Why String Theory? Page 6

  The experimental situation with the strong force was also mysterious, as an abundant harvest of particles upon particles upon particles appeared. Neither force looked anything like quantum electrodynamics. This, and the quantum field theory formalism that accompanied it, was regarded as a special and irrelevant case. A new – perhaps radically different – approach was felt by many to be needed to deal with both the strong and electroweak interactions.

  Their will was weak, and their understanding dark – for the end result was, yet again, the triumph of quantum mechanics and quantum field theory. As the Harvard theorist Sidney Coleman put it,7

  [1966 to 1979] was a great time to be a high-energy theorist, the period of the famous triumph of quantum field theory. And what a triumph it was, in the old sense of the word: a glorious victory parade, full of wonderful things brought back from far places to make the spectator gasp with awe and laugh with joy.

  Those who invested time in the 1960s working on understanding quantum field theory obtained a return that was the scientific equivalent of placing savings in Berkshire Hathaway shares.8 This can be illustrated through the citation record of one of the foundational papers of the Standard Model, Steven Weinberg’s paper ‘A Model of Leptons’, which was published in 1967.9 This measures the number of times a paper is referred to by a different paper.

  Citations in 1968:

  1

  Citations in 1969:

  1

  Citations in 1970:

  1

  Citations in 1971:

  5

  Citations in 1972:

  36

  Citations in 1973:

  48

  Citations in 1974:

  113

  Citations in total:

  > 9000

  The strong and weak interactions are simply further examples of quantum field theories, the logical consequence of the combination of quantum mechanics and special relativity. Despite the best attempts of the best physicists, quantum mechanics really works, and it really does describe the world.

  3.5 NATURE LIKES SYMMETRIES

  If the kaleidoscopic progress of all of 20th century physics had to be summarised by a single word, that word would be symmetry. More than any other single idea, the identification and applications of symmetries have connected the progress made in apparently disparate areas. These disparate areas include both special and general relativity, high energy particle physics and many-body phenomena such as superconductivity.

  The basic concept of a symmetry is simple and familiar. You take an object, make a change, and find that the object still looks the same. For example, a square rotated by ninety degrees looks precisely the same as it did before the rotation. Alternatively, take a hexagon, rotate it this time by sixty degrees, and the same hexagon looks back at you. The form of the symmetry is different – both a hexagon rotated by ninety degrees and a square rotated by sixty degrees do change their shape. Both the square and the hexagon exhibit simple geometric rotational symmetries, and are said to have respectively fourfold or sixfold symmetry.

  Something similar, but subtly and slightly different, happens with the circle. Given a perfect circle, you can rotate it by a tiny amount and find that you still have the same circle. However, this time I do not need to be precise about what a ‘tiny amount’ means. For the square, the rotation needed to be through ninety degrees exactly, and for a hexagon, sixty degrees exactly. For the circle, no precision is needed. Rotations by either one-tenth of a degree or 17.3456 degrees return the circle back to itself. Instead of simply a few discrete choices of rotation angle preserving the shape, there now exists a continuous infinity of choices.

  As ‘number of degrees we can rotate by’ is a continuous quantity for the circle, but one of a discrete set of choices for the square or hexagon, the symmetries are named differently. The symmetry of the circle is said to be a continuous symmetry and the symmetry of the square is said to be a discrete symmetry. Both types of symmetry, continuous and discrete, have played important roles in physics.10 The study of symmetries in and of themselves is a fine topic for mathematicians. The proof of one mathematical result related to symmetries, the so-called classification of finite groups, is reckoned as the longest published proof in all mathematics, coming in at around ten thousand pages once all contributing journal articles are included. We shall return to one aspect of this result in chapter 9. However, to a physicist a symmetry is bereft of relevance unless it has something it can usefully apply itself to, in the way that the rotational symmetries above could act on the square, the hexagon and the circle.

  Given that symmetries have been so important in physics, what then is their target? The target is nothing more nor less than the laws of nature themselves. The symmetries apply to the fundamental laws of nature. There are certain rotations or transformations that we can perform on these laws but that nonetheless leave them unaltered. Two different observers related by symmetry both observe the same laws of physics.

  For example, the laws of physics have been discovered to possess rotational symmetry. Suppose I plant three wooden beams down, all at right angles, using these to define three axes x, y and z. I measure the laws of physics – for example, the statement that force equals mass times acceleration – with respect to these axes and coordinates x, y and z. I now rotate the frame about the z-direction, such that the x-direction points in the y-direction and the y-direction points in the minus-x direction, and repeat the measurements. I will get the same results and find the same laws of physics. The fundamental laws of physics are unaltered when expressed in terms of these new coordinates.

  This result is an amazing discovery. It should come across as shocking and outrageous, a mixture of a ‘what?’ and a ‘wow!’. If it does not, it is a result of both the complacency of hindsight and our inability to recast our minds to an earlier way of thinking. Why is this result so shocking? For rotation in a plane, the result is indeed plausible. We do not fret about whether athletics records should be recorded differently for tracks laid out north-south or east-west. Our ability to walk, run and hunt is unaffected by the direction in which we are facing. We tell the difference between east and west by the sun, not by noticing differences in how objects move. However, this is not at all the case for three-dimensional rotations. In fact, the lessons of physical experience always and everywhere contradict any symmetry between up and across – and anyone doubting this is invited to try and climb a tree and then fall sideways. Gravity, the force that we first meet in the cradle, appears to act only in the vertical direction, and then specifically down rather than up. This singles out both a preferred axis (the up-down one) and a preferred direction (down).

  These results appear so obviously true that they threw natural philosophers, the proto-scientists of their day, for millennia, and an entire false physics was constructed around it. This was the Aristotelian universe, radiating out from the centre of the earth. Everything is made of four elements – earth, air, fire and water – each of which seeks to find their natural location. The centre of the earth is the centre of the universe, and in that objects partake of the element ‘earth’ they want to fall inwards towards this point, whereas the element ‘fire’ wants to rise to the heavens. Aristotelian physics gives an intuitive picture correctly describing our day-to-day experience – and is also totally and irrevocably wrong.

  For, whatever our daily experience may say, the fundamental laws of physics are indeed symmetric under three-dimensional rotations. Gravity has no intrinsic preferred direction. It simply attracts stuff to other stuff, causing objects to clump together. If you put a thousand billion billion tons of stuff – let us call this the earth – down somewhere, by doing so you introduce a preferred direction for everything else towards this great pile of matter. However, this preferred direction is only accidental. Moths flying in a dark room have no preferred direction of flight. Light a candle, though, and they will cluster around it. In a similar way, the apparent preference towards the downwards dir
ection on earth is real but accidental, set by the presence of the earth but not part of the basic rules of the game.

  Even though it alone should be enough to shock, rotational symmetry is only the simplest of the symmetries that apply to the laws of physics. It is also a symmetry that had been recognised prior to the twentieth century. Much of the progress in physics during the last century has come from a broader development of both the types and applications of symmetries.

  As with space, so with time. Special relativity comes from extending the rotational symmetry of space to include time. Instead of simply considering rotations among three spatial coordinates x, y and z, rotations now involve both space and time, involving also a fourth coordinate: time t. In spatial rotations, the roles of x, y and z in equations would be mixed up and interchanged, but leaving the equations with ultimately the same form. In the spatiotemporal rotations of special relativity, the roles of x, y, z and t in equations are interchanged: but as before, the requirement of symmetry is that the laws of physics take the same form both before and after the rotation. This requirement enforces a special form for the physical laws – the structure of special relativity. While this discovery of Einstein is sometimes introduced by thinking about what it would be like to travel near the speed of light, it can also be viewed as the inevitable result of symmetry under spatiotemporal, rather than just spatial, rotations.11

  General relativity is built on a cognate but more audacious use of symmetry. Special relativity makes the laws of physics invariant under some particular choice of coordinates. We set down our wooden spatiotemporal measuring frame, and however we rotate it, the laws of physics will still take the same form. Special relativity does however require the presence of the frame. As one example, the spatial legs of the frame must be at angles of ninety degrees. We can rotate the frame how we wish, but we cannot choose its legs arbitrarily.

  General relativity does away with this. The essence of general relativity is that the laws of physics do not care for coordinates. Coordinates in all their incarnations – atlases, charts, grid references – are mere human constructs, and nature does not give a fig about our choice of coordinate systems. Does the earth know about latitude and longitude? Did the great white whale inwardly rejoice as it crossed the prime meridian? The physical laws are ultimately equally indifferent to all choices of coordinates. Einstein’s greatest insight was to require that the laws of physics be independent of coordinates. Whichever choice of coordinates you lay down, the laws of physics should take the same form. The lift from special to general relativity comes from demanding the symmetry of rigid rotations be extended to any choice of coordinates whatsoever. While this may sound deceptively simple, it is a measure of Einstein’s intellectual depth that this insight also leads to so many consequences.

  This symmetry – whose technical name is general covariance – marks the high point of classical physics. Formulated in 1915 amidst one of the low points in European history, general relativity describes the dynamics of space and time themselves. It is the crowning glory of classical physics – but symmetry principles would be no less relevant for the quantum mechanics that would follow a few years later.

  The symmetries discussed so far have been geometric symmetries, building on intuitions developed for two-dimensional rotations. Their implications may be hard to understand, but their roots are clear. However, there is another type of symmetry that is less easy to intuit, called gauge symmetry. These gauge symmetries are to the Standard Model of particle physics what carbon is to organic molecules: the foundations on which the entire structure is built. While incredibly important, they are also harder to explain than geometric symmetries, as their symmetries are internal to the mathematical equations of the Standard Model rather than inherited from the outside world.

  Gauge symmetries are not easy. One of the many remarks alleged to have been said by Einstein – like Mark Twain, he attracts quotations – is that if you understand something, you can explain it to your grandmother. The logical consequence of this would be that either Granny Einstein was a mathematical whizz, I do not understand gauge symmetry, or Einstein understood gauge theory a whole lot better than I do. It requires no attack of hubris or egomania to say that none of the above are true. Science marches on. In the same manner that any rational person in need of an appendectomy would choose the most ham-fisted product of an NHS teaching hospital in preference to Avicenna, so the least prepared of today’s physics graduate students far surpass Einstein in their understanding of quantum field theory.

  This declaration that gauge symmetry is hard is a prelude to my attempt at explaining it. The following page is however inessential to the narrative and the explanatory attempt can be skipped without harm.

  First of all, what is it not? Gauge symmetry is not a spatiotemporal symmetry. It is an internal symmetry of equations rather than an external symmetry such as rotations. What is meant by an internal symmetry? The equation

  q2 + p2 = r2

  has an internal discrete symmetry whereby q and p are exchanged. Doing so simply gives back the same equation. Furthermore, this equation in fact has a continuous family of symmetries. To see this, we realise that this is the equation of a circle, with q and p treated as the coordinates and r as the radius. There is then an internal family of rotations by which q and p are rotated into one another – although we should think of these rotations as purely mathematical, acting on the objects inside the equations.

  The first requirement for a gauge symmetry is a set of equations that represent physical laws and must hold at every point in space and at every moment in time. An example of these are the laws of electromagnetism. The equations that describe the electric and magnetic fields hold here, there and everywhere; and they hold in the past, present and future. Wherever you are in the universe, these equations hold. What is needed to call these equations a gauge theory is the additional fact that these equations always have, at every instant in space and time, an additional internal rotational symmetry. A theory is then said to possess gauge symmetry if we are able to perform an internal rotation of the equations at every point in space and time, also choosing the rotation to be different at every point (ninety degrees here, ninety-one degrees here, ninety-two degrees here), such that the overall equations still remain unaltered.

  This is the most basic idea of what gauge symmetry is – why is it called so? In the rest of the world, gauges appear most prominently in railway layouts. Here the gauge specifies the separation between the rails, and thereby the spacing between the wheels of the train. Different choices of gauge have brutally different effects, as any attempt to run a train straight from a broad gauge railway onto a narrow gauge railway would demonstrate. In physics, the gauge refers to the degree of internal rotation – and gauge symmetry is the statement that both the equations and the physics remain the same as you alter the gauge so that it constantly changes from point to point.

  This may indeed seem opaque and complicated. However, you should know – you must know – you will know – that the reason gauge symmetries are so important is because they correspond to forces. Electromagnetism is described by a gauge theory. The strong nuclear force is described by a gauge symmetry. The weak nuclear force is described by a gauge symmetry. Three of the four fundamental forces in nature come down to the special type of symmetry known as a gauge symmetry. It is for this reason that gauge theories play the role in the education of aspirant young particle theorists that the Iliad and Odyssey played in the classical educations of an earlier era.

  It is a further remarkable statement that the gauge symmetries dictate not just the forces that are present but also the types of particle that are allowed and the form their interactions must take. One of the beauties of gauge symmetries, which also makes them highly testable, is the way they force many interactions that appear independent into a particular structure. In the same way that the basic unit of light – the basic unit of electromagnetism – is the photon, the basic unit of the stro
ng interactions is called the gluon. The gauge symmetry of the strong interaction tells us that there must be not one but a total of eight gluons, and it furthermore tells us how each of these eight gluons interacts with each of the others. As another example, the W– particle decays to the electron, muon and tau particles with almost equal rates. This equality follows because of gauge symmetry, and could not have been otherwise. The gauge symmetry of the weak interaction also relates the interactions of the electron to the interactions of the neutrino.

  Why is this? It comes down to what gauge symmetries actually do to the equations. For example, in the equations of the Standard Model there are terms that involve both the electron and the neutrino.12 From these equations, one can extract the interactions of the electron and also the interactions of the neutrino. The way the gauge symmetry of the weak interaction works is that it rotates the term corresponding to the electron into the term corresponding to the neutrino. Acting on the innards of the equations, it moves the terms around so that the ‘electron’ term moves to where the ‘neutrino’ term was, and vice-versa. The gauge symmetry is a symmetry, so after this change the equations of the weak interaction must still take the same form – which they do, except what used to be called the ‘electron’ is now called the ‘neutrino’. In this way the weak interactions of the neutrino are fixed by the weak interactions of the electron.13

  Let me also try and explain why gauge symmetries fix the numbers of different types of particles. I have just said that one of the main ways gauge symmetries act on the equations is by moving around the terms corresponding to different particles. Different symmetries do this in different ways. I mentioned earlier the case of the fourfold and sixfold rotational symmetries that belong respectively to the square and the hexagon. Both symmetries return to their starting point, but after producing a different number of ‘copies’: four and six. The number of such copies is set by the mathematics of how the symmetry works. Although the algebra is more complicated, the same principle holds for the gauge symmetries of the Standard Model. The mathematics of symmetry imply that there are only a certain number of independent ways of reordering the particle terms in the equations among each other. The number of such ways that exist tell you the number of each type of particle, all of whose interactions are related to each other by symmetry. It is this mathematics of symmetry that tells us that there are three copies of each quark and eight copies of the gluon.14