Why String Theory? Read online

  Symmetries allow problematic infinities to be removed from calculations. This is good – as far as it goes. The problem is that generally it does not go far enough. In gravitational theories, removing infinities is somewhat analogous to playing whack-a-mole at a mole breeding farm. There are many sources of infinities. In fact, there are an infinite number of infinities to be removed. It is not enough to have a symmetry that will remove some of the infinities – it must be big enough and powerful enough to remove all of them.

  One of the features that is super about supersymmetry is that it has long been recognised as being good at removing infinities from calculations. The possibility of supersymmetry as a symmetry of our four-dimensional spacetime was stumbled upon early in the 1970s, first in the Soviet Union and then independently in the West. Nature partitions particles into two classes, called bosons and fermions,2 and at the simplest level supersymmetry relates the properties of bosons to the properties of fermions.

  Simplifying slightly, in the analysis of infinities both bosons and fermions give independent contributions to these infinities. Supersymmetry is able to relate the calculational divergences caused by bosons to the calculational divergences caused by fermions, in such a way as to make them equal and opposite. The first supersymmetric theories of gravity – or supergravities – were developed at Stony Brook University in New York by Dan Freedman, Peter van Nieuwenhuizen and Sergio Ferrara in 1976, and it was rapidly realised that the presence of supersymmetry improved the situation with respect to divergences.3 Large classes of infinities were removed – but an infinite number still remained. Gravitational theories were constructed with more and more supersymmetry; more and more divergences disappeared. However, even with the fullest and richest form of supersymmetry, the so-called maximal supergravity theory, theoretical arguments still indicated that infinities should remain, although at a level of calculational complexity that was impossible to verify explicitly. It thus appeared that large amounts of supersymmetry, by itself, ameliorated – but did not ultimately cure – the problem of divergences in the direct approach to quantum gravity. This was the situation of quantum gravity in the 1980s, contemporaneously described by Stephen Hawking’s famous book A Brief History of Time.

  Developments in the last ten years have caused this view to be revised. These developments all focus on the maximally supersymmetric theory that was considered most promising during the 1980s. One decisive result is that the old arguments as to when infinities were expected to appear are wrong. These arguments were based on the visible symmetries of the theory, and the expectation that certain infinities would remain was based on those symmetries and those symmetries alone. It has now been discovered that the maximal supergravity theory has additional, hidden symmetries that were previously missed. This extra structure ensures that many terms that had previously been believed to be infinite are actually zero – and this is enforced by the newly discovered hidden symmetries.

  This raises the question of whether all the infinities that may in principle arise do actually in the end vanish. Perhaps, there are further additional hidden symmetries that remain missed. Could it be the case that the visible and hidden symmetries, taken all together, are strong enough to cause the maximally supersymmetric theory to be finite, with no infinities at all? If this were the case, then there might exist a theory of quantum gravity, extending but including Einstein’s theory, with no infinities at all.

  The jury is still out on whether such a theory exists. The question of divergences in maximal supergravity theory must have a definite answer, but it is also not amenable to frontal assault as direct calculations are just too hard. If an answer is found, it will only be through clever arguments.

  However, even if the answer is in the affirmative, it ultimately does not help so much. The prescribed medicine for the ills of infinite divergences is an enormous dose of supersymmetry. Unfortunately, this medicine is far too potent and kills more than just infinities. The same large amounts of supersymmetry that remove the infinities also make it impossible to include the Standard Model of particle physics within the theory. These whopping amounts of supersymmetry impose particular constraints on the particle types that are allowed, and the particles of the Standard Model violate these constraints.

  In particular, the unavoidable consequence of such large amounts of supersymmetry is that the laws of physics must be unable to distinguish left and right. However, one of the most crucial and striking features of the Standard Model is that it violates parity – it is able to distinguish between left and right. What does this mean? As we saw in the previous chapter, this means that it is possible to define the meaning of ‘left’ and ‘right’ by reference only to the laws of physics. Suppose we were meeting for the first time distant humanoid aliens, and that we wished to start our meeting by shaking our right hands together. The laws of the Standard Model are such that we could explain this in an unambiguous way, and we could both agree on which is our right hand.

  Even in principle, this is impossible within maximal supergravity. For this reason, it seems that the theory of maximal supergravity, for all its pristine mathematical beauty, is at best a technical curiosity: it is not the world, and the world is not it.

  The argument of the last few pages has been a classic theoretical argument for why physics is incomplete and something new is needed. It is rather philosophical. It is abstract, founded on general principles, and appeals in at most a minor way to the actual properties of any known particles. The strength of the argument comes simply from the intrinsically quantum-mechanical nature of the world. It identifies the need for something new – a quantum theory of gravity – but offers minimal guidance as to the form it will take or the equations that will govern it. The sole requirement is that this new theory, whatever it may be, must reduce to classical Einstein gravity in the limit where quantum effects are unimportant.

  This represents the ascetic aesthete’s approach to realising that something more is needed. However, the same conclusion can also be reached by another, rather more empirical, route. This is based on examining the known structure of physics, and in particular the known structure of the particles, masses and interactions of the Standard Model.

  These masses cover a large range. The heaviest known fundamental particle, the top quark, has a mass comparable to one atom of lead. The lightest, the neutrinos, have masses roughly one trillion times smaller. The electron is halfway inbetween: around a million times heavier than a neutrino, and around a million times lighter than the top quark.

  The masses and interactions of these particles are described by the Standard Model of particle physics. The Standard Model does impose certain automatic relationships between different interactions – for example, the electron must interact with the photon in precisely the same way as a muon does. However, many of the particle masses and interactions are only parameterised but not explained by the Standard Model. They enter the Standard Model as fundamental constants – and the precise origin of these constants is someone else’s puzzle.

  It is indeed someone else’s puzzle, because these constants have remarkable patterns and structures. Their values are not random. These constants were not drawn by gods playing dice in some pantheonic lottery. Instead, they have patterns that tell us that the ‘constants of nature’ appearing in the Standard Model must really be products of a deeper theory lying underneath the Standard Model.

  What are these patterns? The first pattern is the existence and replication of particle families. Most of the particles of the Standard Model come in groupings called families. Each family contains a particular set of particles, and there are three such families in total. Each grouping involves similar types of particle, with the families differing only by the relative masses of the particles. The first family of the Standard Model contains the electron, the electron neutrino and the up and down quarks. The second family contains the muon, the muon neutrino, and the charm and strange quarks. The final, heaviest, family is made up of
the tau, the tau neutrino and the top and bottom quarks. The particles here have been ordered by their relative type. The tau and the muon are heavier copies of the electron. The top and charm quarks are heavier copies of the up quark, and the bottom and strange quarks heavier copies of the down quark.

  On hearing this, one may think that family replication could occur an infinite number of times. We have observed three families, but perhaps there is also a fourth family that is present but simply too heavy for us to observe with current accelerators.

  This is wrong. It is known experimentally that no fourth family exists: the count stops at three. The simplest argument for this is that all the neutrinos are very light, and so any fourth light neutrino should already have been observed. However, this argument is not watertight – it could be the case that the fourth neutrino exists, but is different and is perhaps just very much heavier than the neutrinos of the first three families.

  This loophole appears contrived but could not be eliminated for a long time. Its definitive elimination occurred with the discovery of the Higgs boson at CERN in 2012 and the analysis of its properties. Although the calculational details are technical, the existence of any fourth family, however heavy, would have drastically modified the properties of the Higgs boson away from the behaviour expected and observed in the Standard Model.4 Fourth family behaviour was not observed, and the existence of a fourth family of particles is now decisively ruled out. There are then exactly three – and no more – particle families. This family replication, included in the Standard Model but not explained by it, is one sign of a deeper layer of physics beyond our current knowledge.

  A second pointer to the incompleteness of the current framework lies in the structure of particle masses. Each jump in family, both from the first family to the second family and again from the second family to the third family, brings a jump in mass by a factor between ten and a hundred. The jump from the electron to the muon is a two-hundred-fold increase; that from the muon to the tau a twenty-fold increase. The jump from down to strange is a factor of thirty, and from strange to bottom a factor of forty. The jump from the up quark to the charm quark, and from the charm quark to the top quark, involve mass ratios of approximately one hundred and fifty. These rapidly increasing particle masses indicate a deeper underlying explanatory structure, but again it is unclear what that structure is. Something new is needed, but what is that something?

  Neutrinos stand out as an exception to the above patterns. Neutrinos are special. They are almost massless and almost non-interacting. Neutrinos do, just about, interact. A neutrino passing through a sheet of lead has a finite probability of being stopped by it. This probability is not large. It requires a lottery winner’s dose of good luck for one kilometre of solid lead to stop a neutrino. However, the probability is not zero. Given enough neutrinos, some will interact, and these interactions can be detected.

  For example, the core of the sun produces neutrinos in abundance as a by-product of the nuclear reactions that fuel the sun. These neutrinos stream outwards in all directions. By the time they reach the earth, their flux is still such that every second one thousand billion neutrinos pass through the palm of your extended hand. Given enough target material – one hundred thousand gallons of dry cleaning fluid in the original experiment – the occasional interactions of these neutrinos can be detected.

  Neutrinos also do, just about, have mass. It is not possible to measure the absolute masses of neutrinos. However, it is possible to measure the difference in masses between different kinds of neutrinos.5 These mass differences can be inferred, although in a manner that is not obvious, from the phenomenon of neutrino oscillation: different kinds of neutrino, such as the electron neutrino and the muon neutrino, oscillate into one another as they travel. Two such mass differences have been measured, and so we know that of the three neutrino species, at most one can be massless. The absolute mass scale of neutrinos is not known, but a combination of direct searches and limits from cosmology tell us that the heaviest neutrino can be no heavier than approximately one millionth of the mass of the electron.

  The particles of the Standard Model therefore reveal a clear structure. They come in three families, and with the exception of the neutrinos, each family is heavier than the previous one by a factor lying between ten and one hundred. The neutrinos sit in almost massless isolation, more than a million times lighter than the other particles. These patterns in the particles are accommodated but not explained by the Standard Model: they are pointers to a better theory, but which theory they point to is unclear.

  There is a final sign within the Standard Model of a better world to come, a sign as clear and as clean an indicator as one could hope for. One of the many parameters of the Standard Model is an angle. This angle is called the theta angle, and it controls certain properties of the strong nuclear force. There is one particle, the neutron, that is especially sensitive to the theta angle. As befits its name, the neutron is neutral, with no overall electric charge. However this neutrality is consistent with an internal charge distribution, where an excess of positive charge on the ‘northern’ side is cancelled by an excess of negative charge on the other ‘southern’ side.

  The theta angle controls this apparent charge distribution. Its extent is determined by the value of the theta angle, as the larger the angle, the more pronounced the charge asymmetry would be. This charge distribution has been searched for – but it does not exist. To within the limits of experimental accuracy, the neutron is entirely neutral: there is no such north-south asymmetry in charge distribution. This measurement determines the value of the theta angle: its magnitude is not larger than one hundred millionth of a degree. The theta angle is, as well as can be measured, zero.

  This is surprising. The theta angle is an unspecified angle, and so a priori can take any value between zero and three hundred and sixty degrees. If this angle were simply a randomly chosen constant of nature, with nothing to guide us we would expect its value to be somewhere in the middle of this interval. Zero at greater than a part-per-billion experimental accuracy looks suspicious.

  At the level of the Standard Model, theta was just an angle, and nothing prefers one particular value over any other. It is hard to look at the measured value of this angle, though, without also suspecting hidden wheels whirring. It does not look random, it almost certainly is not random, and instead this measurement surely tells us about a lumbering hidden structure in the background revealing its shadow through the physics we already know.6

  This second set of arguments for something new has been more practical than the first argument involving quantum gravity. It involves patterns and structures present in existing known theories. These patterns are striking, and it seems highly implausible that they occur purely by chance. Certainly the presence of the patterns implies no breakdown, either theoretical or experimental, in any existing theory, and it is logically possible that the patterns are produced randomly. However, the most economical explanation is that we, as well as our ancestors, are ignorant of some of the important truths of nature.

  There is finally a direct empirical reason why something new is needed and why our existing theories are incomplete. This reason is the existence of dark matter. Dark matter is a form of matter that is implied by astrophysical and cosmological observations to make up around one part in five of the energy budget of the universe. We know a lot about what dark matter is not, but relatively little about what it is. We know it is dark – it neither emits nor absorbs light. We cannot see it, and it interacts by neither the electromagnetic force nor the strong force. Dark matter is also known not to consist of any of the particles of the Standard Model, and at best its interactions with familiar Standard Model particles are exceedingly weak.

  How do we know it is even there? We weigh it. Matter matters. In the last resort, anything with any kind of energy or mass communicates via gravity, and what gravitates can be weighed. Suppose you enter a dark room with a transparent bag known to contain
one kilogram of fluorescent rocks. You place the bag on scales, and the scales read two kilograms. You can see the glowing rocks, but the scales reveal the existence of an additional dark kilogram. You may not be able to see it, and you may not know what it is, but you know it is there.

  The existence of dark matter is established in a similar way. The bag is at least the size of a galaxy. The fluorescent rocks are stars. The principles are the same. The mass that is weighed is bigger than the mass that can be counted, and the difference between the two reveals how much dark matter is present.

  The overall mass of this galactic bag can be found in various ways. The simplest method comes from looking at the motion of stars near the edge of a galaxy. Stars orbit around galaxies in the same way that the planets orbit around the sun. Just as the detailed orbits of the planets tell us the mass of the sun, so the detailed orbits of stars tell us the total mass enclosed within a galaxy. This total mass can be compared with the amount of visible mass present in stars, dust and gas – and is found to be far greater, revealing the need for additional dark matter.

  This technique can also be used on larger scales, applied to large clusters of hundreds or thousands of galaxies. It is now the orbits of galaxies rather than stars that are useful – and these again tell us that the total mass pulling on the galaxies is far larger than the visible mass present in either the galaxies, interstellar gas or dust.

  An entirely complementary way of measuring the total mass present is through examining the bending of light. Einstein’s theory of general relativity tells us that heavy objects attract light, and that the paths of light rays are bent when they pass massive objects. The more massive the object, the more the direction of the light is changing. Through studying the distortions in light paths caused by the gravitational bending and lensing of light, the total mass of an object can be inferred.