixed than the electromagnetic field – which also fluctuates in response to any passing matter. Geometry then affects geometry now. Geometry now determines geometry-to-be. Small rippling perturbations in the geometry of space and time propagate outwards, just as small perturbations of the electromagnetic field – or indeed ripples on a pond – propagate outwards.
Einstein’s theory of general relativity is a fully dynamical theory of gravity-as-geometry. This theory works extraordinarily well and is extraordinarily well tested. It explains the formation of black holes, the expansion of the universe and the gravitational lensing of light around massive objects. It is verified to describe the universe from an age of one second to an age of fourteen billion years. There is not a single observation that is inconsistent with it. It is a fantastic success. There is no experimental reason to mistrust it.
And yet, it must be wrong. General relativity is a theory untouched by quantum mechanics. It represents the last great hurrah of an older and simpler world. Once introduced, quantum mechanics cannot be restrained. The quantum theory is intolerant of its rivals; it inexorably replaces the old ideas and cannot live alongside them.
As we saw in the previous chapter, the transition from classical to quantum mechanics occurred first in the study of the motions and energies of particles, and of the measurements that could be done on them. For example, the quantum uncertainty principle forbids the simultaneous measurement of both the position and momentum of a particle. The non-relativistic quantum mechanical equations of motion were successfully extended to include special relativity. Careful study of the implications of relativity for quantum mechanics led to the description of the strong, weak and electromagnetic forces using the formalism of quantum field theory. Step by step, the laws of physics have transmogrified from classical to quantum formulations. This shift has percolated through all of physics – but not yet to gravity.
This makes it clear that something – at this stage we do not know what – is needed beyond what we already have. The known laws of physics work incredibly well. A theory of quantum gravity is required not for military applications, nor for solving outstanding industrial problems, nor by any unexplained anomaly in experimental physics. The argument that such a theory exists is purely theoretical. Classical general relativity is the last stronghold of the old physics, and all must give way before the imperium of quantum mechanics. This argument requires that the laws of gravitational physics must change, because the laws of gravitational physics must be consistent with quantum mechanics.
Whatever that something is, it must be conceptually new. In particular, we know it is not just a re-application to gravity of the techniques that have been successfully used to quantise the other three forces. A direct quantisation of gravity along this well-trodden path has been tried – and it fails. It is a noble failure, verdant with spectacularly hard and lengthy computations.
It is true that there do exist general, albeit technical, arguments why this direct approach should fail. Einstein’s theory of gravity is, in the parlance of the trade, a ‘non-renormalisable’ theory. Maxwell’s theory of electromagnetism is, in contrast, a ‘renormalisable’ theory. Quantum calculations in both renormalisable and non-renormalisable theories involve infinities. The difference is that in renormalisable theories, the infinities turn up in only a finite number of places. In non-renormalisable theories, the infinities turn up in an infinite number of places. In the former case, it is possible, with some work, to isolate, categorise and eliminate the infinities with the help of a finite number of experimental measurements. With an infinite number of infinities, this is no longer possible. Removing the infinities now requires an infinite number of measurements – which cannot be performed.
This argument then suggests that the head-on approach to quantum gravity should fail. This general argument does not eliminate the possibility of special structures or cancellations that evade this conclusion. This possibility can be checked only by calculation, and these calculations are hard. The problems lie not so much in intrinsic conceptual difficulties with the computation as in the enormous number of terms that need to be included.1 This work included heroic efforts from the Texas physicist Bryce de Witt, in many ways the pioneer of this subject, and even contributions from Richard Feynman – who did not enjoy his time working on gravity, as a letter he sent home to his wife from a gravity conference in Warsaw reveals:
I am not getting anything out of the meeting. I am learning nothing. Because there are no experiments this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here (126) and it is not good for my blood pressure: such inane things are said and seriously discussed here that I get into arguments outside the formal sessions (say, at lunch) whenever anyone asks me a question or starts to tell me about his ‘work’.
This all culminated in an explicit calculation of the relevant divergences by Marc Goroff and Augusto Sagnotti in 1986. The expected divergences were there. The gravity of Albert Einstein is indeed non-renormalisable, and cannot be quantised using standard techniques.
This, more or less, establishes a need for something new. However, such negative results always require scanning for loopholes. The conclusions of any argument are only as strong as its premises, and there are always assumptions that enter a result. There is in particular one promising variation on this approach that I ought to describe, and on which the jury is still out. This variation involves not Einstein’s original theory of gravity but an extended version involving many additional particles. This variation is called ‘maximal supergravity’.
The majority belief is that this variation is only a sticking plaster and not a cure for the ills encountered in quantising gra
vity. Most probably, it does not work. If it does work, it describes a version of gravity that must be purely theoretical and cannot, even in principle, be merged with what we know of particle physics.
This variation is based on something called supersymmetry. If infinities are going to disappear, they need a good reason to disappear. In the last chapter I enthused about symmetries. Symmetries can provide a good reason for something to disappear. Symmetries make things equal. If a picture is symmetric, we know that the area of the left-hand part of the image exactly equals the area of the right-hand part, without having to work out either area precisely. A symmetry is capable of taking two halves of a calculation and forcing each part to give an answer that is exactly equal and opposite to the other part. The resulting sum of the two halves vanishes exactly, irrespective of how large each individual part may be.
It even holds if each part is infinitely large. This may sound troubling at first – the sum of positive infinity and negative infinity does not appear very well defined. It is here where a symmetry is so useful. Instead of allowing a divergence to continue all the way to infinity, we could cut off all sums at a value of ten to the power of ten to the power of one hundred thousand: a number unimaginably larger than the total number of particles in the universe. Every number in the calculation is now finite and well defined. There are no ambiguities. The symmetry still ensures that the two halves of the calculation add up to zero, and the answer has no pathological features for any value of our cutoff. As we remove this cutoff and make it even larger, the overall sum remains at zero. We can then finally take a formal limit in which the cutoff is removed altogether. As on every step of the way we have sensible answers, this last step of removing the cutoff to infinity also gives a sensible and well-defined answer.