The Frontier of Physics

Introduction

Modern physics is based on two great intellectual achievements of the 20th century: quantum mechanics and general relativity.

Quantum mechanics is the most successful theory ever invented. It describes everything from the interactions between elementary particles, like electrons and quarks, to the dynamics of the DNA molecule. In principle, we expect it to describe all of chemistry and  perhaps even all of biology.

General relativity is an extremely successful theory of gravity. It can, for example, describe the motion of binary neutron stars in exquisite detail, including the gravitational radiation the binary system emits. Few doubt that gravitational radiation will be discovered and, when it is, will be described precisely by general relativity. 

General relativity describes the behavior of matter on a large scale while quantum mechanics describes matter on small scales. Therefore, most of the time we can use each theory, on its own, in its own domain. However, there are circumstances in which both theories must be used simultaneously. An example is the singularity within black holes, or at the big bang. 

Unfortunately, both theories when used together produce nonsense.  

The two theories seem to be mathematically incompatible. Quantum theory predicts that on very small scales all physical quantities fluctuate violently including, presumably, space and time! However, general relativity assumes that space and time form a smooth manifold. 

A goal of 21st century physics is to create a new consistent theory that will provide a unified description of space, time, matter and energy on all length scales. Work towards this goal is under way.

The Standard Model of Particle Physics

The standard model of particle physics is a theory, based on quantum mechanics, that describes our world in terms of a set of elementary, point-like, particles that can be thought of as interacting with each other by exchanging other messenger particles, as shown in the figure. The particles fall into two basic classes: bosons and fermions. 

Bosons are particles that can occupy a quantum state that is already occupied. Indeed, any number of bosons can occupy the same quantum state. 

Fermions, however, are not quite as chummy: at most one fermion can occupy a given quantum state. This principle is called the Pauli Exclusion Principle. One crucial consequence of the exclusion principle is that fermions cannot occupy the same space; therefore, they tend to keep apart (as if there were a repulsive force between them). Matter is made of fermionic particles whereas the messenger particles, that generate the force between particles, are bosonic. 

It is fortunate that Nature chose matter to be fermionic. The fermionic nature of matter allows for the development of structure in the world. It makes possible different kinds of nuclei and atoms, and hence galaxies, stars, planets and people. If matter, in bulk, were bosonic all the bosons in the universe would eventually condense into the lowest energy state. Matter would become a large structure-less blob! The tendency of bosons to condense into the lowest energy state is called Bose-Einstein condensation.  

Here is a pictorial representation of our current understanding of the spectrum of the elementary particles.


 

The Arabic numbers in the figure are the charges of the particles in units of the electric charge of the proton. The Roman numbers label the three generations of particles.

The top quark, which my colleagues and I discovered in 1995, is of the third generation. Ordinary matter is made up of first generation particles: the up-quark, down-quark, the electron and its associated neutrino. Two up-quarks and one down-quark make up a proton, while two down-quarks and one up-quark make up the neutron. 

Notice that each quark comes in three ``colors''. Needless to say, color here has nothing whatever to do with what we mean, ordinarily, by color!  What we really mean is that each quark has a property, a kind of charge, that we call color charge, that can take three values. Color is just another physical property like mass, charge and spin. Although, one can think of fermions  and bosons as if they were rapidly spinning objects, in truth, the word spin is misleading: spin is just another strange property of particles, the basic unit of which is 1/2.

Quarks, electrons and neutrinos are examples of fermions, that is, objects with spin that is an odd multiple of 1/2 unit. Photons and gluons are examples of bosons, that is, objects whose spin is zero or an even multiple of 1/2 unit. For example, an electron has spin 1/2, whereas a photon has spin 1. 

In the standard model no particular relationship exists between fermions and bosons.

Renormalization - Since the 1920s a rather curious, and curiously successful, procedure has been used to arrive at new quantum-based theories: start with a classical theory, for example Maxwell's theory of electromagnetism, and quantize it; that is, apply certain well-defined, albeit mysterious, rules to convert the classical theory to a quantum theory. This is a curious procedure because it seems to be the wrong way round. After all, quantum theory is supposed to be more fundamental than classical theory; the latter should, in principle, be derived from quantum theory. Nonetheless, this procedure has proven highly successful, culminating in the Standard Model of particle physics.

Many physicists have tried to quantize general relativity, in the hope of producing a quantum theory of gravity, by applying the procedure that has worked so well for other classical theories. The procedure, alas, failed horribly: the answers obtained were nonsensical; they gave probabilities that were greater than one, and infinite values for physical quantities. 

Getting infinities is a clear sign that a theory makes no sense. No one has succeeded in quantizing gravity, by applying the usual rules. It seems that quantum theory and general relativity are truly incompatible. So we have a curious situation: we have two amazingly accurate theories that work remarkably well in their own domains but that make no sense when applied together. 

Most physicists have come to accept that a quantum theory of gravity must involve wholly new principles of physics, and that most likely this new theory will radically alter our understanding of both quantum theory and gravity. Superstring theory, or string theory for short, is the best attempt to date of arriving at this new theory. But before we present a sketch of string theory let's try to understand where the infinities come from.

The Inverse Square Law

Newton's law of gravity,
(1) F = GmM/r2
and consequently Einstein's---since the former is an approximation to the latter, contains a fundamental problem. It predicts that the force is infinite at r = 0. The force between two electric charges has the same inverse square form. So one might expect that whatever problems gravity may have the same problem would arise for electromagnetism. Not so! For electromagnetism the singularity at r = 0 is tempered by the uncertainty introduced by quantum theory. Roughly speaking, the point r = 0 is "smeared" out, thereby ridding the theory of the infinity at r = 0. But for gravity the quantum "fuzziness" is insufficient to banish the infinities from gravitation theory. Indeed, it makes the problem much worse.

The Fundamental Length

A clue that something quite extraordinary must happen before r = 0 is reached comes from playing with the three constants of Nature: h (Planck's constant), c (the speed of light) and G (Newton's constant of gravity). The following combination of these constants defines a length
(2) Lp = (hG/c3)1/2
called the Planck length, whose value is about 10-35 m. Corresponding to this incredibly minute length is a huge energy given by
(3) Mp = (hc/G)1/2 ~ 1019 GeV.
Mp is called the Planck mass. This energy is huge compared with the energy scale of the proton, which is about 1 GeV---one billion electron-volts, and even huge compared with the almost 2 trillion electron-volts (2 TeV) of the Fermilab particle accelerator, near Chicago. The Planck mass corresponds to a temperature of about 1028 K, that is, about ten thousand trillion trillion degrees above absolute zero!

Because this length (or mass) is defined in terms of the fundamental constants of Nature we expect it must have some deep significance. Indeed it has. It is the length scale at which the strength of gravity is equal to that of the other forces (the nuclear, weak and electromagnetic). It is also the scale at which we are forced to reconcile, somehow, the quantum nature of matter with the space-time nature of gravity. Note that the fundamental length is 20 orders of magnitude smaller than the size of a nucleus; that is, about one hundred million trillion times smaller! We are a long way from being able to resolve such distances. The shortest distances that have been resolved to date at Fermilab is about 10-18 m, about one thousandth the diameter of a nucleus.

String Theory: A Bit of History

In the 1960s there were many attempts to describe the strong nuclear forces. These are the forces, conveyed by the gluons, that bind quarks together in the proton or protons and neutrons together in nuclei. String theory was one such attempt, in which the nuclear particles (hadrons) were imagined to be bits of vibrating string. This simple idea didn't quite work, however. Besides, a more successful theory of the hadrons and nuclear forces was created called Quantum Chromodynamics (QCD)---a theory describing the interaction between quarks and gluons.  So most physicists abandoned string theory in favor of QCD.

In 1974, however, Joel Scherk and John H. Schwarz proposed that string theory is better thought of as a theory describing, not nuclear particles, but fundamental strings on the scale of the Planck length. This change in perspective was revolutionary because it led to a theory that has the promise, as yet only partly realized, of being able to unify all the known forces of Nature, of which there are four:

Gravity - long-ranged; acts between all forms of matter and    energy (strength ~ 10-40)

Weak - short-ranged; acts between leptons (electrons, neutrinos etc.) and quarks; responsible for radioactivity. (strength ~ 10-5)

Electromagnetic - long-ranged; acts between electrically charged particles (strength ~ 10-2)

Strong - short-ranged; acts between quarks; binds nucleus together (strength ~ 1)

What is a string?

A string is a one-dimensional object that travels through space-time. Strings can be open (that is, have free ends) or closed (that is, form loops). As a string travels through space-time it sweeps out a 2-dimensional surface, a membrane, called a world-sheet.

Strings can merge, split, and vibrate. Indeed, there is strong evidence that the vibrations of a string correspond to the observed particles of Nature! The patterns of vibration can explain not only the types of particles, but also their properties. This is quite breathtaking because from a single entity, a string, it may be that all the (perhaps infinite number of) particles (electrons, quarks, photons, gluons etc.) are merely different harmonics of the string. Pythagoras would have been very pleased with this development: he believed that number and music were the basis of all creation!

Thus, strings promise to unify all particles and therefore all forces. There is a particular vibration mode of the string, the quantum of which is called the graviton. When the mathematics of this harmonic mode is worked out it is found that the equations governing the large-scale behavior of a collective of gravitons are exactly those of general relativity! Here, finally, is a theory that requires the existence of gravity as described by Einstein.

When point-like particles are replaced by strings we are able to replace the singular interaction between the point-like particles by the smooth, seamless, interaction between strings:

The smoothness of string merging and splitting (that is, interaction) results in a theory that is finite: that is, it contains no infinities. The principal reason is that there is nothing particularly different about points in region A and B and points on other parts of the world-sheet. The local geometry around every point of the world-sheet is identical, namely, it is 2-dimensional. In a sense the force at r = 0 has been smoothed out.

Supersymmetry

Not only does string theory require the existence of gravity, it also requires the existence of something called gauge symmetry, which describes how different particles (that is, different string harmonics) are related to each other. We know that gravity and gauge symmetry exist in Nature. Gauge symmetry is the foundation of the Standard Model of particle physics. However, there is one more thing that must exist, according to string theory: supersymmetry.

Supersymmetry is a symmetry that relates bosons and fermions. One consequence of this symmetry is that every boson must have a fermionic partner and every fermion a bosonic partner. So the electron, a fermion, must have a bosonic partner, called the supersymmetric electron or selectron, for short. Likewise, quarks are associated with bosons called squarks. Photons, which are bosons, have fermionic partners called photinos; gluons are linked with gluinos and so on.

Another, deeper, way to think about supersymmetry is to suppose that in addition to the usual dimensions of space-time (t,x,y,z) there are four, rather peculiar, extra dimensions q1 to q4. Whereas t, x, y and z are ordinary numbers that commute, that is, that obey the rule xy = yx, the dimensions q1 to q4 do not! In fact, they obey a different algebra, for example: q1q2 = -q2q1. Mathematical entities that behave this way are said to anticommute. Space-time augmented with these non-commuting dimensions is called super-space.

It is very hard to visualize what this means geometrically because we have no direct experience of anti-commutative geometry. At any rate, I don't! However, these extra dimensions manifest themselves by the presence of the boson/fermion symmetry. But if there are super-particles out there, where are they?

According to supersymmetry the selectron's mass should be identical to that of the electron. But no one has found a selectron. Maybe this means that the world is not supersymmetric, and therefore string theory is a fantasy. The other possibility is simply that the masses of the super-partners are larger than those of the regular particles, and our accelerators are not powerful enough to create the super-particles. If this is true it implies that the perfect symmetry between bosons and fermions, called supersymmetry, does not manifest itself at our very low energies (or large distance scales).  If Nature is indeed supersymmetric, that aspect of Nature may only become apparent at fantastically high energies, when we expect all the different forces to lose their identity and be subsumed into a unified force.

These energies are totally beyond the reach of any technology we can currently imagine. It is therefore unlikely that we shall ever see supersymmetry in its full glory. But, remarkably, there are reasons to believe that a vestige of the supersymmetry, namely, the super-particles, may be within reach of the next round of experiments at Fermilab and at CERN---the European high energy laboratory, in Geneva, Switzerland. That is, even though supersymmetry itself may be beyond our reach the super-particles themselves may not be.

This is very exciting because we have the opportunity to discover something about Nature that is truly remarkable. Were we to discover selectrons, for example, or gluinos, we would have discovered that the world is fundamentally supersymmetric, at least at some sufficiently high energy. This would be one of the most profound discoveries of science, for it would be the first evidence that space-time has more structure than we perceive directly.

Superstring Theory

When superstring theory was born, people were very excited because it was believed that this theory was unique, in the sense that it allowed no room for any adjustment. In fact, string theory has only one fundamental constant: the string tension, called the Planck tension. In principle, this one number is sufficient to explain all the other constants of Nature! 

The string tension is enormous: it is about one thousand billion billion billion billion (1039) tons! Strings are incredibly rigid compared with any conceivable material. An immediate implication of this rigidity is that the vibrational energies of strings must likewise be enormous; in fact, on the order of the Planck mass! (The vibrational energy of any string increases with the string tension. For example, piano strings are under far greater tension than violin strings and therefore can vibrate with a lot more energy.) 

But wait a minute, if the natural string energies are so enormous how can string vibrations possibly explain things as light as quarks and electrons?  The clue to resolving this potentially devastating difficulty lies in the observation that strings must be subject to violent quantum fluctuations.  In the 1970s it was discovered that the quantum fluctuations of the string have enormous negative energy, which almost exactly cancels the huge positive energy of the string vibrations!  When the cancellation is exact, we get the zero-mass graviton. For less than exact cancellation we get quarks and other particles. 

Uniqueness - In the 1980s string theory appeared, alas, not to be unique. Scientists discovered 5 superstring theories, that is, theories of fundamental strings that describe both fermions and bosons in a symmetric way; hence the ``super". These theories are known as

Ok, so you're non the wiser!  Happily, though, the basic ideas are easy to understand.

A remarkable aspect of the superstring theories is that they only make sense in 11 space-time dimensions: 10 space dimensions and 1 time. So, those of you who were looking for an extra time dimension are out of luck! But, those of us looking for interesting things to do with extra spatial dimensions have 7 extra dimensions to play with. 

On the scale we can directly perceive space-time is of dimensionality 4: 3 space plus 1 time dimension. Where are the remaining dimensions? If they exist, perhaps they are too small to be seen. It could be that the extra dimensions are compactified to a size of the same order as that of a string. However, this has not been demonstrated mathematically.

The compactified space has the remarkable consequence that its geometry and topology (see below) could determine the nature of the particles we see in the 4-dimensional world we directly perceive. I find it astonishing that something so utterly removed from direct experience, a 7-dimensional space, can inform the world of that experience. It is such wonderful connections that have made superstring theory, or whatever it might evolve into, a cause of so much recent excitement in physics.

Unfortunately, although superstring theory is reasonably unique (there are, after all, only 5 theories), which in itself is remarkable, there are thousands upon thousands of ways to mold the geometry of  the extra 7 dimensions. At present no one knows how to decide which (if any) geometry is correct. Also, no one knows what triggers the collapse of 11-dimensional spacetime to one that is effectively 4-dimensional. Moreover, because of the breakdown of the notion of space-time when one goes to the scale of strings it is conceivable that the topology of the extra dimensions may change.

The End of Spacetime

We have used the word topology a couple of times, already. What is topology? This is the mathematics of shape without geometry. Topology deals with the connectedness of shapes. An example, will make this clear. A cup and a doughnut have the same connectedness, that is, topology. This means that, in principle, one could deform, without tearing, a cup into the shape of a doughnut, and vice versa. Both shapes are characterized by having a single hole. On the other hand a globe of the earth is topologically distinct from a frisbee: there is no way to deform the globe into a frisbee, or vice versa. But, make a single hole in the globe and we shall have transformed its topology to match that of the frisbee, because now we can widen the hole and flatten out the globe into a frisbee, without tearing. The edge of the hole becomes the outer edge of the frisbee.

Until the advent of superstring theory physicists had assumed that the topology of space is a fixed thing. If the global shape of the universe were that of a torus (a doughnut, if you will) then its topology (characterized by the presence of the hole) would not change. That is, the topology would not evolve into that, say, of a sphere. General relativity suggests this to be true on large scales. (By the way, if the global topology of space were that of a 3-dimensional doughnut what kind of effects would you see? What would travel be like in such a space? Note: a regular doughnut is a 2-dimensional surface; not 3-d!)

On the scale of strings, however, it was shown in 1993 by Witten, Greene, Aspinwall and Morrison that, according to string theory, the topology of space can change: space can be torn apart and reconnected!  The tearings of space are called topology-changing transitions. 

However, we note that the shape of the compactified space determines the properties of the forces and particles we observe. Therefore,  if at the present epoch, far from the big bang, space is undergoing a tear it must be happening very slowly, otherwise we should observe changes in the various "constants" of Nature. But presumably near the big bang, or at the center of a black hole, space must suffer rapid topology changes, perhaps of a very drastic nature, and consequently rapid changes in the forces and particles (that is, string harmonics) that exist. 

One possible consequence of these rapid topology changes could be that at the center of a black hole the forces and particles with which we are familiar (electrons, quarks, photons, gravitons etc.) may cease to exist and be replaced with strange new forces and particles. It could be that gravity itself is destroyed at the center of a black hole and that the singularity predicted by general relativity does not exist!

One intriguing question one might ask is: Can tearing occur in the large spatial dimensions that we perceive? That is, can ordinary 3-d space rip apart and re-connect? No one knows. 

These fluctuations of space-time have profound implications. One is that space-time must be an approximate concept, appropriate only on scales that are large compared to the fluctuations. It's rather as if when we draw a space-time diagram the time and space axes look smooth when viewed from afar; but when we zoom in on the axes we see that, in reality, the axes resemble a fantastically corrugated coastline. Once we have a complete theory of quantum gravity (perhaps superstring theory is its precursor) we shall be able to understand what happens to matter at the center of a black hole, and more far-reaching, what happened at the big bang when the universe was no larger than the size of a string.

A Theory of Everything?

So, where are we today? Is superstring theory this holy grail of physics, a so-called Theory Of Everything (TOE)? Is such a thing really possible? We do not know. But, what is known (and which became clear only in the last few years) is that the 5 superstring theories are really only different approximations of a single underlying theory. No one knows what this underlying theory is, nor its basic principles. We are like the proverbial blind men exploring different parts of an elephant, struggling to discern what it is we are exploring. It's much worse than this, of course---at least the sighted have seen an elephant, but those on the frontier are groping at something that may be far beyond anything we've seen before in physics. Even if some brilliant mind finds this theory, will mankind be blessed with minds brilliant enough to solve it and deduce from it all that we wish to understand?

This unknown theory is called M-Theory. (M for Mother, Magic, Mystery, Matrix or whatever you like!) The search for M-Theory is at the sharpest and most difficult cutting edge of contemporary physics. 

As Professor Edward Witten, the foremost proponent of string theory, puts it:

String theory is twenty-first century physics that fell accidentally into the twentieth century.

Last updated, Wednesday 14 April, 2002, Harrison B. Prosper