Gauge Symmetry, Unification and Strings 1 Symmetries in Nature

Home , Amitava Raychaudhuri, Rajesh Gopakumar, Sunil Mukhi

Gauge Symmetry, Uniﬁcation and Strings

Sunil Mukhi Tata Institute of Fundamental Research, Mumbai

1 Symmetries in nature

The term “symmetry” refers to the even balance and proportion of an object. If this definition looks somewhat vague, a more precise one can easily be provided. Symmetry of a shape is the property that its form remains unchanged after a transformation is performed on it. As simple examples, the human figure looks roughly the same when inverted in a mirror, and there are objects like star-shaped fish that look the same when we rotate them about an axis such that each of the arms gets mapped onto the next one. In these examples there is a definite “symmetry operation” that takes the object and trans- forms it such that the final result looks the same as the original. For a regular five-pointed starfish, such an operation is rotation by a fixed angle of 72 degrees (one-fifth of a full turn), and multiples of that. A different rotation, for example a rotation by 5 degrees, will not leave the starfish looking the same as before, but puts it in a different configuration that can be distinguished from the original one.

72◦ rotation −→

Figure 1: Symmetry operation on a starﬁsh. (Original photo c Nevit Dilmen.)

Things are diﬀerent for a smooth round ball. Here we can rotate it as much or little as we like and it looks the same as before (we assume nothing is drawn on its surface, so that all points look exactly the same as all others). Moreover there are three independent axes about which it can be rotated. For a smooth egg-shaped surface there is only one axis of symmetry, which – if we orient the egg so that its narrow end points upwards – is the vertical axis.

1 But in either case, the symmetry is present for arbitrarily tiny rotation angles. Rotations by larger angles can be “built up” by combining a large number of tiny rotations.

Figure 2: Symmetry axis of an egg.

Since an egg only admits rotations about a single axis, two successive symmetry operations have to both be rotations (by two different angles) about the same axis. The end result corresponds to a rotation by the sum of those angles, and the order does not matter. In this case the symmetry is called Abelian. For a sphere, however, we have three different symmetry axes. Rotating any object by some angle around the x-axis and then by another angle around the y-axis, yields a different result from performing the same acts in the reverse order. A symmetry for which the order matters is called non-Abelian. The above examples illustrate some basic features of symmetry transformations on an object. The transformations can be “discrete” as for the starfish, or “continous” as for the ball. We will be interested primarily in the latter kind. The collection of all symmetry transformations on a given system, along with the inverse transformations and the rule for composing two transformations to get a third one, is a mathematical structure called a “group”.

2 Local gauge symmetry

While symmetries of the kind described above are quite familiar in our everyday experience, there is a refinement of the idea that is very strange at first sight. Suppose we break up the surface of an egg into a hundred little parts, for example by drawing a grid on it. Now we rotate the egg about the vertical axis as before, by a small angle, but let each element of the grid rotate by a different amount. The result, if the surface of the egg were stretchable, would

2 be a rather deformed egg (and with a real egg, the result would surely be much worse!). Thus “local” rotations in space – here “local” means “independently at each point” – are not a symmetry of our world in our everyday experience. A local rotation can be thought of as a rotation by an angle which is a function of where one is located. Surprisingly, this strange operation, along with its generalisations, is actually a symmetry of the world at the microscopic level. We will explore this fact in the rest of this article, and will encounter the even more surprising fact that such local symmetries dictate the way our theories of nature are built, as embodied in the “Standard Model” of particle physics. It is expected that future theories that will take us beyond the domain of the Standard Model will also be governed by such local symmetries, which for historical reasons are more commonly known as “gauge symmetries”.

3 Gauge symmetries and general relativity

Before engaging with the Standard Model, let us consider the one force in nature that lies outside the domain of this model: gravity. For centuries it was believed that gravity was governed by the inverse-square force law due to Newton. However, Einstein’s radical General Theory of Relativity overthrew that in favour of a more complex description. In this description, spacetime is a curved manifold and the force of gravity is a consequence of the geometry of this manifold. Einstein’s theory embodies a principle called “general coordinate invariance” according to which coordinates of spacetime are mere tools that we use to label and study physical phenomena, but have no intrinsic significance. We are free to change from one coordinate system to another with no change in the intrinsic geometrical structure and according to Einstein this can lead to no change in the physical laws (the “principle of equivalence”). Now an arbitrary change of coordinate systems amounts to defining new coordinates as arbitrary functions of the old ones. Therefore general coordinate invariance is a kind of local symmetry. The analogy with local rotations of an egg, as discussed in the previous section, can be made stronger. At each point of a manifold one can erect a “tangent space”. For this, imagine a tangent plane to each point of a spherical surface, and then (this is the difficult part!) extend the notion to 3+1 dimensional spacetime. Now starting from the general coordinate invariance of general relativity, it can be shown that the laws of physics are unchanged if we perform a spatial rotation or a boost on the tangent space independently at every point

3 of spacetime. Together, rotations and boosts make up the set of “Lorentz transformations” and the Special Theory of Relativity teaches us that these are symmetries of nature. But because these can be performed independently at each point of spacetime, we really have a local, or gauge, symmetry. In this example, and the others to follow, local symmetry is not merely a consequence of the physical laws. In a deﬁnite sense it predicts those laws. What this means is that if we require local Lorentz invariance and try to ﬁnd any theory whose equations possess this invariance, we will be inevitably led to Einstein’s theory of gravity – including the very existence of the force as well as the precise set of possible interactions. So, from a modern point of view one can treat local Lorentz invariance as the origin of gravity. In a quantum treatment of gravity, there would be a particle communicating the gravitational force, called the graviton. It is supposed have 2 units of intrinsic angular momentum1 or spin. Although there is no direct evidence for it, it is widely expected that such a particle does exist and is exactly massless, as evidenced by the fact that the gravitational force travels from end to end of the universe. Masslessness of the graviton is actually a prediction from local Lorentz invariance. We will see that gauge symmetry always predicts massless particles in a certain sense, though nature has found interesting and subtle ways to re-interpret that statement!

4 Gauge symmetry in electrodynamics and strong interactions

After the famous Maxwell equations were discovered, it took a long time for their symmetry properties to be appreciated. Today we know that these equations have a local symmetry under which the wave function of each matter particle acquires a phase, and the photon ﬁeld shifts by a corresponding amount. The important thing is that this phase is an arbitrary function of spacetime. In the complex plane a phase is just a simple rotation, and in this sense the gauge symmetry of electrodynamics is the one we encountered earlier when we rotated each part of an egg independently2. The Abelian group of phase rotations is called U(1). Here the lesson expounded in the previous section works equally well. By merely postu- lating that the laws of physics are invariant under local phase rotations we ﬁnd that the

1in units of the scaled Planck constanth ¯. 2We could even rotate these parts diﬀerently at diﬀerent instants of time.

4 theory requires the existence of charged particles as well as the force-carrying particle that communicates between them, the photon. But unlike the graviton, the photon has spin 1. Again gauge invariance implies that the photon must be massless and the electromagnetic force consequently long-range. This is experimentally verified to a very high degree of ac- curacy. So from a modern point of view one can treat local phase rotations as the origin of electrodynamics. Today we know that something very similar is true for the strong interactions3. All known “hadrons” (strongly interacting particles) come in three colours each, and the field theory describing them (discovered in 1970 by Gross, Wilczek and Politzer) is invariant under local rotations of these colours. If the fields describing quarks had been real these rotations would have been exactly like the group of ordinary rotations in space, which we call SO(3) (the “O” stands for “orthogonal” which is a property of the matrices generating such rotations). However the quarks are complex, so we have something like a complex rotation group in three dimensions and this goes by the name of SU(3) (the “U” stands for “unitary” which describes the matrices generating these transformations). Local SU(3) symmetry generates the forces required for the strong interactions, just as local U(1) symmetry generated the photon required to mediate electrodynamics. The analogue of the photon here is a set of eight particles called “gluons”, which like the photon have 1 unit of spin. But there is a puzzle. Local gauge invariance requires that these gluons be massless, just like photons and gravitons. However massless particles mediate long-range forces, and the only known long-range forces are precisely those mediated by the photon and the graviton. If nature really had a set of eight strongly interacting massless particles then we would have seen them long ago and the world would have been a different place. How do the gluons evade the restriction placed on them by local SU(3) gauge symmetry? As with so many other paradoxes in science, the resolution is that the assumptions leading to the paradox are not valid in the present case. The correct statement is that local gauge invariance requires a certain set of fields to be “massless”, which is a well-defined mathematical statement embodied in the equations of the theory. However a massless field does not necessarily lead to a massless particle! The catch is that the correspondence between fields and propagating particles actually breaks down at large distances. There is in fact no physically propagating particle corresponding to the gluon field, except at subnuclear distances. Gluons do mediate strong interactions, but they (along with the quarks between which they communicate) stay tightly locked up inside protons, neutrons and other

3For an explanation of these interactions, see the article by Amitava Raychaudhuri in this volume.

5 Figure 3: Conﬁned quarks interacting via the colour gauge force. (Original photo c Arpad Horvath)

“hadrons”. This property is known as “confinement” and can be roughly understood as a consequence of the growth of the strong force with distance. Due to confinement, we have the benefit of local gauge symmetry and its predictions without the attendant prediction of massless particles that are not found in nature4. It must be emphasised here that local gauge symmetry in this context was not imposed as a fanciful requirement for reasons of beauty and elegance, but came to be established as an experimental fact about the strong interactions, just as it is a fact about electrodynamics and gravity.

5 The electroweak theory and the Higgs mechanism

We have seen that local gauge symmetry is a key property of the electromagnetic, strong nuclear and gravitational interactions. That leaves just one more interaction in nature: the weak interaction. Here too gauge symmetry is realised, but in another surprising way. In fact there are several important new features in this case. One new feature is that the relevant theory, due to Glashow, Salam and Weinberg in the late 1960’s, describes the weak and electromagnetic interactions together using the non-Abelian gauge group SU(2) × U(1). In a sense the two forces, although experienced as totally independent in nature, arise from a common theory in which they are “mixed up” with each other, or uniﬁed, with a total of four force carriers between them (four being the dimension of the group SU(2) × U(1)). This structure, called the “electro-weak theory”, partially realises Einstein’s dream that all forces in nature arise from a single common force.

4Gauge invariance of the strong interactions had a complex and confusing history which we have avoided discussing here.

6 Einstein had actually hoped to unify electromagnetic with gravitational interactions (these were the only interactions known during his youth!) but his project got off to a concrete start only some years after his death with the above electro-weak theory. As we will see, the remaining forces are not yet unified with these two and this remains a tantalising and essential goal for researchers in particle physics. The second new feature of the electro-weak theory is that it violates parity, the symmetry between left-handed and right-handed objects that most people take for granted. It was known for a long time that the weak interactions violate parity, and this requirement was therefore built in to the electro-weak gauge theory. Parity violation was achieved by having left-handed particles transform differently under local gauge transformations from their right- handed counterparts. The third new feature of electromagnetic interactions is that the gauge symmetry is “spontaneously broken”. This property arose in response to a question that has already come up in the previous section: gauge symmetry apparently requires the carriers of the corresponding force to be exactly massless. Applying this to the gauge group SU(2) × U(1) we would conclude that the photon as well as the three additional carriers of the weak force must be exactly massless. That statement is true enough for the photon, but emphatically false for the weak force as evidenced by its short-range nature. Given the parity violating nature of the particle spectrum, an additional puzzle in formulating the theory was that, for mathematical reasons, mass terms for the matter particles (quarks and leptons) respecting the gauge symmetry seemed impossible to write down. Then not only would the force carriers be massless, so would matter particles like the electron – in blatant contradiction with experiment! Both puzzles were solved in one step by adding a set of extra scalar (spinless) particles to the theory. These particles interact with each other through a potential energy term. A suitable choice of this potential forces the scalar field to take a nonzero value in the vacuum in order to minimise the energy. As a result, SU(2) × U(1) transformations do not leave the vacuum invariant and we have the phenomenon of spontaneous symmetry breaking. The end result, according to a beautiful mechanism discovered by Peter Higgs, is that three of the four scalar fields disappear from the spectrum of the theory (i.e. they do not correspond to physical particles) while the three carriers of the weak interaction simultaneously acquire a mass through their coupling to these scalar fields. Moreover, the interactions of quarks and leptons with these scalar fields lead to masses for the quarks and leptons themselves (overcoming the difficulty of writing mass terms in a parity-violating theory). At the end of the day, we have three massive particles called the W +, W − and Z bosons. There is also

7 one surviving scalar as a result of the Higgs mechanism, and it goes by the name of “Higgs boson”. The massive W ± and Z-particles predicted by the electro-weak theory have thereafter been abundantly produced and studied, while the Higgs particle itself still awaits discovery. How- ever, there is a sense in which the existence of the Higgs is already tested, since it contributes to the precise values of other experimentally measurable quantities which have been tested to high precision, and since the entire successful picture of electroweak interactions would fail to hold (or at least become far more complicated and unnatural) without it. The Higgs is like the last remaining piece in a jigsaw puzzle.

6 Uniﬁcation: many from one

The nature of fundamental physical theories depends on the energy scale at which we make our observations. This energy-scale dependence is realised concretely through the “running” of coupling constants which govern the strengths of interactions, namely the fact that they are energy-dependent. A nice example arises in the strong interactions. We saw that quarks and gluons are confined inside hadrons such as the proton and neutron. However, paradoxically, measurements with high-energy probes that penetrate deep inside these hadrons reveal that the quarks are relatively free of each other, like balls rattling around inside a bag. A better analogy is that of balls connected by rubber bands. As we attempt to separate the balls, the bands stretch and cause confinement. However if we let the balls sit close to each other, the bands remain slack and there is effectively no interaction at all. With quarks in a hadron, this kind of behaviour arises from the fact that the strong-interaction coupling strength gs is large at low energies (quantum-mechanically equivalent to long distances) and small at high energies (equivalent to short distances). With the electroweak interactions the situation is different. Above the weak-interaction scale of about 200 GeV, the spontaneously broken gauge symmetry gets “restored”. Thereafter, the two factors of the SU(2) × U(1)) gauge symmetry group evolve with energy at different rates. As we discussed above, the SU(3) colour gauge coupling strength is also evolving with energy. It would be a miracle if all three coupling strengths became equal at some common high value of energy, suggesting that they might originate from a common source. Indeed, this miracle is – almost – realised. What actually happens is that the three evolving coupling constants meet at an enormously high energy around 1016 GeV, far higher than any energy that can be liberated in a terrestrial accelerator. However they fail to meet precisely

8 at a single point – as indicated in Figure 4, each pair of couplings meets at a different point in the same rough vicinity. Thus, unification of the three gauge interactions (electromagnetic, weak, strong) cannot take place within the Standard Model. Some new phenomenon, not present in this model, has to favourably alter the running of couplings. The curves representing the evolution of the coupling constants with energy can change their slope if extra particles exist with masses intermediate between the presently explored energies, around 100 GeV, and the potential unification scale of 1016 GeV. Indeed, for quite independent reasons to which we will return shortly, it has been proposed that a whole set of “supersymmetric partners” of the usual particles exist at a scale just above 1 TeV. Many different models have been made that incorporate these super-particles, but the simplest or “minimal” version exhibits a remarkable property: in this theory, instead of crossing at different points the three gauge interactions now appear to meet at a single point! This is illustrated in Figure 4. It is too early to be sure, but this fact could be evidence that at the meeting point of around 1016 GeV there is a single (supersymmetric) unified gauge theory with a single gauge group. Experiments performed at this scale would then see just one force rather than three5, while at lower energies the gauge symmetry would “spontaneously break” into SU(3)×SU(2)×U(1).

2 Figure 4: Evolution of inverse coupling constants 1/αi ∼ 1/gi in the Standard Model (left) and Minimal Supersymmetric Standard Model (right). The subscripts 1,2,3 refer to the gauge groups U(1),SU(2),SU(3). Taken from D.I. Kazakov, hep-ph/0012288.

5Unfortunately such incredibly high-energy experiments will not be possible to directly undertake in the foreseeable future, which is why we must look for indirect signatures of this phenomenon at lower energies. Experimental hints of uniﬁcation may, however, emerge from cosmological observations.

9 If this picture is consistent then we expect there to be a field theory valid at 1016 GeV and based on a single gauge group. The first such theories, called Grand Unified Theories or GUTs, were based on the group SU(5), which “contains” the Standard Model gauge group SU(3) × SU(2) × U(1) within it. Another unified gauge group is the orthogonal group SO(10). In such a unified theory there are new “gauge bosons” called X-bosons, beyond the usual W and Z particles, photons and gluons. The Standard Model would emerge from the unified theory by spontaneous breaking of the unified group into SU(3) × SU(2) × U(1) and at low energies the X particles would not be observed since their masses would be of the order of the unification scale and they would require unthinkable energies to produce. In GUTs, quarks and leptons occur together in “multiplets” of the gauge symmetry group, and are related to each other by gauge transformations. It follows that the interactions in the theory will allow quarks and leptons to transmute into each other, something that is impossible in the Standard Model. Since quarks are confined, this transmutation would physically be observed as the decay of hadrons such as protons into leptons and other decay products, due to the decay of one or more quarks within the hadron. The typical time-scale of such a process is related to the mass of the X-boson. Proton-decay experiments have so far found no evidence of decay, which translates into a lifetime of at least 1035 years. Some unified theories are not consistent with this bound and are therefore thought to be ruled out by experiment, although other versions (particularly supersymmetric GUTs) are still consistent with the bound. At this point the alert reader might be wondering what becomes of gravity, the fourth force. Surely it too evolves with energy and can meet the other forces at some scale? Unfortunately we cannot answer this question with any precision, because despite efforts for decades, we have not been able to find a satisfactory field theory of quantum gravity6. This is due to a mathematical difficulty rooted in the fact (mentioned earlier) that the graviton, has a spin of 2 units, unlike all the other force carriers: photons, W and Z bosons and gluons, each of which has a spin of 1 unit. Today there seems little hope to unify the fourth force, gravity, with the other three forces if we remain within the framework of quantum field theory. Lacking a quantum field theory of gravity, in particular we cannot ask about the detailed energy-scale dependence of coupling constants. Nevertheless we have good reason to believe that the gravitational interaction will become “strong” (coupling strength of order 1) at a scale around 1019 GeV, known as the Planck scale. This is close, though not very close, to the unification scale for the other three forces in a minimal supersymmetric theory which as

6See Rajesh Gopakumar’s article in this volume.

10 we saw is 1016 GeV.

7 The hierarchy problem and supersymmetry

Before moving on to describe how unification of all forces including gravity might be achieved, we need to address the question of why supersymmetry was proposed in the first place. This is related to the proposal that the fundamental forces unify at high energies, and has to do specifically with the Higgs particle, the only spinless particle in the Standard Model (all 1 other elementary particles, besides the graviton of course, have spins of either 2 or 1 in units ofh ¯). In quantum field theory, spinless particles have the property that their mass is naturally set to be of the order of the highest mass scale in the theory. Now in order to be consistent with experimental observations, the Higgs should have a mass roughly in the range 100 − 1000 TeV. However the highest mass scale in a unified theory of all interactions will be in the region where the coupling strengths merge, namely 1016 GeV. This is an enormous disparity, called the “hierarchy problem”, and threatens the possible emergence of the Standard Model of particle physics from a high-scale unified theory. Supersymmetry was initially studied as a purely theoretical symmetry that can exist when a theory has equal numbers of bosons and fermions occurring in pairs, along with precise relations among their interactions. It has several interesting and beautiful properties, one of which was eventually used to address the hierarchy problem as follows. Due to what are called “non-renormalisation theorems” arising from cancelling contributions between bosons and fermions, the quantum-theoretic mechanism which causes spinless particles to have a mass of the order of the highest mass scale actually fails in the presence of supersymmetry. Hence a supersymmetric theory can consistently have a high unification scale as well as a low (around 1 TeV or less) Higgs mass. Notice that this motivation for supersymmetry is predicated on the assumption that the coupling constants will evolve in such a way as to merge at a roughly common value. As we saw in the previous section, it is an unexpected bonus that after including the effects of supersymmetry the merging of couplings is itself affected and actually improved, as seen in Figure 4, so that the coupling constants really do meet at a common point within small errors. The proposed solution of the hierarchy problem based on supersymmetry is very appealing, but also comes with its own problems. A supersymmetric version of the Standard Model

11 has a host of new particles in it and supersymmetry requires each one to have the same mass as its conventional partner. If this were true we would surely have detected many super-particles already. Conversely, non-detection of super-particles seems to prove that the idea of supersymmetry is wrong. This objection can be overcome by assuming that supersymmetry is present in an approxi- mate or “broken” form. In this case the masses of the superpartners can be split from those of the conventional particles by as much as 1 TeV, and there is no contradiction with experiment. Indeed one can then hope to find the superpartners at the Large Hadron Collider, and this is one of many goals of this machine! At the same time, the proposed amount of supersymmetry breaking is small compared to the unification scale of 1016 GeV or so, therefore even in its broken form supersymmetry would still solve the hierarchy problem. Today supersymmetric is a purely speculative idea. But it is a truly exciting fact that speculations about supersymmetry, as well as the unresolved question of the Higgs mass, are poised to be resolved within the next 2-5 years at the time of writing this article! Not many fields of research can expect to be revolutionised in such a short time. This is thanks to the Large Hadron Collider having commenced operations, and subject to its continued successful performance.

8 Gauge symmetries and uniﬁcation from string theory

We now address the question of unifying gravity with the other forces. While progress on this question using conventional (quantum field theoretic) formulations has proved almost impossible, striking progress was made using a somewhat different formulation of physics. Around 1970, in the process of trying to understand the “rubber-band” like forces that confine quarks, theoretical physicists decided to study fundamental objects that were not pointlike but extended – stringlike. Such objects provide a nice physical picture for confinement, as we have briefly seen in a previous section. With this initial motivation, interest began to grow in the study of fundamental strings for their own sake, and their theoretical formulation was intensively investigated. A string can have many excitation modes with increasing energies. These are interpreted as distinct elementary particles located at the centre of mass of the string, and having a mass equal to the excitation energy of the string. It was discovered that when closed strings are quantised, their lightest state is a massless particle of spin 2. This is quite striking, for as we

12 _ q q flux tube

_ qstring q

Figure 5: Colour ﬂux interaction replaced by a fundamental string. Taken from Sunil Mukhi and Probir Roy, arXiv:0905.1793.

1 have seen, most particles in nature have spin 2 or 1, the only exception being the mediator of gravity, the graviton which has spin 2. And the graviton is the only force carrier besides the photon to be exactly massless. So it seemed that the spin-2 particle state of the closed string might be interpreted as a graviton. The analogous calculation performed on strings that were open (i.e. having endpoints) led to a massless particle of spin 1. Moreover with open strings it proved possible to obtain “multiplets” of spin 1 particles much like the 8-plet of gluons that arise in the strong interactions. So far, these discoveries were merely tantalising hints. But when string theorists computed the scattering probabilities of these string states, using the systematic and highly constrained rules of string theory, they received a pleasant surprise. The theory of open strings gives rise to the precise interactions among spin-1 particles that are predicted by gauge invariance in conventional ﬁeld theory. Moreover, the theory of closed strings gives rise to the precise interactions among spin-2 particles predicted by general coordinate invariance.

Figure 6: Open and closed strings.

That means that in theories of strings, gauge symmetry (including general coordinate in-

13 variance) is a consequence and not an assumption. Given the very successful role these symmetries play in defining today’s fundamental theories, this makes a compelling case for the relevance of string theory to nature. Along with gauge symmetry, string theory has several other appealing features, one of which is closely related to unification. In string theory there is just one species of open string and one species of closed string. Moreover open strings and closed strings can transmute into each other: a closed string can split open at a point, or an open string can close up when its end-points join, so even open and closed strings are just variations on the same basic object. It follows that string theory is intrinsically a unified theory: all particles and all forces arise from the same object. How does this tie in with our previous discussion of merging coupling constants? String theory has an intrinsic scale, related to the intrinsic size of the string. This is embodied in a parameter with dimensions of energy known as the “string scale”. Because of the relation to gravity, this scale is likely to be close to the Planck scale or 1019 GeV. At much lower energies, string theory behaves effectively as a conventional quantum field theory whose coupling constants will evolve with a change in energy scale. Stringy excitations then simply appear as different particles. So starting from low energies, the evolution of couplings makes sense but by the time they merge into one the stringy nature of the theory will become manifest. Then we expect that all the low-energy forces and interactions will merge into the basic “stringy” interactions wherein strings simply split and join. This promises to provide a richer and deeper variant of unification than anything before it. In order to carry out this enterprise we need to understand in great detail how strings can provide a description of the world we live in. Despite much progress, this is still largely an open problem. Strings can propagate in spacetimes that are very different from the one we live in, even having different dimensionalities. It has turned out extremely difficult to find an example where our own world is very accurately reproduced. But if efforts under way do succeed then we will quite possibly end up with a completely unified theory of all particles and forces in nature.