SPACE AND TIME
Infinite Abyss
By AlphaZealot
Reevaluation. Yes, you didn't think it was possible, but it is. I keep going back and forth on my thoughts in this theory, but I have come to a conclusion that will last another week until someone disproves it with basic logic that I am too lazy to think of right now. Anyway, back to the point. Have you ever wondered why space is black? Of course not, it's such a basic question that thinking about it would be absurd. However, I'm asking you to think about it for a just a moment. If you were to turn out all of the lights in your room, then there would be blackness. It's very simple, and basic third grade science tells us that the Black is the complete absence of light. However it does not really specify where this blackness exists. Shadows fall onto objects. As does blackness. It has a location. In other words, if you stare at the wall that is black because there is no light, the location of the blackness is on the wall, not in the space between the wall and yourself. If you put your hand up (in front of your face), the blackness falls onto your hand, not the space in between you and your hand. Still following? Lets apply the same logic (as screwed up as it is) to space. Now, if we are to believe that there is nothing in space, thus the term space would make sense, and then it should be no color. What color is no color? I have no idea, because the is nothing that exists to create no color, because the very fact that something exists deems that it has a color. Its like gods voice, you can’t imagine it, its beyond out brain capacity. Anyway, we know that there is nothing located between our planet and a star, because we can see a star. So the ether, or space between our planet and a given star, is not black, but just non-existent. The fact remains that in order for something to have color, or be black; there must be something for this color to fall on. There must be some curtain beyond our edge of the universe that is has no light, and causes the color of black. Maybe we are a new universe, and an older previous universe died out and condensed into and spherical black shell beyond the edge of our current universe. Or maybe I'm just crazy...
Flying Time
By Vertigo
As the saying goes, "Time flies when you're having fun!" This can be eerily true sometimes, but can it really be that time seems to flow faster when we are energetic and enjoying ourselves? To answer this we must first ask ourselves what time is. Scientifically defined time is the motion of atoms. A second is the amount of time that atoms and other particles of matter move from point A to point B. What may be a second for some matter may be entirely different for others. Back to the point, when a person is energetic, actively thinking and enjoying themselves, "having fun", the matter that makes up their body and brain may move faster than normal. Likewise, when that same person is bored and depressed, thinking only negatively or anxiously, and not being comfortable and enjoying themselves, they move slower in general causing that same second to be longer and more drawn out. Thus when you observe time from the inside, you can theoretically control...
Aliens
By AlphaZealot
Many people believe that we are alone in the universe, but is there life beyond our galaxy, beyond what we can see and is just waiting to be discovered? Well, many people believe that there are such things as aliens. They spend their whole life looking for aliens when the truth is all around us. Aliens walk and talk like they're one of us, but when they are alone, are their secrets revealed. Aliens have been studying mankind for as far back as when we first made permanent settlement on the earth. In fact many Egyptian gods were originally the result of Alien sightings. But will the aliens attack and if so why? Well the government has known of aliens for a long time. They are afraid that if they admit it, it will cause public hysteria. So they cover up what is already the result of alien doings. The red Sea in the bible was actually the work of aliens. Trying early attempts to poison our water supply. But now the government knows enough to cover up such attacks, and thus alien sightings are less and less frequent. Now UFOs in the sky are actually highly advanced government planes that they don't want the common citizen to know about. So there you have it, the truth. But will we ever know the truth for sure? We may never know...
Mars
By AlphaZealot
Mars... For years now we've looked at her and wondered what lies beneath her surface: vegetation, water, and life? The truth may be that that an ancient civilization used to live there. Why is it not there now? Simple: about a thirty-five million years ago an asteroid the half the size of Alaska crashed into the planet's surface. The land quickly became uninhabitable because the atmosphere was destroyed and the water was quickly being sucked into space. What is there to keep this from happening to us? Simple again: Jupiter is usually in the right alignment all year round so that its gravity pulls in any wondering asteroids. That's one of the reasons that Jupiter has about thirty different moons. Anyway, when the asteroid crashed into the planet, the Martians dug beneath the surface to where the water would not escape. They manipulated their environment to suit their needs. About six years ago we sent the first unmanned droid to Mars. This made the Martians aware that we had advanced further than they had expected. In fear that we might discover and ruin their home world they destroyed many of the probes sent to pick stuff up on mars. They are now aware that we wish to inhabit their home world. So, they are preparing a strike before we know it. We could be facing extinction ourselves. Or maybe they'll let us kill us off and ally to claim our home world. Maybe...
Gravity
By Slash
Assuming the Big Bang is true, but also assuming that God caused it, his only physical power over the universe could be gravity, since gravity is nothing more than a defining law. We can't explain why it's there, why matter is attracted to other matter, it's just there. So once we narrow down all these other mysterious things (like how the universe came to be) as explicable gravity remains most certainly inexplicable. But, Einstein determined that space is a lot like a flat cloth. Putting a large object somewhere creates dent and sucks smaller objects in, thus there is gravity. However, if there were a universal fabric, wouldn't there be a noncircular pattern based around heavier celestial objects? I used to believe in fabric, but I no longer do. I think it lacks a definitive pattern that would have to exist with fabric. The only pattern I see is the normal little-surrounding-big pattern. Of course, there is no definite "photograph" of the universe, so we get back to our lack of evidence on both sides. If there was weight, and then wouldn't there be "indentations" with the heavy objects, therefore forming noncircular patterns for stuff like solar systems and galaxies? It's possible that that is so, how would we really know? Galaxies, atoms, and stuff like that would most likely take on a funnel shape... unless gravity is far different outside planetary gravity wells... yet it seems so unlikely... The space fabric may only be an example of what our minds can understand, although there are indentations, they may not exist in normal space, thus no patterns to be seen. This only leaves more open to question, and asks more questions about the universal force that connects us...
Rare Earth
By AlphaZealot
Earth is our home, our only suitable home. We look to other planets to see if they too could be a home for us or are already home to other life forms. Now most people think that with billions of planets in a galaxy and billions of galaxies in the universe that it is impossible for they're not to be hundreds of millions of different species. There is actually a formula to determine the number of planets that could hold intelligent life. Since there are a number of factors in the formula that must be considered I'll only mention a couple of the major factors that contribute to intelligent life. First there a planet most has a suitable solar system with a sun. Then there is a "spot" in the solar system where life is most likely to be. This is due to the size of the planet, its location from the sun, and its location in relationship to other planets. If a planet is to far from the sun then it will be too cold for life to survive. If it is to close, it will be too hot for life to survive. If the planet does not have gas giants to protect it from asteroids and other objects it will not survive for very long. Some other major factors are the minerals on the planet and resources on a planet. You need certain elements such as water and energy (Lightning). With out these elements life will never start. Then once life starts you must have a stable atmosphere for about a billion years for life to progress to complex organisms. Now that there are creatures like frogs, dinosaurs, and other organisms that live on land there needs to be genetic mutations to create intelligent life, or the ability for creature to evolve. Every person remembers how dinosaurs became extinct right? Well if that hadn't happened then, intelligent life may never have existed. But that's off the subject. In conclusion there are most likely hundreds of millions of single celled organisms, but maybe only a couple thousand intelligent life forms and when you add distance into the formula then it seems unlikely for us to discover any of them. But who knows what the future may hold?Some factors of this theory taken from the book Rare Earth
The Matter/Antimatter Universe
By Vertigo
Antimatter intrigues me. I find it extremely interesting that such a thing exists, alike in some ways to matter, but opposite in others. One day I was mentally considering the properties of antimatter, such as: weather or not anti-neutrons are the same as neutrons because they have no charge an then remembering that neutrons and protons are made up of 3 quarks each and concluding that up and down quarks must have opposite charges in antimatter and that an anti-neutron was different from a neutron because it must have two up quarks with -1/3 charge and a down quark with a +2/3 charge, making it neutral, whereas a neutron have two +1/3 up quarks and one -2/3 down. So are up quarks down quarks in antimatter, or just anti-up quarks? Anyway...I devised a new theory of the universe. Each paragraph makes things more confusing. Read with caution. Let's start with the big bang. In my model of what I call the figure eight or hourglass universe, in which there are two more or less enclosed sections, the big bang occurred at the point of intersection. The point of pure energy, upon explosion, released an equal about of matter and antimatter. Why didn't the universe annihilate itself then? Well, the matter went one way on through the hourglass, while the antimatter went another way. However, only a small amount was released, but this spread out to form the matter universe (ours) and a parallel antimatter universe. The remainder of the mass at the big bang then collapsed on itself, leaving a black hole, but the matter and antimatter universes kept expanding to their present size. So why is there antimatter in our universe? Well, some of it got stuck there, but I'll get to that later. I believe that this black hole serves as a medium to transfer small amounts of matter and antimatter into the opposite universe. So antimatter that gets too close to the "intersection of the figure eight" gets drawn in, but then deposited on the other side. Likewise, matter exists in limited quantities in the antimatter universe. On second thought, maybe we are a matter planet trapped in the anti matter universe, and at imminent risk of being destroyed by all the antimatter around us. Anyway... So what pulls apart the matter and antimatter at the big bang? My guess: more matter and antimatter, respectively. This universe diagram is more like the continuous stream of eights or hourglasses. If matter is attracted to antimatter, then when the big bang occurs, the matter and antimatter created are pushed apart by the force. The direction in which they go after going outward is to the next source of their counterpart. Matter to antimatter and vice versa. Because inertia prevents most from going backwards, they go forwards to the next line in the chain. This is accomplished before the remaining mass collapses, i.e. before the black hole. This process is not perfect however, so matter can be stuck to the "wrong side", which is another explanation for antimatter in our universe. What happens when the matter and antimatter in different eights reach each other? Well, the annihilation creates enough energy to then create another big bang, and a new universe. This cycle is continuous, because matter and antimatter universes are always next to each other. So what about the parallel universe thing I said? The concept of parallel universes is popular in science fiction and is reasonable in my theory, because symmetry makes mass move in equal ways. The matter and antimatter universes are more or less the same in form (with the exception of particle charges, etc.) There is an equal amount of matter in the antimatter universe as there is antimatter in our universe. However, since the universe is periodically reborn from the same matter and energy, all universes are similar. These alternate realities may have slightly different factors which change them. The one closest together are most similar, while far away ones are far apart. These may be the "alternate realities" for ourselves, and the effects of choices on our lives, but I wont get into that now. One more thought and then I'm done. So what happens to the universe at the end of the chain? The chain doesn't end, it's a circle. That one was easy to explain. Or was it? The number of universes is infinite, but the best model is a circle, because they are infinite, yet connected. This just leads to more questions. Could the circle expand? Is there a "figure eight" of figure eight universes? The possibilities are endless...
Does the Universe Really Exist: The Flat Line of Genesis?
By Sergio Machine and Dave George
Added April 19, 2003
The key issues rose when talking about the Universe is- which is older, matter or energy. We will try in short to point to the problem occurring with this way of thinking. If our starting point is that one of the two aforementioned must be given priority, then the new question will arise. For example, if we that energy is older than matter, the question are where did it come from and vice versa? Something had to precede something else so that it might appear. But here we face new problem. Suppose that matter preceded energy, the logical question would be where matter came from. The answer might be that matter has always existed and there is no use in going any further into questions and possible answers, for it leads us into a vicious circle without any possibility of ever being closed. Was matter the first one from which later Universe appeared in the cataclysmic explosion, the Big Bang, or was energy the first one, from which matter was gradually created. If we take the model in which the space originates from pre-matter in the Big Bang, and which results in the expanding spaces in all its luxurious appearance, the question imposed from there is> where did pre-matter come from. Has it always existed or was it created. If we start from the point that it has always existed, this would point to its existence out of time and out of space, that is Eternity. Why would something perfect, eternal and infinite transform into something far from perfection- finite and transitory. Is it possible at all that something, anything would exist without being created previously? The answer might be that the pre-matter was created, and then the Universe was created from it in the Big Explosion. So, let us take the act of creation of pre-matter itself. This will lead us to the inevitable conclusion that something existed prior to it. Was it energy and is it possible at all that energy precedes matter. Energy is created by interaction of particles, but it cannot exist per se. Transformation of energy into matter, if we take the energy to be the beginning of all things, is simply impossible. The origin of Universe from nothingness is impossible, for if there is neither matter nor energy, there is nothing from which the Universe might be created. Would the outer space really be aware of its existence were it not for the human race. The conclusions on the existence of the Universe are made by ourselves, relying on our senses. So far there is nothing unrelated to the human race that would corroborate the existence of all things. In accordance with all this the famous class is thought I think, therefore I exist could be rewritten into We think we exist.
The Perpetual Universe
By Daren Sayre
Added September 26, 2004
The universe is a perpetual motion machine, like the circle of life and the death and rebirth of stars. The universe is based around a sphere. Like the planets their orbits galaxies stars, they are all spherical. I believe the universe is as well. It in a sense feeds back into itself due to gravity. Some high intensity black holes spew matter from northern and southern poles like our own planet's weather. As the matter gets shot out from these points, the intense gravity in the center of the black hole pulls the matter back to it, creating a perpetual motion machine, never ending, spawned by the original explosion i.e. "The Big Bang". The mere size of the universe would make this effect billions and billions of years for the matter getting shot out to be pulled back in. Judging that we seem to be moving away from other galaxies I theorize that our galaxy must be nearer to the poles where the matter is moving away from itself, not the center where matter would be moving toward the huge black hole.
The Universe in a Nutshell
By Gerard Morton
Added February 20, 2004
According to my imagine model this is what I believe gravity is and how it is created thus leading to how the known universe works. Gravity is a force created when matter is being removed from a seal container. Hence I believe the universe is surrounded by a shell of matter. This is not an original idea, however partially based on a theory by a Italian scientist named Renzo Boscoli who applying the Ranque Effect (that is gas that rotates on it's own axis creates a cooler temperature around the center of the gas and hotter along the outer circumference of the gas) says that the sun is really powered by cold fusion, I believe I can explain how it works. Take matter freely detached from a shell of matter that surrounds it and is sealed. Movement however it is started is present thus energy is present and when energy comes in contact with matter not only does it change it but it causes it to break down. The very movement of this detached matter via energy causes it to break down thus began the act of creating more space in a sealed area and a suction is created! Something to try yourself to better illustrate this. Open your mouth and place your tongue against the inside of your lower teeth. Now close your mouth and slowly try to pull your tongue back, that resistance you feel is caused by the base of your tongue blocking off air from flowing into the mouth as your breath and the act of you trying to pull the tip of your tongue back, inadvertly making more space and thus creating a suction. Once the suction is created it serves to shape this detached matter and the shell around it, based on how it is moving. The suction does this because as it's name suggest is sucking or pulling at matter, both the detached matter from it's center outward toward the Shell Wall and from the depths of the Shell Wall inward toward the detached matter. For this model we will say this detached matter is moving upon it's own axis and is shaped into a spherical shape of matter (via the suction upon it's surface)or Sphere for short. The inside of the shell too is shaped according to the movement of the Sphere, in this case it's inner wall is shaped into a huge spherical shell around the Sphere (concaved). If the suction is pulling at matter on every level then what you have is the suction pulling at the Shell's walls and like wise pulling at the Sphere as well continuously. Looking at the Sphere itself, it's own rotating mass will create a suction around itself pulling any loose matter (particles or atoms or such) to it's surface and the same will occur near the Shell's inner wall as it rotates around the Sphere creating it's own suction being pulled toward the inner wall surface. Suction thus is equated in this model to Gravity and heat as we are aware moves away from the pull of Gravity. This means that the heat from the friction of the Sphere and that of the Shell wall via gravity will meet at a middle point equal distance between the two. Perhaps this is what we see as the Cosmic Background Radiation! Though the Sphere creates it's own Gravity the surface area of the Shell wall is greater and so will still have an effect on the sphere even if on a saddle level or microscopic level (atoms and particles). This means that continuously over time cavities will form within the Sphere that will via the Sphere's own gravity collapse in on itself. In doing so it's core as it collapses in on itself becomes denser and the outer layer becomes less dense or gaseous. The reason; when matter becomes denser it has a greater surface area thus a stronger gravitational pull, that stronger gravitational pull causes heat to move not only faster away but in a much larger quantity. This means that the core of this Sphere becomes cold and solid, the middle layer warm and liquid, and the outer layer hot and gaseous, you thus have a star! In this model the first star is born. Eventually so many cavities form over a large enough area within the core that the entire core collapses in on itself in one swift and powerful event; this would be equated to a supernova. Depending on the size of the Sphere at it's start will determine how big the nova will be. But if Gravity affecting the Sphere causes such an action what of the Shell wall itself. We know that when a star goes supernova what you have left is a neutron star, that is smaller then the original star. What has occurred is when the core collapses the heat released is far greater in speed and quantity then what it was previously generating and thus literally pushes it's outer liquid and gas layers off it giving rise to a nebula. The same would thus occur to the Shell wall, but there are some major differences. First the Shell wall would have a Gravitational pull, and anything close to it would of course be pulled to it. But what makes a star heat up and shine is the saddle effect of the Shell's Wall own Gravitational pull (keep in mind a suction in a seal container pulls both on the inside of the container and that which is inside the container - which ever has a greater surface area has a greater pull, in this case the Shell Wall has a greater pull). This means that the effect from the Star upon the Shell Wall would be far less and thus the Shell Wall may heat up and even give off light, but in comparison would be so minute that it would be hard to see. Especially since the CMBR would obscure any view of it! That aside, it's own supernova would have the same effect, however because of it shape and the way gravity is pulling at it would collapse in on itself or against itself! When the star goes supernova it becomes denser, hence smaller and more compact, the wall does the same as well, however. When the wall collapses against itself all at once (supernova) it constricts and does become denser but it's shape dictates it's movement during the supernova. So the entire shell would rapidly expand outward with the same explosive action as a star going supernova, only on a truly cosmic scale! Thus the cavity or area of space becomes larger because the entire Shell not just it's inner wall has expanded and become larger. But this action would take far longer to occur as oppose to a star going supernova due to it's greater surface area then the star that is effecting it. Note: In this model only one star was use to simplify the description and illustration of how it functions but can be applied to the known universe as a whole with the same effects occurring. What lies on the other side of this wall...? Since it can't be directly observed your guess is as good as the next! If my model is correct then a star never dies or it's core never dies and continues to collapse in on itself. Perhaps a black hole is not a hole at all but the core of the star so dense and compact that we just can't see it! If this is the case and this is how the universe started, though it's initial start can never be fully understood, perhaps there is another explanation of how a planet is created. Most importantly, this process is perpetual, there would be no such thing as a big crunch or such, the wall would become thinner and thinner but at the same time denser and denser, kind of like taking half of every step you take in a gymnasium you would never reach the other side, nor would you technically ever stop moving!
Sunday, August 31, 2008
WHAT IS STRING THEORY?
We live in a wonderfully complex universe, and we are curious about it by nature. Time and again we have wondered--- why are we here? Where did we and the world come from? What is the world made of? It is our privilege to live in a time when enormous progress has been made towards finding some of the answers. String theory is our most recent attempt to answer the last (and part of the second) question.
So, what is the world made of? Ordinary matter is made of atoms, which are in turn made of just three basic components: electrons whirling around a nucleus composed of neutrons and protons. The electron is a truly fundamental particle (it is one of a family of particles known as leptons), but neutrons and protons are made of smaller particles, known as quarks. Quarks are, as far as we know, truly elementary.
Our current knowledge about the subatomic composition of the universe is summarized in what is known as the Standard Model of particle physics. It describes both the fundamental building blocks out of which the world is made, and the forces through which these blocks interact. There are twelve basic building blocks. Six of these are quarks--- they go by the interesting names of up, down, charm, strange, bottom and top. (A proton, for instance, is made of two up quarks and one down quark.) The other six are leptons--- these include the electron and its two heavier siblings, the muon and the tauon, as well as three neutrinos.
There are four fundamental forces in the universe: gravity, electromagnetism, and the weak and strong nuclear forces. Each of these is produced by fundamental particles that act as carriers of the force. The most familiar of these is the photon, a particle of light, which is the mediator of electromagnetic forces. (This means that, for instance, a magnet attracts a nail because both objects exchange photons.) The graviton is the particle associated with gravity. The strong force is carried by eight particles known as gluons. Finally, the weak force is transmitted by three particles, the W+, the W- , and the Z.
The behavior of all of these particles and forces is described with impeccable precision by the Standard Model, with one notable exception: gravity. For technical reasons, the gravitational force, the most familiar in our every day lives, has proven very difficult to describe microscopically. This has been for many years one of the most important problems in theoretical physics-- to formulate a quantum theory of gravity.
In the last few decades, string theory has emerged as the most promising candidate for a microscopic theory of gravity. And it is infinitely more ambitious than that: it attempts to provide a complete, unified, and consistent description of the fundamental structure of our universe. (For this reason it is sometimes, quite arrogantly, called a 'Theory of Everything').
The essential idea behind string theory is this: all of the different 'fundamental ' particles of the Standard Model are really just different manifestations of one basic object: a string. How can that be? Well, we would ordinarily picture an electron, for instance, as a point with no internal structure. A point cannot do anything but move. But, if string theory is correct, then under an extremely powerful 'microscope' we would realize that the electron is not really a point, but a tiny loop of string. A string can do something aside from moving--- it can oscillate in different ways. If it oscillates a certain way, then from a distance, unable to tell it is really a string, we see an electron. But if it oscillates some other way, well, then we call it a photon, or a quark, or a ... you get the idea. So, if string theory is correct, the entire world is made of strings!
Perhaps the most remarkable thing about string theory is that such a simple idea works--- it is possible to derive (an extension of) the Standard Model (which has been verified experimentally with incredible precision) from a theory of strings. But it should also be said that, to date, there is no direct experimental evidence that string theory itself is the correct description of Nature. This is mostly due to the fact that string theory is still under development. We know bits and pieces of it, but we do not yet see the whole picture, and we are therefore unable to make definite predictions. In recent years many exciting developments have taken place, radically improving our understanding of what the theory is.
Reasons for Origin of String theory
Resolution of Contradictions
Major advances in understanding of the physical world have been achieved during the past century by focusing on apparent contradictions between well-established theoretical structures. In each case the reconciliation required a better theory, often involving radical new concepts and striking experimental predictions. Four major advances of this type are indicated in Figure 1. [1] These advances were the discoveries of special relativity, quantum mechanics, general relativity, and quantum field theory. Let us briefly recall the contradictions that each of these resolved and some of the experimental confirmations that followed.
Special Relativity
The Quantum Theory
General Relativity
Quantum Field Theory
The Final Contradiction
Special Relativity
Galilean invariance, which is embodied in Newton's mechanics, for example, was considered intuitively obvious and observationally successful for several centuries. Among other things it implies that velocities add; i.e., if A observes B moving with speed v1 and B observes C moving with speed v2 in the same direction, then A observes C moving with speed v12 = v1 + v2 in that direction. A contradiction arose as a consequence of the development of a very successful theory of electricity and magnetism in the nineteenth century, which is embodied in a set of differential equations known as Maxwell's equation. One of the implications of Maxwell's equations is the existence of various waves, such as radio waves, all of which are just light waves at various different frequencies. The equations imply that all of these waves travel with the speed of light (about 300,000 km/sec) regardless of the motion of the observer. This is a violation of Galilean invariance. Einstein resolved this paradox by recognizing that Galilean invariance is just an approximation, valid for speeds much smaller than the speed of light. The principles that apply for all speeds are embodied in the special theory of relativity. In this theory the rule for the addition of velocities is
,
Where c denotes the speed of light
Quantum Theory
The theory of electromagnetism embodied in Maxwell's equations also conflicted with the understanding of thermodynamics -- the behavior of systems in thermal equilibrium. In particular, a hot object (a so-called `black-body') emits electromagnetic radiation with a certain well-defined spectrum (intensity as a function of the frequency of the radiation). The problem with this is that if one adds up the energy carried off by radiation at all the different frequencies, the formulas imply that the total is infinite, which is an absurd result. Just before the turn of the century, Max Planck realized that if the energy was emitted in discrete packets (or `quanta'), rather than in a continuous distribution, the total energy would be finite. He postulated that radiation of frequency n comes in quanta of energy E = hn, where h is a fundamental constant of nature, known as Planck's constant. The individual quanta of light are called ``photons''. This was the beginning of the idea of wave-particle duality and the quantum theory. ht.
General Relativity
Einstein was not content with the special theory of relativity, because it was in contradiction with Newton's theory of gravitation. Newtonian gravitation was a very successful theory. In particular, it accounted for the motion of the planets to high precision. However, it has a peculiar property that had even bothered Newton. It implies instantaneous transmission of the gravitation force between two objects across great distances. Einstein knew this had to be wrong, because special relativity implies that no signal can be transmitted faster than the speed of light. Einstein provided the resolution around 1915 with a new theory of gravitation, which he called the general theory of relativity. It agrees with the Newtonian theory for low speeds and weak gravitational fields, but differs from it at high speeds and strong fields. The theory had several immediate observational successes. First it implied a small correction to the orbit of the planet Mercury that accounted for a small discrepancy between the orbit implied by the Newtonian theory and the observed orbit. (The effect is too small to be observed for the other planets.) Second, it predicted that light from a distant star passing near to the limb of sun would be bent by a small but measurable angle. The measurement was made by Eddington during a solar eclipse. The observations confirmed the theoretical prediction, and Einstein became an international celebrity. Other predictions of the theory -- such as the existence of black holes and of gravitational radiation -- have been confirmed much more recently. The thing that impressed other physicists most about the general theory of relativity is that it is based on very general physical principles -- the equivalence principle and general coordinate invariance -- and very beautiful mathematical concepts. The relevant mathematics is called differential geometry (specifically, Riemannian geometry). The idea is that gravity is a manifestation of the curvature of space-time. Also, the geometry of space-time is determined by the distribution of energy and momentum. The basic equation of motion is
In this equation Gmn describes the space-time geometry, G is Newton's constant characterizing the strength of gravitation, and Tmn describes the distribution of energy and momentum. Quantum Field Theory
The next contradiction that physicists faced was between quantum mechanics (which had been developed over the thirty years following Planck's seminal insight) and the special theory of relativity. Most of the work in quantum mechanics was in the Galilean (or non-relativistic) approximation. To be sure, Dirac had developed a relativistic wave equation for the electron, which was an important advance, but there was still a basic contradiction that needed to be resolved. The new feature that is required in a successful union of quantum mechanics and special relativity is the possibility of the creation and annihilation of quanta (or `particles'). The non-relativistic theory does not have this feature. The framework in which quantum mechanics and special relativity are successfully reconciled is called quantum field theory. It is based on three basic principles: two of them, of course, are quantum mechanics and special relativity. The third one, which I wish to emphasize, is the postulate that elementary particles are point-like objects of zero intrinsic size. In practice, they are smeared over a region of space due to quantum effects, but their description in the basic equations is as mathematical points. Now the general principles on which quantum field theory are based actually allow for many different consistent theories to be constructed. (The consistency has not been established with mathematical rigour, but this is not a concern for most physicists.) Among these various possible theories there is a class of theories, called `gauge theories' or `Yang-Mills theories' that turn out to be especially interesting and important. These are characterized by a symmetry structure (called a Lie group) and the assignment of various matter particles to particular symmetry patterns (called group representations). There is an infinite set of possibilities for the choice of the symmetry group, and for each group there are many possible choices of group representations for the matter particles. One of this infinite array of theories has been experimentally singled out. It is called the ``standard model''. It is based on a Lie group called SU (3) X SU (2) X U (1). The matter particles consist of three families of quarks and leptons. (I will not describe the representations that they are assigned to here.) There are also addition matter particles called ``Higgs particles'', which are required to account for the fact that part of the symmetry is spontaneously broken. The standard model contains some 20 adjustable parameters, whose values are determined experimentally. Still, there are many more things that can be measured than that, and the standard model is amazingly successful in accounting for a wide range of experiments to very high precision. Indeed, at the time this is written, there is only one clear-cut piece of experimental evidence that the standard model is not an exactly correct theory. This evidence is the fact that the standard model does not contain gravity!
The Final Contradiction
The results described above constitute quite an achievement for one century, but it leaves us with one fundamental contradiction that still needs to be resolved. General relativity and quantum field theory are incompatible. Many theorical physicists are convinced that superstring theory will provide the answer. There have been major advances in our understanding of this subject, which I consider to constitute the ``second superstring revolution,'' during the past few years. After presenting some more background, I will describe the recent developments and their implications. There are various problems that arise when one attempts to combine general relativity and quantum field theory. The field theorist would point to the breakdown of the usual procedure for eliminating infinities from calculations of physical quantities. This procedure is called "renormalization", and when it fails the theory is said to be "non-renormalizable." In such theories the short-distance behavior of interactions is so singular that it is not possible to carry out meaningful calculations. By replacing point-like particles with one-dimensional extended strings, as the fundamental objects, superstring theory overcomes the problem of non-renormalizability. An expert in general relativity might point to a different set of problems such as the issue of how to understand the causal structure of space-time when the geometry has quantum-mechanical excitations. There are also a host of problems associated to black holes such as the fundamental origin of their thermodynamic properties and their apparent incompatibility with quantum mechanics. The latter, if true, would mean that a modification in the basic structure of quantum mechanics is required. In fact, superstring theory does not modify quantum mechanics; rather, it modifies general relativity. The relativist's set of issues cannot be addressed properly in the usual approach to quantum field theory (perturbation theory), but the recent discoveries are leading to non-perturbative understandings that should help in addressing them. Most string theorists expect that the theory will provide satisfying resolutions of these problems without any revision in the basic structure of quantum mechanics. Indeed, there are indications that someday quantum mechanics will be viewed as an implication (or at least a necessary ingredient) of superstring theory. When a new theoretical edifice is proposed, it is very desirable to identify distinctive testable experimental predictions. In the case of superstring theory there have been no detailed computations of the properties of elementary particles or the structure of the universe that are convincing, though many valiant attempts have been made. In my opinion, success in such enterprises requires a better understanding of the theory than has been achieved as yet. It is very difficult to assess whether this level of understanding is just around the corner or whether it will take many decades and several more revolutions. In the absence of this kind of confirmation, we can point to three qualitative "predictions" of superstring theory. The first is the existence of gravitation, approximated at low energies by general relativity. No other quantum theory can claim to have this property (and I suspect that no other ever will). The second is the fact that superstring solutions generally include Yang--Mills gauge theories like those that make up the "standard model" of elementary particles. The third general prediction is the existence of super symmetry at low energies (the electroweak scale). Since super symmetry is the major qualitative prediction of superstring theory not already known to be true before the prediction, let us look at it a little more closely. (One could imagine that in some other civilization, the sequence of discoveries is different.)
Super symmetry
Super symmetry is a theoretically attractive possibility for several reasons. Most important from my viewpoint, is the fact that it is required by superstring theory. Beyond that is the remarkable fact that it is the unique possibility for a non-trivial extension of the known symmetries of space and time (which are described in special relativity by the Poincare group). Mathematically, it can be described in terms of extra dimensions that are rather peculiar. Whereas ordinary space and time dimensions are described by ordinary numbers, which have the property that they commute: X·Y = Y·X, the super symmetry directions are described by numbers that anti-commute: X·Y = -Y·X.
Super partners
Experimental Prospects
Super partners
As we have said, super symmetry (also known as SUSY) is the major prediction of superstring theory at experimentally accessible energies that has not yet been confirmed. If correct, it implies that every known elementary particle must have a "super partner." We are quite sure, for reasons I won't go into, that no pair of the known particles are super symmetry partners of one another. So super symmetry requires the existence of a new elementary particle for every known one. As ofter happens, the names are somewhat whimsical: the partners of quarks are called "squarks", the partners of electrons are called "selectrons", the partners of gluons (the Yang-Mills particles that carry the strong nuclear force) are called "gluinos" and so forth. It is believed that the reason that these particles have not yet been observed is because super symmetry is a broken symmetry, and as a result the super partners are heavier than the known elementary particles. Experiments carried out so far have not had particle beams of sufficient energy and intensity to produce them in observable numbers. Unfortunately, current theoretical ideas are insufficient to accurately predict the super partner masses, though the way in which these particles interact with one another and with the known particles is predicted precisely. Even though accurate predictions of the super partner masses do not exist, there are three distinct arguments that make qualititative predictions of the masses. All three of them lead to the conclusion that a typical super partner mass should be in the range of 100 Gee to 1000 Gee. In other words, they should be about 100--1000 times heavier than a proton. The three arguments are the following: First, super symmetry leads to a softening of the short distance singularities of quantum field theory. If we require a sufficient softening so that the Higgs mechanism can break the electroweak symmetry (SU(2) X U(1)) at the observed 300 GeV scale, in which case the Higgs particles have masses of the same order of magnitude, then the scale of super symmetry breaking must also be approximately the same. The second argument concerns the unification of the electroweak and strong nuclear forces at very high energy (around $10^ {16} $ GeV). One can argue that such a unification is inconsistent with the current experimental data, if one includes the effects of only known particles in the extrapolation, but that it works if super symmetry partner particles with masses in the 100 GeV to 1000 GeV range are included. The third argument concerns the possibility that the lightest SUSY particle could be a form of dark matter accounting for a substantial fraction of the mass of the universe. This also requires the same range of masses!
Experimental Prospects
All of the above makes a very suggestive case for SUSY. It is also very exciting because the mass range that is predicted is just what the new generation of particle accelerators is beginning to explore. Regrettably, the American superconducting supercollider project (SSC) was cancelled, but a European accelerator called the large hadron collider (LHC) will begin operating at a lab in Geneva, Switzerland (called CERN) around 2005. Its energy will be about 8000 GeV per beam, whereas the SSC energy would have been 20,000 GeV per beam. In my opinion, The LHC energy is high enough so that if it does not find super symmetry after a few years of operation, we can safely conclude that it does not exist in the vicinity of the electroweak scale. If the LHC (or another machine) does find SUSY, on the other hand, this would be one of the most profound achievements in the history of humankind. It would be more profound, in my opinion, than the discovery of life on Mars, for example.
A Brief History of Superstrings
The history of string theory is very fascinating, with many bizarre twists and turns. It has not yet received the attention it deserves from historians of science. Here we will settle for a very quick sketch.
String Theory of the Strong Nuclear Force
Superstring Unification
The First Superstring Revolution
String Theory of the Strong Nuclear Force
The subject of string theory arose in the late 1960's in an attempt to describe strong nuclear forces. This approach to the description of strong nuclear forces was a quite active subject for about five years, until it was abandoned because it ran into various theoretical difficulties and because a better theory came along. There were two main problems that stymied us in our attempts to use string theory to describe the strong nuclear forces. One was that the theory required the existence of a kind of particle that we didn't want -- namely a particle with no mass and two units of spin. Its existence was very generic and very frustrating. The second problem was that the theory required that space-time have ten dimensions (nine space and one time), whereas the correct answer is clearly four (three space and one time). As if all that weren't bad enough, around 1973 quantum chromodynamics (QCD) -- the SU (3) part of the standard model -- merged as a convincing theory of the strong nuclear force. One curious fact about string theory in the 1968--1973 period is that it took two years studying various complicated mathematical formulas before several people realized that these were formulas describing the interactions of extended one-dimensional objects, which were named ``strings.'' Once this was realized, it became clear that this type of theory was outside the framework of conventional quantum field theory, which as we have emphasized, is based on point-like elementary particles. Another important development during this period (in 1971) was the discovery that to incorporate a class of elementary particles called fermions (electrons and quarks are examples) string theory requires a two-dimensional version of super symmetry.[2] This led to the development of space-time super symmetry, which was eventually recognized to be a generic feature of all consistent string theories (hence the name superstrings").
Superstring Unification
In 1974 Joel Scherk and I proposed that the problems that string theory had encountered could be turned into virtues if it were used as a framework for realizing Einstein's old dream of a ``unified theory'' of fundamental forces and elementary particles, rather than as a theory of hadrons (the strongly interacting nuclear particles). [3] Specifically, we pointed out that it would provide a theory that incorporates general relativity without the characteristic short-distance infinities of quantum field theory. The massless spin two particle, which we had tried so hard to get rid of, would be identified as the graviton -- the quantum of gravitation! One implication of this change in viewpoint was that, to account for the observed strength of the gravitational force, the characteristic size of a string had to be roughly the Planck length
(the symbol h is Planck's constant.) This was a big change, since this distance is some 20 orders of magnitude smaller than the characteristic size of hadrons previously envisaged. More refined analyses lead to a string scale Lst that is about two orders of magnitude larger than the Planck length. In any case, experiments at existing accelerators cannot resolve distances shorter than about 10-16 cm, which explains why the point-particle approximation of ordinary quantum field theories is so successful. The second problem -- the extra dimensions -- could also be addressed in this setting. Once one has a theory containing gravitation and generalizing general relativity, one knows that that the geometry of space-time is dynamically determined. One could imagine that, as a consequence of the dynamics, the extra six dimensions form a small compact space attached to each point in ordinary four-dimensional space-time. If the size of the extra dimensions is sufficiently small, there would be no conflict with observations. I found these ideas very exciting and have been pursuing them ever since. However, for the ten year period 1974--1984 only a few colleagues and I pursued these ideas. One who did was Joel Scherk; tragically, he passed away in 1980.
The First Superstring Revolution
In 1984-85 there was a series of discoveries[4] that convinced many theorists that superstring theory is a very promising approach to unification. Almost overnight, the subject was transformed from an intellectual backwater to one of the most active areas of theoretical physics, which it has remained ever since. By the time the dust settled in 1985, it seemed clear that there are five different superstring theories, each requiring ten dimensions (nine space and one time), and that each of them has a consistent description in term of a power series expansion in the coupling constant (perturbation expansion). The five theories, about which I'll say more later, are denoted type I, type IIA, type IIB, E8 X E8 heterotic (HE, for short), and SO(32) heterotic (HO, for short). The type II theories have two super symmetries in the ten-dimensional sense, while the other three have just one. The type I theory is special in that it is based on unoriented open and closed strings, whereas the other four are based on oriented closed strings. The IIA theory is special in that it is non-chiral (i.e., it is parity conserving), whereas the other four are chiral (parity violating). At this point I'll end the historical discussion and turn to superstring theory itself.
Basic Ideas of Superstring Theory
A string's space-time history is described by functions Xm(s,t) which describe how the string's two-dimensional "world sheet," represented by coordinates (s,t), is mapped into space-time Xm. There are also functions defined on the two-dimensional world-sheet that describe other degrees of freedom, such as those associated with super symmetry and gauge symmetries. Surprisingly, classical string theory dynamics is described by a conformally invariant 2D quantum field theory. (Roughly, conformal invariance is symmetry under a change of length scale.) What distinguishes one-dimensional strings from higher dimensional analogs is the fact that this 2D theory is renormalizable (no bad short-distance infinities). By contrast, objects with p dimensions, called "p-branes," have a (p+1)-dimensional world volume theory. For p > 1, those theories are non-renormalizable. This is the feature that gives strings a special status, even though, as we will discuss later, higher-dimensional p-branes do occur in superstring theory.
Perturbation Theory
Compactification of Extra Dimensions
A Theory of Everything?
Perturbation Theory
As was briefly mentioned earlier, a useful way of studying theories that cannot be solved exactly is by computing power series expansions in a small parameter. For example, quantum electrodynamics has a small parameter, called the fine structure constant, which is given by
Thus, if T(a) denotes some physical quantity of interest, one computes
and the first few terms can give a very good approximation. This approach, which is called perturbation theory, is the way superstring theories were studied until recently. The problem is that in superstring theory there is no reason that the expansion parameter a should be small. More significantly, there are important qualitative phenomena that are missed in perturbation theory. The reason is that there are non-perturbative contributions to many physically interesting quantities that have the structure
Such a contribution is completely invisible in perturbation theory. Perturbative quantum string theory can be formulated by the Feynman sum-over-histories method. This amounts to associating a genus h Riemann surface, which can be visualized as a sphere with h handles attached to it, to the hth term in the string theory perturbation expansion. The genus h surface is identified as the corresponding string theory Feynman diagram. The attractive features of this approach are that there is just one diagram at each order of the perturbation expansion and that each diagram represents an elegant (though complicated) mathematical expression that is ultraviolet finite (no short-distance infinities). The main drawback of this approach is that it gives no insight into how to go beyond perturbation theory.
Compactification of Extra Dimensions
As has already been mentioned, to have a chance of being realistic, the six extra space dimensions must curl up into a tiny geometrical space, whose size should be comparable to the string length Lst.. Since space-time geometry is determined dynamically (as in general relativity), only geometries that satisfy the dynamical equations are allowed. The HE string theory, compactified on a particular kind of six-dimensional space, called a Calabi--Yau manifold, has many qualitative features at low energies that resemble the standard model. In particular, the low mass fermions (identified as quarks and leptons) occur in families, whose number is controlled by the topology of the CY manifold. These successes have been achieved in a perturbative framework, and are necessarily qualitative at best, since non-perturbative phenomena are essential to an understanding of super symmetry breaking and other important matters of detail.
A Theory of Everything?
In the euphoria following the first superstring revolution in 1985, some of the less experienced participants in the enterprise thought that we were on the verge of constructing a complete fundamental theory of the physical world. To put it mildly, I found this naive. In this setting, the phrase "Theory of Everything" was introduced and propagated by the public media. This was very unfortunate for several reasons. The TOE phrase is very misleading on several counts. First of all, the theory is not yet fully formulated, and when it is (which might still take decades) it is not entirely clear that it will be the last word in fundamental physics. Furthermore, even if the theory is a complete description of quantum dynamics, it seems unlikely that it will also provide a theory of initial conditions, which is another key ingredient required to explain why we observe the particular universe that we do. But even if a theory of initial conditions is also obtained, there will still be much about this universe that cannot be explained. Many things, such as our very existence, are a consequence of the inherent quantum indeterminacy of nature. I believe that cannot be overcome. Maybe that is just as well, because if we had old-fashioned classical determinism, the future would be fully determined, which would undermine our humanity. There is also a more mundane sort of unpredictability that is also to be expected. Many of the things that the theory predicts unambiguously in principle could require intractable calculations. Part of the art of physics is to identify those things that can be calculated. The other reason the TOE phrase upset me is that it alienated many of our physics colleagues, some of whom had serious doubts about the subject anyway. Quite understandably, it gave them the impression that people who work in this field are a very arrogant bunch. Actually, we are all very charming and delightful.
The Second Superstring Revolution
Just One Theory!
T Duality
S Duality
M Theory
D-Branes
Conclusions
Just One Theory!
The second superstring revolution (1994-??) has brought non-perturbative string physics within reach. The key discoveries were the recognition of amazing and surprising "dualities." They have taught us that what we viewed previously as five distinct superstring theories is in fact five different perturbative expansions of a single underlying theory about five different points. It is now clear that there is a unique theory, though it may allow many different quantum mechanical solutions. For example, a sixth special quantum solution implies the existence of an 11-dimensional space-time. Another lesson we have learned is that, non-perturbatively, objects of more than one dimension (membranes and higher "p-branes") play a central role. In most respects they appear just as fundamental as the strings (which can now be called one-branes), except that a perturbation expansion cannot be based on p-branes with p >1. Three kinds of dualities, called S,T, and U, have been identified. It can sometimes happen that theory A with a large strength of interaction (or `strong coupling') is equivalent to theory B at weak coupling, in which case they are said to be S dual. Similarly, if theory A compactified on a space of large volume is equivalent to theory B compactified on a space of small volume, then they are called T dual. Combining these ideas, if theory A compactified on a space of large (or small) volume is equivalent to theory B at strong (or weak) coupling, they are called U dual. If theories A and B are the same, then the duality becomes a self-duality, and it can be viewed as a kind of symmetry. T duality, unlike S or U duality, can be understood perturbatively, and therefore it was discovered between the two string revolutions.
T Duality
The basic idea of T duality(for a recent discussion see [5]) can be illustrated by considering a compact dimension consisting of a circle of radius R. In this case there are two kinds of excitations to consider. The first, which is not special to string theory, are Kaluza--Klein momentum excitations on the circle, which contribute (n/R)2 to the energy squared, where n is an integer. Winding-mode excitations, due to a closed string winding m times around the circular dimension, are special to string theory. If
denotes the string tension (energy per unit length), the contribution to the energy squared is Em=2pmRT.T duality exchanges these two kinds of excitations by exchanging m with n and
This is part of an exact map between a T-dual pair A and B. One implication is that usual geometric concepts break down at short distances, and classical geometry is replaced by "quantum geometry," which is described mathematically by 2D conformal field theory. It also suggests a generalization of the Heisenberg uncertainty principle according to which the best possible spatial resolution Dx is bounded below not only by the reciprocal of the momentum spread, Dp, but also by the string scale Lst. (Including non-perturbative effects, it may be possible to do a little better and reach the Planck scale.) Two important examples of superstring theories that are T-dual when compactified on a circle are the IIA and IIB theories and the HE and HO theories. These two dualities reduce the number of distinct theories from five to three.
S Duality
Suppose now that a pair of theories A and B is S-dual. This means that if f denotes any physical observable and l denotes the coupling constant, then
(The expansion parameter a introduced earlier corresponds to l). This duality, whose recognition was the first step in the current revolution, [6] generalizes the electric-magnetic symmetry of Maxwell theory. Since the Dirac quantization condition implies that the basic unit of magnetic charge is inversely proportional to the unit of electric charge, their interchange amounts to an inversion of the charge (which is the coupling constant). S duality relates the type I theory to the HO theory and the IIB theory to itself. This explains the strong coupling behavior of those three theories.
M Theory
The understanding of how the IIA and HE theories behave at strong coupling, which is by now well-established, came as quite a surprise. In each of these cases there is an 11th dimension that becomes large at strong coupling, the scaling law being
In the IIA case the 11th dimension is a circle, whereas in the HE case it is a line interval (so that the eleven-dimensional space-time has two ten-dimensional boundaries). The strong coupling limit of either of these theories gives an 11-dimensional space-time. The eleven-dimensional description of the underlying theory is called "M theory." As yet, it is less well understood than the five 10-dimensional string theories. The connections among theories that we've mentioned are sketched below.
(S1 denotes a circle and I denotes a line interval.) There are many additional dualities that arise when more dimensions are compactified, which will not be described here.
D-Branes
Another source of insight into non-perturbative properties of superstring theory has arisen from the study of a special class of p-branes called Dirichlet p-branes (or D-branes for short). The name derives from the boundary conditions assigned to the ends of open strings. The usual open strings of the type I theory satisfy a condition (Neumann boundary condition) that ensures that no momentum flows on or of the end of a string. However, T duality implies the existence of dual open strings with specified positions (Dirichlet boundary conditions) in the dimensions that are T-transformed. More generally, in type II theories, one can consider open strings with specified positions for the end-points in some of the dimensions, which implies that they are forced to end on a preferred surface. At first sight this appears to break the relativistic invariance of the theory, which is paradoxical. The resolution of the paradox is that strings end on a p-dimensional dynamical object -- a D-brane. D-branes had been studied for a number of years, but their significance was explained by Polchinski only recently[7] The importance of D-branes stems from the fact that they make it possible to study the excitations of the brane using the renormalizable 2D quantum field theory of the open string instead of the non-renormalizable world-volume theory of the D-brane itself. In this way it becomes possible to compute non-perturbative phenomena using perturbative methods. Many (but not all) of the previously identified p-branes are D-branes. Others are related to D-branes by duality symmetries, so that they can also be brought under mathematical control. D-branes have found many interesting applications, but the most remarkable of these concerns the study of black holes. Strominger and Vafa[8] (and subsequently many others) have shown that D-brane techniques can be used to count the quantum microstates associated to classical black hole configurations. The simplest case, which was studied first, is static extremely charged black holes in five dimensions. Strominger and Vafa showed that for large values of the charges the entropy (defined by S = log N, where N is the number of quantum states that system can be in) agrees with the Bekenstein-Hawking prediction (1/4 the area of the event horizon). This result has been generalized to black holes in 4D as well as to ones that are near extremely (and radiate correctly) or rotating. In my opinion, this is a truly dramatic advance. It has not yet been proved that there is no breakdown of quantum mechanics due to black holes, but I expect that result to follow in due course.
Conclusion
I have touched on some of the highlights of the current revolution, but there is much more that does not fit here. For example, I have not discussed the dramatic discoveries of Seiberg and Written [9] for super symmetric gauge theories and their extensions to string theory. Another important development due to Vafa[10] (called F theory) has made it possible to construct large new classes of non-perturbative vacua of Type IIB superstrings. For a somewhat more technical discussion of these recent developments I recommend ref[11]. Despite all the progress that has taken place in our understanding of superstring theory, there are many important questions whose answers are still unknown. It is not clear how many important discoveries still remain to be made before it will be possible to answer the ultimate question that we are striving towards -- why does the universe behave the way it does? Short of that, we have some other pretty big questions. What is the best way to formulate the theory? How and why is super symmetry broken? Why is the cosmological constant, which characterizes the energy density of the vacuum, so small (or zero)? How is a realistic solution of the theory chosen from the myriad of possibilities? What are the cosmological implications of the theory? What testable predictions can we make? Stay tuned.
We live in a wonderfully complex universe, and we are curious about it by nature. Time and again we have wondered--- why are we here? Where did we and the world come from? What is the world made of? It is our privilege to live in a time when enormous progress has been made towards finding some of the answers. String theory is our most recent attempt to answer the last (and part of the second) question.
So, what is the world made of? Ordinary matter is made of atoms, which are in turn made of just three basic components: electrons whirling around a nucleus composed of neutrons and protons. The electron is a truly fundamental particle (it is one of a family of particles known as leptons), but neutrons and protons are made of smaller particles, known as quarks. Quarks are, as far as we know, truly elementary.
Our current knowledge about the subatomic composition of the universe is summarized in what is known as the Standard Model of particle physics. It describes both the fundamental building blocks out of which the world is made, and the forces through which these blocks interact. There are twelve basic building blocks. Six of these are quarks--- they go by the interesting names of up, down, charm, strange, bottom and top. (A proton, for instance, is made of two up quarks and one down quark.) The other six are leptons--- these include the electron and its two heavier siblings, the muon and the tauon, as well as three neutrinos.
There are four fundamental forces in the universe: gravity, electromagnetism, and the weak and strong nuclear forces. Each of these is produced by fundamental particles that act as carriers of the force. The most familiar of these is the photon, a particle of light, which is the mediator of electromagnetic forces. (This means that, for instance, a magnet attracts a nail because both objects exchange photons.) The graviton is the particle associated with gravity. The strong force is carried by eight particles known as gluons. Finally, the weak force is transmitted by three particles, the W+, the W- , and the Z.
The behavior of all of these particles and forces is described with impeccable precision by the Standard Model, with one notable exception: gravity. For technical reasons, the gravitational force, the most familiar in our every day lives, has proven very difficult to describe microscopically. This has been for many years one of the most important problems in theoretical physics-- to formulate a quantum theory of gravity.
In the last few decades, string theory has emerged as the most promising candidate for a microscopic theory of gravity. And it is infinitely more ambitious than that: it attempts to provide a complete, unified, and consistent description of the fundamental structure of our universe. (For this reason it is sometimes, quite arrogantly, called a 'Theory of Everything').
The essential idea behind string theory is this: all of the different 'fundamental ' particles of the Standard Model are really just different manifestations of one basic object: a string. How can that be? Well, we would ordinarily picture an electron, for instance, as a point with no internal structure. A point cannot do anything but move. But, if string theory is correct, then under an extremely powerful 'microscope' we would realize that the electron is not really a point, but a tiny loop of string. A string can do something aside from moving--- it can oscillate in different ways. If it oscillates a certain way, then from a distance, unable to tell it is really a string, we see an electron. But if it oscillates some other way, well, then we call it a photon, or a quark, or a ... you get the idea. So, if string theory is correct, the entire world is made of strings!
Perhaps the most remarkable thing about string theory is that such a simple idea works--- it is possible to derive (an extension of) the Standard Model (which has been verified experimentally with incredible precision) from a theory of strings. But it should also be said that, to date, there is no direct experimental evidence that string theory itself is the correct description of Nature. This is mostly due to the fact that string theory is still under development. We know bits and pieces of it, but we do not yet see the whole picture, and we are therefore unable to make definite predictions. In recent years many exciting developments have taken place, radically improving our understanding of what the theory is.
Reasons for Origin of String theory
Resolution of Contradictions
Major advances in understanding of the physical world have been achieved during the past century by focusing on apparent contradictions between well-established theoretical structures. In each case the reconciliation required a better theory, often involving radical new concepts and striking experimental predictions. Four major advances of this type are indicated in Figure 1. [1] These advances were the discoveries of special relativity, quantum mechanics, general relativity, and quantum field theory. Let us briefly recall the contradictions that each of these resolved and some of the experimental confirmations that followed.
Special Relativity
The Quantum Theory
General Relativity
Quantum Field Theory
The Final Contradiction
Special Relativity
Galilean invariance, which is embodied in Newton's mechanics, for example, was considered intuitively obvious and observationally successful for several centuries. Among other things it implies that velocities add; i.e., if A observes B moving with speed v1 and B observes C moving with speed v2 in the same direction, then A observes C moving with speed v12 = v1 + v2 in that direction. A contradiction arose as a consequence of the development of a very successful theory of electricity and magnetism in the nineteenth century, which is embodied in a set of differential equations known as Maxwell's equation. One of the implications of Maxwell's equations is the existence of various waves, such as radio waves, all of which are just light waves at various different frequencies. The equations imply that all of these waves travel with the speed of light (about 300,000 km/sec) regardless of the motion of the observer. This is a violation of Galilean invariance. Einstein resolved this paradox by recognizing that Galilean invariance is just an approximation, valid for speeds much smaller than the speed of light. The principles that apply for all speeds are embodied in the special theory of relativity. In this theory the rule for the addition of velocities is
,
Where c denotes the speed of light
Quantum Theory
The theory of electromagnetism embodied in Maxwell's equations also conflicted with the understanding of thermodynamics -- the behavior of systems in thermal equilibrium. In particular, a hot object (a so-called `black-body') emits electromagnetic radiation with a certain well-defined spectrum (intensity as a function of the frequency of the radiation). The problem with this is that if one adds up the energy carried off by radiation at all the different frequencies, the formulas imply that the total is infinite, which is an absurd result. Just before the turn of the century, Max Planck realized that if the energy was emitted in discrete packets (or `quanta'), rather than in a continuous distribution, the total energy would be finite. He postulated that radiation of frequency n comes in quanta of energy E = hn, where h is a fundamental constant of nature, known as Planck's constant. The individual quanta of light are called ``photons''. This was the beginning of the idea of wave-particle duality and the quantum theory. ht.
General Relativity
Einstein was not content with the special theory of relativity, because it was in contradiction with Newton's theory of gravitation. Newtonian gravitation was a very successful theory. In particular, it accounted for the motion of the planets to high precision. However, it has a peculiar property that had even bothered Newton. It implies instantaneous transmission of the gravitation force between two objects across great distances. Einstein knew this had to be wrong, because special relativity implies that no signal can be transmitted faster than the speed of light. Einstein provided the resolution around 1915 with a new theory of gravitation, which he called the general theory of relativity. It agrees with the Newtonian theory for low speeds and weak gravitational fields, but differs from it at high speeds and strong fields. The theory had several immediate observational successes. First it implied a small correction to the orbit of the planet Mercury that accounted for a small discrepancy between the orbit implied by the Newtonian theory and the observed orbit. (The effect is too small to be observed for the other planets.) Second, it predicted that light from a distant star passing near to the limb of sun would be bent by a small but measurable angle. The measurement was made by Eddington during a solar eclipse. The observations confirmed the theoretical prediction, and Einstein became an international celebrity. Other predictions of the theory -- such as the existence of black holes and of gravitational radiation -- have been confirmed much more recently. The thing that impressed other physicists most about the general theory of relativity is that it is based on very general physical principles -- the equivalence principle and general coordinate invariance -- and very beautiful mathematical concepts. The relevant mathematics is called differential geometry (specifically, Riemannian geometry). The idea is that gravity is a manifestation of the curvature of space-time. Also, the geometry of space-time is determined by the distribution of energy and momentum. The basic equation of motion is
In this equation Gmn describes the space-time geometry, G is Newton's constant characterizing the strength of gravitation, and Tmn describes the distribution of energy and momentum. Quantum Field Theory
The next contradiction that physicists faced was between quantum mechanics (which had been developed over the thirty years following Planck's seminal insight) and the special theory of relativity. Most of the work in quantum mechanics was in the Galilean (or non-relativistic) approximation. To be sure, Dirac had developed a relativistic wave equation for the electron, which was an important advance, but there was still a basic contradiction that needed to be resolved. The new feature that is required in a successful union of quantum mechanics and special relativity is the possibility of the creation and annihilation of quanta (or `particles'). The non-relativistic theory does not have this feature. The framework in which quantum mechanics and special relativity are successfully reconciled is called quantum field theory. It is based on three basic principles: two of them, of course, are quantum mechanics and special relativity. The third one, which I wish to emphasize, is the postulate that elementary particles are point-like objects of zero intrinsic size. In practice, they are smeared over a region of space due to quantum effects, but their description in the basic equations is as mathematical points. Now the general principles on which quantum field theory are based actually allow for many different consistent theories to be constructed. (The consistency has not been established with mathematical rigour, but this is not a concern for most physicists.) Among these various possible theories there is a class of theories, called `gauge theories' or `Yang-Mills theories' that turn out to be especially interesting and important. These are characterized by a symmetry structure (called a Lie group) and the assignment of various matter particles to particular symmetry patterns (called group representations). There is an infinite set of possibilities for the choice of the symmetry group, and for each group there are many possible choices of group representations for the matter particles. One of this infinite array of theories has been experimentally singled out. It is called the ``standard model''. It is based on a Lie group called SU (3) X SU (2) X U (1). The matter particles consist of three families of quarks and leptons. (I will not describe the representations that they are assigned to here.) There are also addition matter particles called ``Higgs particles'', which are required to account for the fact that part of the symmetry is spontaneously broken. The standard model contains some 20 adjustable parameters, whose values are determined experimentally. Still, there are many more things that can be measured than that, and the standard model is amazingly successful in accounting for a wide range of experiments to very high precision. Indeed, at the time this is written, there is only one clear-cut piece of experimental evidence that the standard model is not an exactly correct theory. This evidence is the fact that the standard model does not contain gravity!
The Final Contradiction
The results described above constitute quite an achievement for one century, but it leaves us with one fundamental contradiction that still needs to be resolved. General relativity and quantum field theory are incompatible. Many theorical physicists are convinced that superstring theory will provide the answer. There have been major advances in our understanding of this subject, which I consider to constitute the ``second superstring revolution,'' during the past few years. After presenting some more background, I will describe the recent developments and their implications. There are various problems that arise when one attempts to combine general relativity and quantum field theory. The field theorist would point to the breakdown of the usual procedure for eliminating infinities from calculations of physical quantities. This procedure is called "renormalization", and when it fails the theory is said to be "non-renormalizable." In such theories the short-distance behavior of interactions is so singular that it is not possible to carry out meaningful calculations. By replacing point-like particles with one-dimensional extended strings, as the fundamental objects, superstring theory overcomes the problem of non-renormalizability. An expert in general relativity might point to a different set of problems such as the issue of how to understand the causal structure of space-time when the geometry has quantum-mechanical excitations. There are also a host of problems associated to black holes such as the fundamental origin of their thermodynamic properties and their apparent incompatibility with quantum mechanics. The latter, if true, would mean that a modification in the basic structure of quantum mechanics is required. In fact, superstring theory does not modify quantum mechanics; rather, it modifies general relativity. The relativist's set of issues cannot be addressed properly in the usual approach to quantum field theory (perturbation theory), but the recent discoveries are leading to non-perturbative understandings that should help in addressing them. Most string theorists expect that the theory will provide satisfying resolutions of these problems without any revision in the basic structure of quantum mechanics. Indeed, there are indications that someday quantum mechanics will be viewed as an implication (or at least a necessary ingredient) of superstring theory. When a new theoretical edifice is proposed, it is very desirable to identify distinctive testable experimental predictions. In the case of superstring theory there have been no detailed computations of the properties of elementary particles or the structure of the universe that are convincing, though many valiant attempts have been made. In my opinion, success in such enterprises requires a better understanding of the theory than has been achieved as yet. It is very difficult to assess whether this level of understanding is just around the corner or whether it will take many decades and several more revolutions. In the absence of this kind of confirmation, we can point to three qualitative "predictions" of superstring theory. The first is the existence of gravitation, approximated at low energies by general relativity. No other quantum theory can claim to have this property (and I suspect that no other ever will). The second is the fact that superstring solutions generally include Yang--Mills gauge theories like those that make up the "standard model" of elementary particles. The third general prediction is the existence of super symmetry at low energies (the electroweak scale). Since super symmetry is the major qualitative prediction of superstring theory not already known to be true before the prediction, let us look at it a little more closely. (One could imagine that in some other civilization, the sequence of discoveries is different.)
Super symmetry
Super symmetry is a theoretically attractive possibility for several reasons. Most important from my viewpoint, is the fact that it is required by superstring theory. Beyond that is the remarkable fact that it is the unique possibility for a non-trivial extension of the known symmetries of space and time (which are described in special relativity by the Poincare group). Mathematically, it can be described in terms of extra dimensions that are rather peculiar. Whereas ordinary space and time dimensions are described by ordinary numbers, which have the property that they commute: X·Y = Y·X, the super symmetry directions are described by numbers that anti-commute: X·Y = -Y·X.
Super partners
Experimental Prospects
Super partners
As we have said, super symmetry (also known as SUSY) is the major prediction of superstring theory at experimentally accessible energies that has not yet been confirmed. If correct, it implies that every known elementary particle must have a "super partner." We are quite sure, for reasons I won't go into, that no pair of the known particles are super symmetry partners of one another. So super symmetry requires the existence of a new elementary particle for every known one. As ofter happens, the names are somewhat whimsical: the partners of quarks are called "squarks", the partners of electrons are called "selectrons", the partners of gluons (the Yang-Mills particles that carry the strong nuclear force) are called "gluinos" and so forth. It is believed that the reason that these particles have not yet been observed is because super symmetry is a broken symmetry, and as a result the super partners are heavier than the known elementary particles. Experiments carried out so far have not had particle beams of sufficient energy and intensity to produce them in observable numbers. Unfortunately, current theoretical ideas are insufficient to accurately predict the super partner masses, though the way in which these particles interact with one another and with the known particles is predicted precisely. Even though accurate predictions of the super partner masses do not exist, there are three distinct arguments that make qualititative predictions of the masses. All three of them lead to the conclusion that a typical super partner mass should be in the range of 100 Gee to 1000 Gee. In other words, they should be about 100--1000 times heavier than a proton. The three arguments are the following: First, super symmetry leads to a softening of the short distance singularities of quantum field theory. If we require a sufficient softening so that the Higgs mechanism can break the electroweak symmetry (SU(2) X U(1)) at the observed 300 GeV scale, in which case the Higgs particles have masses of the same order of magnitude, then the scale of super symmetry breaking must also be approximately the same. The second argument concerns the unification of the electroweak and strong nuclear forces at very high energy (around $10^ {16} $ GeV). One can argue that such a unification is inconsistent with the current experimental data, if one includes the effects of only known particles in the extrapolation, but that it works if super symmetry partner particles with masses in the 100 GeV to 1000 GeV range are included. The third argument concerns the possibility that the lightest SUSY particle could be a form of dark matter accounting for a substantial fraction of the mass of the universe. This also requires the same range of masses!
Experimental Prospects
All of the above makes a very suggestive case for SUSY. It is also very exciting because the mass range that is predicted is just what the new generation of particle accelerators is beginning to explore. Regrettably, the American superconducting supercollider project (SSC) was cancelled, but a European accelerator called the large hadron collider (LHC) will begin operating at a lab in Geneva, Switzerland (called CERN) around 2005. Its energy will be about 8000 GeV per beam, whereas the SSC energy would have been 20,000 GeV per beam. In my opinion, The LHC energy is high enough so that if it does not find super symmetry after a few years of operation, we can safely conclude that it does not exist in the vicinity of the electroweak scale. If the LHC (or another machine) does find SUSY, on the other hand, this would be one of the most profound achievements in the history of humankind. It would be more profound, in my opinion, than the discovery of life on Mars, for example.
A Brief History of Superstrings
The history of string theory is very fascinating, with many bizarre twists and turns. It has not yet received the attention it deserves from historians of science. Here we will settle for a very quick sketch.
String Theory of the Strong Nuclear Force
Superstring Unification
The First Superstring Revolution
String Theory of the Strong Nuclear Force
The subject of string theory arose in the late 1960's in an attempt to describe strong nuclear forces. This approach to the description of strong nuclear forces was a quite active subject for about five years, until it was abandoned because it ran into various theoretical difficulties and because a better theory came along. There were two main problems that stymied us in our attempts to use string theory to describe the strong nuclear forces. One was that the theory required the existence of a kind of particle that we didn't want -- namely a particle with no mass and two units of spin. Its existence was very generic and very frustrating. The second problem was that the theory required that space-time have ten dimensions (nine space and one time), whereas the correct answer is clearly four (three space and one time). As if all that weren't bad enough, around 1973 quantum chromodynamics (QCD) -- the SU (3) part of the standard model -- merged as a convincing theory of the strong nuclear force. One curious fact about string theory in the 1968--1973 period is that it took two years studying various complicated mathematical formulas before several people realized that these were formulas describing the interactions of extended one-dimensional objects, which were named ``strings.'' Once this was realized, it became clear that this type of theory was outside the framework of conventional quantum field theory, which as we have emphasized, is based on point-like elementary particles. Another important development during this period (in 1971) was the discovery that to incorporate a class of elementary particles called fermions (electrons and quarks are examples) string theory requires a two-dimensional version of super symmetry.[2] This led to the development of space-time super symmetry, which was eventually recognized to be a generic feature of all consistent string theories (hence the name superstrings").
Superstring Unification
In 1974 Joel Scherk and I proposed that the problems that string theory had encountered could be turned into virtues if it were used as a framework for realizing Einstein's old dream of a ``unified theory'' of fundamental forces and elementary particles, rather than as a theory of hadrons (the strongly interacting nuclear particles). [3] Specifically, we pointed out that it would provide a theory that incorporates general relativity without the characteristic short-distance infinities of quantum field theory. The massless spin two particle, which we had tried so hard to get rid of, would be identified as the graviton -- the quantum of gravitation! One implication of this change in viewpoint was that, to account for the observed strength of the gravitational force, the characteristic size of a string had to be roughly the Planck length
(the symbol h is Planck's constant.) This was a big change, since this distance is some 20 orders of magnitude smaller than the characteristic size of hadrons previously envisaged. More refined analyses lead to a string scale Lst that is about two orders of magnitude larger than the Planck length. In any case, experiments at existing accelerators cannot resolve distances shorter than about 10-16 cm, which explains why the point-particle approximation of ordinary quantum field theories is so successful. The second problem -- the extra dimensions -- could also be addressed in this setting. Once one has a theory containing gravitation and generalizing general relativity, one knows that that the geometry of space-time is dynamically determined. One could imagine that, as a consequence of the dynamics, the extra six dimensions form a small compact space attached to each point in ordinary four-dimensional space-time. If the size of the extra dimensions is sufficiently small, there would be no conflict with observations. I found these ideas very exciting and have been pursuing them ever since. However, for the ten year period 1974--1984 only a few colleagues and I pursued these ideas. One who did was Joel Scherk; tragically, he passed away in 1980.
The First Superstring Revolution
In 1984-85 there was a series of discoveries[4] that convinced many theorists that superstring theory is a very promising approach to unification. Almost overnight, the subject was transformed from an intellectual backwater to one of the most active areas of theoretical physics, which it has remained ever since. By the time the dust settled in 1985, it seemed clear that there are five different superstring theories, each requiring ten dimensions (nine space and one time), and that each of them has a consistent description in term of a power series expansion in the coupling constant (perturbation expansion). The five theories, about which I'll say more later, are denoted type I, type IIA, type IIB, E8 X E8 heterotic (HE, for short), and SO(32) heterotic (HO, for short). The type II theories have two super symmetries in the ten-dimensional sense, while the other three have just one. The type I theory is special in that it is based on unoriented open and closed strings, whereas the other four are based on oriented closed strings. The IIA theory is special in that it is non-chiral (i.e., it is parity conserving), whereas the other four are chiral (parity violating). At this point I'll end the historical discussion and turn to superstring theory itself.
Basic Ideas of Superstring Theory
A string's space-time history is described by functions Xm(s,t) which describe how the string's two-dimensional "world sheet," represented by coordinates (s,t), is mapped into space-time Xm. There are also functions defined on the two-dimensional world-sheet that describe other degrees of freedom, such as those associated with super symmetry and gauge symmetries. Surprisingly, classical string theory dynamics is described by a conformally invariant 2D quantum field theory. (Roughly, conformal invariance is symmetry under a change of length scale.) What distinguishes one-dimensional strings from higher dimensional analogs is the fact that this 2D theory is renormalizable (no bad short-distance infinities). By contrast, objects with p dimensions, called "p-branes," have a (p+1)-dimensional world volume theory. For p > 1, those theories are non-renormalizable. This is the feature that gives strings a special status, even though, as we will discuss later, higher-dimensional p-branes do occur in superstring theory.
Perturbation Theory
Compactification of Extra Dimensions
A Theory of Everything?
Perturbation Theory
As was briefly mentioned earlier, a useful way of studying theories that cannot be solved exactly is by computing power series expansions in a small parameter. For example, quantum electrodynamics has a small parameter, called the fine structure constant, which is given by
Thus, if T(a) denotes some physical quantity of interest, one computes
and the first few terms can give a very good approximation. This approach, which is called perturbation theory, is the way superstring theories were studied until recently. The problem is that in superstring theory there is no reason that the expansion parameter a should be small. More significantly, there are important qualitative phenomena that are missed in perturbation theory. The reason is that there are non-perturbative contributions to many physically interesting quantities that have the structure
Such a contribution is completely invisible in perturbation theory. Perturbative quantum string theory can be formulated by the Feynman sum-over-histories method. This amounts to associating a genus h Riemann surface, which can be visualized as a sphere with h handles attached to it, to the hth term in the string theory perturbation expansion. The genus h surface is identified as the corresponding string theory Feynman diagram. The attractive features of this approach are that there is just one diagram at each order of the perturbation expansion and that each diagram represents an elegant (though complicated) mathematical expression that is ultraviolet finite (no short-distance infinities). The main drawback of this approach is that it gives no insight into how to go beyond perturbation theory.
Compactification of Extra Dimensions
As has already been mentioned, to have a chance of being realistic, the six extra space dimensions must curl up into a tiny geometrical space, whose size should be comparable to the string length Lst.. Since space-time geometry is determined dynamically (as in general relativity), only geometries that satisfy the dynamical equations are allowed. The HE string theory, compactified on a particular kind of six-dimensional space, called a Calabi--Yau manifold, has many qualitative features at low energies that resemble the standard model. In particular, the low mass fermions (identified as quarks and leptons) occur in families, whose number is controlled by the topology of the CY manifold. These successes have been achieved in a perturbative framework, and are necessarily qualitative at best, since non-perturbative phenomena are essential to an understanding of super symmetry breaking and other important matters of detail.
A Theory of Everything?
In the euphoria following the first superstring revolution in 1985, some of the less experienced participants in the enterprise thought that we were on the verge of constructing a complete fundamental theory of the physical world. To put it mildly, I found this naive. In this setting, the phrase "Theory of Everything" was introduced and propagated by the public media. This was very unfortunate for several reasons. The TOE phrase is very misleading on several counts. First of all, the theory is not yet fully formulated, and when it is (which might still take decades) it is not entirely clear that it will be the last word in fundamental physics. Furthermore, even if the theory is a complete description of quantum dynamics, it seems unlikely that it will also provide a theory of initial conditions, which is another key ingredient required to explain why we observe the particular universe that we do. But even if a theory of initial conditions is also obtained, there will still be much about this universe that cannot be explained. Many things, such as our very existence, are a consequence of the inherent quantum indeterminacy of nature. I believe that cannot be overcome. Maybe that is just as well, because if we had old-fashioned classical determinism, the future would be fully determined, which would undermine our humanity. There is also a more mundane sort of unpredictability that is also to be expected. Many of the things that the theory predicts unambiguously in principle could require intractable calculations. Part of the art of physics is to identify those things that can be calculated. The other reason the TOE phrase upset me is that it alienated many of our physics colleagues, some of whom had serious doubts about the subject anyway. Quite understandably, it gave them the impression that people who work in this field are a very arrogant bunch. Actually, we are all very charming and delightful.
The Second Superstring Revolution
Just One Theory!
T Duality
S Duality
M Theory
D-Branes
Conclusions
Just One Theory!
The second superstring revolution (1994-??) has brought non-perturbative string physics within reach. The key discoveries were the recognition of amazing and surprising "dualities." They have taught us that what we viewed previously as five distinct superstring theories is in fact five different perturbative expansions of a single underlying theory about five different points. It is now clear that there is a unique theory, though it may allow many different quantum mechanical solutions. For example, a sixth special quantum solution implies the existence of an 11-dimensional space-time. Another lesson we have learned is that, non-perturbatively, objects of more than one dimension (membranes and higher "p-branes") play a central role. In most respects they appear just as fundamental as the strings (which can now be called one-branes), except that a perturbation expansion cannot be based on p-branes with p >1. Three kinds of dualities, called S,T, and U, have been identified. It can sometimes happen that theory A with a large strength of interaction (or `strong coupling') is equivalent to theory B at weak coupling, in which case they are said to be S dual. Similarly, if theory A compactified on a space of large volume is equivalent to theory B compactified on a space of small volume, then they are called T dual. Combining these ideas, if theory A compactified on a space of large (or small) volume is equivalent to theory B at strong (or weak) coupling, they are called U dual. If theories A and B are the same, then the duality becomes a self-duality, and it can be viewed as a kind of symmetry. T duality, unlike S or U duality, can be understood perturbatively, and therefore it was discovered between the two string revolutions.
T Duality
The basic idea of T duality(for a recent discussion see [5]) can be illustrated by considering a compact dimension consisting of a circle of radius R. In this case there are two kinds of excitations to consider. The first, which is not special to string theory, are Kaluza--Klein momentum excitations on the circle, which contribute (n/R)2 to the energy squared, where n is an integer. Winding-mode excitations, due to a closed string winding m times around the circular dimension, are special to string theory. If
denotes the string tension (energy per unit length), the contribution to the energy squared is Em=2pmRT.T duality exchanges these two kinds of excitations by exchanging m with n and
This is part of an exact map between a T-dual pair A and B. One implication is that usual geometric concepts break down at short distances, and classical geometry is replaced by "quantum geometry," which is described mathematically by 2D conformal field theory. It also suggests a generalization of the Heisenberg uncertainty principle according to which the best possible spatial resolution Dx is bounded below not only by the reciprocal of the momentum spread, Dp, but also by the string scale Lst. (Including non-perturbative effects, it may be possible to do a little better and reach the Planck scale.) Two important examples of superstring theories that are T-dual when compactified on a circle are the IIA and IIB theories and the HE and HO theories. These two dualities reduce the number of distinct theories from five to three.
S Duality
Suppose now that a pair of theories A and B is S-dual. This means that if f denotes any physical observable and l denotes the coupling constant, then
(The expansion parameter a introduced earlier corresponds to l). This duality, whose recognition was the first step in the current revolution, [6] generalizes the electric-magnetic symmetry of Maxwell theory. Since the Dirac quantization condition implies that the basic unit of magnetic charge is inversely proportional to the unit of electric charge, their interchange amounts to an inversion of the charge (which is the coupling constant). S duality relates the type I theory to the HO theory and the IIB theory to itself. This explains the strong coupling behavior of those three theories.
M Theory
The understanding of how the IIA and HE theories behave at strong coupling, which is by now well-established, came as quite a surprise. In each of these cases there is an 11th dimension that becomes large at strong coupling, the scaling law being
In the IIA case the 11th dimension is a circle, whereas in the HE case it is a line interval (so that the eleven-dimensional space-time has two ten-dimensional boundaries). The strong coupling limit of either of these theories gives an 11-dimensional space-time. The eleven-dimensional description of the underlying theory is called "M theory." As yet, it is less well understood than the five 10-dimensional string theories. The connections among theories that we've mentioned are sketched below.
(S1 denotes a circle and I denotes a line interval.) There are many additional dualities that arise when more dimensions are compactified, which will not be described here.
D-Branes
Another source of insight into non-perturbative properties of superstring theory has arisen from the study of a special class of p-branes called Dirichlet p-branes (or D-branes for short). The name derives from the boundary conditions assigned to the ends of open strings. The usual open strings of the type I theory satisfy a condition (Neumann boundary condition) that ensures that no momentum flows on or of the end of a string. However, T duality implies the existence of dual open strings with specified positions (Dirichlet boundary conditions) in the dimensions that are T-transformed. More generally, in type II theories, one can consider open strings with specified positions for the end-points in some of the dimensions, which implies that they are forced to end on a preferred surface. At first sight this appears to break the relativistic invariance of the theory, which is paradoxical. The resolution of the paradox is that strings end on a p-dimensional dynamical object -- a D-brane. D-branes had been studied for a number of years, but their significance was explained by Polchinski only recently[7] The importance of D-branes stems from the fact that they make it possible to study the excitations of the brane using the renormalizable 2D quantum field theory of the open string instead of the non-renormalizable world-volume theory of the D-brane itself. In this way it becomes possible to compute non-perturbative phenomena using perturbative methods. Many (but not all) of the previously identified p-branes are D-branes. Others are related to D-branes by duality symmetries, so that they can also be brought under mathematical control. D-branes have found many interesting applications, but the most remarkable of these concerns the study of black holes. Strominger and Vafa[8] (and subsequently many others) have shown that D-brane techniques can be used to count the quantum microstates associated to classical black hole configurations. The simplest case, which was studied first, is static extremely charged black holes in five dimensions. Strominger and Vafa showed that for large values of the charges the entropy (defined by S = log N, where N is the number of quantum states that system can be in) agrees with the Bekenstein-Hawking prediction (1/4 the area of the event horizon). This result has been generalized to black holes in 4D as well as to ones that are near extremely (and radiate correctly) or rotating. In my opinion, this is a truly dramatic advance. It has not yet been proved that there is no breakdown of quantum mechanics due to black holes, but I expect that result to follow in due course.
Conclusion
I have touched on some of the highlights of the current revolution, but there is much more that does not fit here. For example, I have not discussed the dramatic discoveries of Seiberg and Written [9] for super symmetric gauge theories and their extensions to string theory. Another important development due to Vafa[10] (called F theory) has made it possible to construct large new classes of non-perturbative vacua of Type IIB superstrings. For a somewhat more technical discussion of these recent developments I recommend ref[11]. Despite all the progress that has taken place in our understanding of superstring theory, there are many important questions whose answers are still unknown. It is not clear how many important discoveries still remain to be made before it will be possible to answer the ultimate question that we are striving towards -- why does the universe behave the way it does? Short of that, we have some other pretty big questions. What is the best way to formulate the theory? How and why is super symmetry broken? Why is the cosmological constant, which characterizes the energy density of the vacuum, so small (or zero)? How is a realistic solution of the theory chosen from the myriad of possibilities? What are the cosmological implications of the theory? What testable predictions can we make? Stay tuned.
THEORY OF EVERYTHING
A theory of everything (TOE) is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories. For example, a great-grandfather of Ijon Tichy — a character from a cycle of Stanisław Lem's science fiction stories of 1960s — was known to work on the "General Theory of Everything". Physicist John Ellis claims[1] to have introduced the term into the technical literature in an article in Nature in 1986.[2] Over time, the term stuck in popularizations of quantum physics to describe a theory that would unify or explain through a single model the theories of all fundamental interactions of nature.
There have been many theories of everything proposed by theoretical physicists over the last century, but none have been confirmed experimentally. The primary problem in producing a TOE is that the accepted theories of quantum mechanics and general relativity are hard to combine.
Based on theoretical holographic principle arguments from the 1990s, many physicists believe that 11-dimensional M-theory, which is described in many sectors by matrix string theory, in many other sectors by perturbative string theory is the complete theory of everything. Other physicists disagree
[edit] Historical antecedents
Laplace famously suggested that a sufficiently powerful intellect could, if it knew the velocity of every particle at a given time, along with the laws of nature, calculate the position of any particle at any other time:
An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
– Essai philosophique sur les probabilités, Introduction. 1814
Although modern quantum mechanics suggests that uncertainty is inescapable, a "single formula" may nevertheless exist.
[edit] Ancient Greece to Einstein
Since ancient Greek times, philosophers have speculated that the apparent diversity of appearances conceals an underlying unity, and thus that the list of forces might be short, indeed might contain only a single entry. For example, the mechanical philosophy of the 17th century posited that all forces could be ultimately reduced to contact forces between tiny solid particles.[3] This was abandoned after the acceptance of Isaac Newton's long-distance force of gravity; but at the same time, Newton's work in his Principia provided the first dramatic empirical evidence for the unification of apparently distinct forces: Galileo's work on terrestrial gravity, Kepler's laws of planetary motion, and the phenomonenon of tides were all quantitatively explained by a single law of universal gravitation.
In 1820, Hans Christian Oersted discovered a connection between electricity and magnetism, triggering decades of work that culminated in James Clerk Maxwell's theory of electromagnetism. Also during the 19th and early 20th centuries, it gradually became apparent that many common examples of forces—contact forces, elasticity, viscosity, friction, pressure—resulted from electrical interactions between the smallest particles of matter. In the late 1920s, the new quantum mechanics showed that the chemical bonds between atoms were examples of (quantum) electrical forces, justifying Dirac's boast that "the underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known".[4]
Attempts to unify gravity with electromagnetism date back at least to Michael Faraday's experiments of 1849–50.[5] After Albert Einstein's theory of gravity (general relativity) was published in 1915, the search for a unified field theory combining gravity with electromagnetism began in earnest. At the time, it seemed plausible that no other fundamental forces exist. Prominent contributors were Gunnar Nordström, Hermann Weyl, Arthur Eddington, Theodor Kaluza, Oskar Klein, and most notably, many attempts by Einstein and his collaborators. In his last years, Albert Einstein was intensely occupied in finding such a unifying theory. None of these attempts were successful.[6]
[edit] New discoveries
The search for a unifying theory was interrupted by the discovery of the strong and weak nuclear forces, which could not be subsumed into either gravity or electromagnetism. A further hurdle was the acceptance that quantum mechanics had to be incorporated from the start, rather than emerging as a consequence of a deterministic unified theory, as Einstein had hoped. Gravity and electromagnetism could always peacefully coexist as entries in a list of Newtonian forces, but for many years it seemed that gravity could not even be incorporated into the quantum framework, let alone unified with the other fundamental forces. For this reason, work on unification for much of the twentieth century, focused on understanding the three "quantum" forces: electromagnetism and the weak and strong forces. The first two were unified in 1967–8 by Sheldon Glashow, Steven Weinberg, and Abdus Salam.[7] The strong and electroweak forces peacefully coexist in the standard model of particle physics, but remain distinct. Several Grand Unified Theories (GUTs) have been proposed to unify them. Although the simplest GUTs have been experimentally ruled out, the general idea, especially when linked with supersymmetry, remains strongly favored by the theoretical physics community.
[edit] Modern physics
In current mainstream physics, a Theory of Everything would unify all the fundamental interactions of nature, which are usually considered to be four in number: gravity, the strong nuclear force, the weak nuclear force, and the electromagnetic force. Because the weak force can transform elementary particles from one kind into another, the TOE should yield a deep understanding of the various different kinds of particles as well as the different forces. The expected pattern of theories is:
Theory of Everything
Gravity
Electronuclear force (GUT)
Strong force su(3)
Electroweak forcesu(2) x u(1)
Weak force su(2)
Electromagnetism u(1)
Electric force
Magnetic force
In addition to the forces listed here, modern cosmology might require an inflationary force, dark energy, and also dark matter composed of fundamental particles outside the scheme of the standard model. The existence of these has not been proven and there are alternative theories such as modified Newtonian dynamics.
Electroweak unification is a broken symmetry: the electromagnetic and weak forces appear distinct at low energies because the particles carrying the weak force, the W and Z bosons have a mass of about 100 GeV, whereas the photon, which carries the electromagnetic force, is massless. At higher energies Ws and Zs can be created easily and the unified nature of the force becomes apparent. Grand unification is expected to work in a similar way, but at energies of the order of 1016 GeV, far greater than could be reached by any possible Earth-based particle accelerator. By analogy, unification of the GUT force with gravity is expected at the Planck energy, roughly 1019 GeV.
It may seem premature to be searching for a TOE when there is as yet no direct evidence for an electronuclear force, and while in any case there are many different proposed GUTs. In fact the name deliberately suggests the hubris involved. Nevertheless, most physicists believe this unification is possible, partly due to the past history of convergence towards a single theory. Supersymmetric GUTs seem plausible not only for their theoretical "beauty", but because they naturally produce large quantities of dark matter, and the inflationary force may be related to GUT physics (although it does not seem to form an inevitable part of the theory). And yet GUTs are clearly not the final answer. Both the current standard model and proposed GUTs are quantum field theories which require the problematic technique of renormalization to yield sensible answers. This is usually regarded as a sign that these are only effective field theories, omitting crucial phenomena relevant only at very high energies. Furthermore, the inconsistency between quantum mechanics and general relativity implies that one or both of these must be replaced by a theory incorporating quantum gravity.
Unsolved problems in physics: Is string theory, superstring theory, or M-theory, or some other variant on this theme, a step on the road to a "theory of everything", or just a blind alley?
The mainstream theory of everything at the moment is superstring theory / M-theory; current research on loop quantum gravity may eventually play a fundamental role in a TOE, but that is not its primary aim. These theories attempt to deal with the renormalization problem by setting up some lower bound on the length scales possible. String theories and supergravity (both believed to be limiting cases of the yet-to-be-defined M-theory) suppose that the universe actually has more dimensions than the easily observed three of space and one of time. The motivation behind this approach began with the Kaluza-Klein theory in which it was noted that applying general relativity to a five dimensional universe (with the usual four dimensions plus one small curled-up dimension) yields the equivalent of the usual general relativity in four dimensions together with Maxwell's equations (electromagnetism, also in four dimensions). This has led to efforts to work with theories with large number of dimensions in the hopes that this would produce equations that are similar to known laws of physics. The notion of extra dimensions also helps to resolve the hierarchy problem, which is the question of why gravity is so much weaker than any other force. The common answer involves gravity leaking into the extra dimensions in ways that the other forces do not.
In the late 1990s, it was noted that one problem with several of the candidates for theories of everything (but particularly string theory) was that they did not constrain the characteristics of the predicted universe. For example, many theories of quantum gravity can create universes with arbitrary numbers of dimensions or with arbitrary cosmological constants. Even the "standard" ten-dimensional string theory allows the "curled up" dimensions to be compactified in an enormous number of different ways (one estimate is 10500) each of which corresponds to a different collection of fundamental particles and low-energy forces. This array of theories is known as the string theory landscape.
A speculative solution is that many or all of these possibilities are realised in one or another of a huge number of universes, but that only a small number of them are habitable, and hence the fundamental constants of the universe are ultimately the result of the anthropic principle rather than a consequence of the theory of everything. This anthropic approach is often criticised in that, because the theory is flexible enough to encompass almost any observation, it cannot make useful (as in original, falsifiable, and verifiable) predictions. In this view, string theory would be considered a pseudoscience, where an unfalsifiable theory is constantly adapted to fit the experimental results.
[edit] With reference to Gödel's incompleteness theorem
A small number of scientists claim that Gödel's incompleteness theorem proves that any attempt to construct a TOE is bound to fail. Gödel's theorem states that any non-trivial mathematical theory that has a finite description is either inconsistent or incomplete. In his 1966 book The Relevance of Physics, Stanley Jaki pointed out that, because any "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything.[8]
Freeman Dyson has stated that
“
Gödel’s theorem implies that pure mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. [...] Because of Gödel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them.
”
Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable.
“
Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind.
”
This view has been argued against by Jürgen Schmidhuber (1997), who pointed out that Gödel's theorems are irrelevant even for computable physics.[9] In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose pseudo-randomness based on undecidable, Gödel-like halting problems is extremely hard to detect but does not at all prevent formal TOEs describable by very few bits of information.[10][11]
Related critique was offered by Solomon Feferman,[12] among others. Douglas S. Robertson offers Conway's game of life as an example:[13] The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.
Since most physicists would consider the statement of the underlying rules to suffice as the definition of a "theory of everything", these researchers argue that Gödel's Theorem does not mean that a TOE cannot exist. On the other hand, the physicists invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.[14] This definitional discrepancy may explain some of the disagreement among researchers.
[edit] Potential status of a theory of everything
No physical theory to date is believed to be precisely accurate. Instead, physics has proceeded by a series of "successive approximations" allowing more and more accurate predictions over a wider and wider range of phenomena. Some physicists believe that it is therefore a mistake to confuse theoretical models with the true nature of reality, and hold that the series of approximations will never terminate in the "truth". Einstein himself expressed this view on occasions.[15] On this view, we may reasonably hope for a theory of everything which self-consistently incorporates all currently known forces, but should not expect it to be the final answer. On the other hand it is often claimed that, despite the apparently ever-increasing complexity of the mathematics of each new theory, in a deep sense associated with their underlying gauge symmetry and the number of fundamental physical constants, the theories are becoming simpler. If so, the process of simplification cannot continue indefinitely.
There is a philosophical debate within the physics community as to whether a theory of everything deserves to be called the fundamental law of the universe.[16] One view is the hard reductionist position that the TOE is the fundamental law and that all other theories that apply within the universe are a consequence of the TOE. Another view is that emergent laws (called "free floating laws" by Steven Weinberg), which govern the behavior of complex systems, should be seen as equally fundamental. Examples are the second law of thermodynamics and the theory of natural selection. The point being that, although in our universe these laws describe systems whose behaviour could ("in principle") be predicted from a TOE, they would also hold in universes with different low-level laws, subject only to some very general conditions. Therefore it is of no help, even in principle, to invoke low-level laws when discussing the behavior of complex systems. Some argue that this attitude would violate Occam's Razor if a completely valid TOE were formulated. It is not clear that there is any point at issue in these debates (e.g., between Steven Weinberg and Philip Anderson) other than the right to apply the high-status word "fundamental" to their respective subjects of interest.
Although the name "theory of everything" suggests the determinism of Laplace's quote, this gives a very misleading impression. Determinism is frustrated by the probabilistic nature of quantum mechanical predictions, by the extreme sensitivity to initial conditions that leads to mathematical chaos, and by the extreme mathematical difficulty of applying the theory. Thus, although the current standard model of particle physics "in principle" predicts all known non-gravitational phenomena, in practice only a few quantitative results have been derived from the full theory (e.g., the masses of some of the simplest hadrons), and these results (especially the particle masses which are most relevant for low-energy physics) are less accurate than existing experimental measurements. The true TOE would almost certainly be even harder to apply. The main motive for seeking a TOE, apart from the pure intellectual satisfaction of completing a centuries-long quest, is that all prior successful unifications have predicted new phenomena, some of which (e.g., electrical generators) have proved of great practical importance. As in other cases of theory reduction, the TOE would also allow us to confidently define the domain of validity and residual error of low-energy approximations to the full theory which could be used for practical calculations.
[edit] Theory of everything and philosophy
Main article: Theory of everything (philosophy)
The status of a physical TOE is open to philosophical debate. For example, if physicalism is true, a physical TOE would coincide with a philosophical theory of everything. Some philosophers (Aristotle, Plato, Hegel, Whitehead, et al) have attempted to construct all-encompassing systems. Others are highly dubious about the very possibility of such an exercise.
A theory of everything (TOE) is a putative theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories. For example, a great-grandfather of Ijon Tichy — a character from a cycle of Stanisław Lem's science fiction stories of 1960s — was known to work on the "General Theory of Everything". Physicist John Ellis claims[1] to have introduced the term into the technical literature in an article in Nature in 1986.[2] Over time, the term stuck in popularizations of quantum physics to describe a theory that would unify or explain through a single model the theories of all fundamental interactions of nature.
There have been many theories of everything proposed by theoretical physicists over the last century, but none have been confirmed experimentally. The primary problem in producing a TOE is that the accepted theories of quantum mechanics and general relativity are hard to combine.
Based on theoretical holographic principle arguments from the 1990s, many physicists believe that 11-dimensional M-theory, which is described in many sectors by matrix string theory, in many other sectors by perturbative string theory is the complete theory of everything. Other physicists disagree
[edit] Historical antecedents
Laplace famously suggested that a sufficiently powerful intellect could, if it knew the velocity of every particle at a given time, along with the laws of nature, calculate the position of any particle at any other time:
An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
– Essai philosophique sur les probabilités, Introduction. 1814
Although modern quantum mechanics suggests that uncertainty is inescapable, a "single formula" may nevertheless exist.
[edit] Ancient Greece to Einstein
Since ancient Greek times, philosophers have speculated that the apparent diversity of appearances conceals an underlying unity, and thus that the list of forces might be short, indeed might contain only a single entry. For example, the mechanical philosophy of the 17th century posited that all forces could be ultimately reduced to contact forces between tiny solid particles.[3] This was abandoned after the acceptance of Isaac Newton's long-distance force of gravity; but at the same time, Newton's work in his Principia provided the first dramatic empirical evidence for the unification of apparently distinct forces: Galileo's work on terrestrial gravity, Kepler's laws of planetary motion, and the phenomonenon of tides were all quantitatively explained by a single law of universal gravitation.
In 1820, Hans Christian Oersted discovered a connection between electricity and magnetism, triggering decades of work that culminated in James Clerk Maxwell's theory of electromagnetism. Also during the 19th and early 20th centuries, it gradually became apparent that many common examples of forces—contact forces, elasticity, viscosity, friction, pressure—resulted from electrical interactions between the smallest particles of matter. In the late 1920s, the new quantum mechanics showed that the chemical bonds between atoms were examples of (quantum) electrical forces, justifying Dirac's boast that "the underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known".[4]
Attempts to unify gravity with electromagnetism date back at least to Michael Faraday's experiments of 1849–50.[5] After Albert Einstein's theory of gravity (general relativity) was published in 1915, the search for a unified field theory combining gravity with electromagnetism began in earnest. At the time, it seemed plausible that no other fundamental forces exist. Prominent contributors were Gunnar Nordström, Hermann Weyl, Arthur Eddington, Theodor Kaluza, Oskar Klein, and most notably, many attempts by Einstein and his collaborators. In his last years, Albert Einstein was intensely occupied in finding such a unifying theory. None of these attempts were successful.[6]
[edit] New discoveries
The search for a unifying theory was interrupted by the discovery of the strong and weak nuclear forces, which could not be subsumed into either gravity or electromagnetism. A further hurdle was the acceptance that quantum mechanics had to be incorporated from the start, rather than emerging as a consequence of a deterministic unified theory, as Einstein had hoped. Gravity and electromagnetism could always peacefully coexist as entries in a list of Newtonian forces, but for many years it seemed that gravity could not even be incorporated into the quantum framework, let alone unified with the other fundamental forces. For this reason, work on unification for much of the twentieth century, focused on understanding the three "quantum" forces: electromagnetism and the weak and strong forces. The first two were unified in 1967–8 by Sheldon Glashow, Steven Weinberg, and Abdus Salam.[7] The strong and electroweak forces peacefully coexist in the standard model of particle physics, but remain distinct. Several Grand Unified Theories (GUTs) have been proposed to unify them. Although the simplest GUTs have been experimentally ruled out, the general idea, especially when linked with supersymmetry, remains strongly favored by the theoretical physics community.
[edit] Modern physics
In current mainstream physics, a Theory of Everything would unify all the fundamental interactions of nature, which are usually considered to be four in number: gravity, the strong nuclear force, the weak nuclear force, and the electromagnetic force. Because the weak force can transform elementary particles from one kind into another, the TOE should yield a deep understanding of the various different kinds of particles as well as the different forces. The expected pattern of theories is:
Theory of Everything
Gravity
Electronuclear force (GUT)
Strong force su(3)
Electroweak forcesu(2) x u(1)
Weak force su(2)
Electromagnetism u(1)
Electric force
Magnetic force
In addition to the forces listed here, modern cosmology might require an inflationary force, dark energy, and also dark matter composed of fundamental particles outside the scheme of the standard model. The existence of these has not been proven and there are alternative theories such as modified Newtonian dynamics.
Electroweak unification is a broken symmetry: the electromagnetic and weak forces appear distinct at low energies because the particles carrying the weak force, the W and Z bosons have a mass of about 100 GeV, whereas the photon, which carries the electromagnetic force, is massless. At higher energies Ws and Zs can be created easily and the unified nature of the force becomes apparent. Grand unification is expected to work in a similar way, but at energies of the order of 1016 GeV, far greater than could be reached by any possible Earth-based particle accelerator. By analogy, unification of the GUT force with gravity is expected at the Planck energy, roughly 1019 GeV.
It may seem premature to be searching for a TOE when there is as yet no direct evidence for an electronuclear force, and while in any case there are many different proposed GUTs. In fact the name deliberately suggests the hubris involved. Nevertheless, most physicists believe this unification is possible, partly due to the past history of convergence towards a single theory. Supersymmetric GUTs seem plausible not only for their theoretical "beauty", but because they naturally produce large quantities of dark matter, and the inflationary force may be related to GUT physics (although it does not seem to form an inevitable part of the theory). And yet GUTs are clearly not the final answer. Both the current standard model and proposed GUTs are quantum field theories which require the problematic technique of renormalization to yield sensible answers. This is usually regarded as a sign that these are only effective field theories, omitting crucial phenomena relevant only at very high energies. Furthermore, the inconsistency between quantum mechanics and general relativity implies that one or both of these must be replaced by a theory incorporating quantum gravity.
Unsolved problems in physics: Is string theory, superstring theory, or M-theory, or some other variant on this theme, a step on the road to a "theory of everything", or just a blind alley?
The mainstream theory of everything at the moment is superstring theory / M-theory; current research on loop quantum gravity may eventually play a fundamental role in a TOE, but that is not its primary aim. These theories attempt to deal with the renormalization problem by setting up some lower bound on the length scales possible. String theories and supergravity (both believed to be limiting cases of the yet-to-be-defined M-theory) suppose that the universe actually has more dimensions than the easily observed three of space and one of time. The motivation behind this approach began with the Kaluza-Klein theory in which it was noted that applying general relativity to a five dimensional universe (with the usual four dimensions plus one small curled-up dimension) yields the equivalent of the usual general relativity in four dimensions together with Maxwell's equations (electromagnetism, also in four dimensions). This has led to efforts to work with theories with large number of dimensions in the hopes that this would produce equations that are similar to known laws of physics. The notion of extra dimensions also helps to resolve the hierarchy problem, which is the question of why gravity is so much weaker than any other force. The common answer involves gravity leaking into the extra dimensions in ways that the other forces do not.
In the late 1990s, it was noted that one problem with several of the candidates for theories of everything (but particularly string theory) was that they did not constrain the characteristics of the predicted universe. For example, many theories of quantum gravity can create universes with arbitrary numbers of dimensions or with arbitrary cosmological constants. Even the "standard" ten-dimensional string theory allows the "curled up" dimensions to be compactified in an enormous number of different ways (one estimate is 10500) each of which corresponds to a different collection of fundamental particles and low-energy forces. This array of theories is known as the string theory landscape.
A speculative solution is that many or all of these possibilities are realised in one or another of a huge number of universes, but that only a small number of them are habitable, and hence the fundamental constants of the universe are ultimately the result of the anthropic principle rather than a consequence of the theory of everything. This anthropic approach is often criticised in that, because the theory is flexible enough to encompass almost any observation, it cannot make useful (as in original, falsifiable, and verifiable) predictions. In this view, string theory would be considered a pseudoscience, where an unfalsifiable theory is constantly adapted to fit the experimental results.
[edit] With reference to Gödel's incompleteness theorem
A small number of scientists claim that Gödel's incompleteness theorem proves that any attempt to construct a TOE is bound to fail. Gödel's theorem states that any non-trivial mathematical theory that has a finite description is either inconsistent or incomplete. In his 1966 book The Relevance of Physics, Stanley Jaki pointed out that, because any "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything.[8]
Freeman Dyson has stated that
“
Gödel’s theorem implies that pure mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. [...] Because of Gödel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them.
”
Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable.
“
Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind.
”
This view has been argued against by Jürgen Schmidhuber (1997), who pointed out that Gödel's theorems are irrelevant even for computable physics.[9] In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose pseudo-randomness based on undecidable, Gödel-like halting problems is extremely hard to detect but does not at all prevent formal TOEs describable by very few bits of information.[10][11]
Related critique was offered by Solomon Feferman,[12] among others. Douglas S. Robertson offers Conway's game of life as an example:[13] The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.
Since most physicists would consider the statement of the underlying rules to suffice as the definition of a "theory of everything", these researchers argue that Gödel's Theorem does not mean that a TOE cannot exist. On the other hand, the physicists invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.[14] This definitional discrepancy may explain some of the disagreement among researchers.
[edit] Potential status of a theory of everything
No physical theory to date is believed to be precisely accurate. Instead, physics has proceeded by a series of "successive approximations" allowing more and more accurate predictions over a wider and wider range of phenomena. Some physicists believe that it is therefore a mistake to confuse theoretical models with the true nature of reality, and hold that the series of approximations will never terminate in the "truth". Einstein himself expressed this view on occasions.[15] On this view, we may reasonably hope for a theory of everything which self-consistently incorporates all currently known forces, but should not expect it to be the final answer. On the other hand it is often claimed that, despite the apparently ever-increasing complexity of the mathematics of each new theory, in a deep sense associated with their underlying gauge symmetry and the number of fundamental physical constants, the theories are becoming simpler. If so, the process of simplification cannot continue indefinitely.
There is a philosophical debate within the physics community as to whether a theory of everything deserves to be called the fundamental law of the universe.[16] One view is the hard reductionist position that the TOE is the fundamental law and that all other theories that apply within the universe are a consequence of the TOE. Another view is that emergent laws (called "free floating laws" by Steven Weinberg), which govern the behavior of complex systems, should be seen as equally fundamental. Examples are the second law of thermodynamics and the theory of natural selection. The point being that, although in our universe these laws describe systems whose behaviour could ("in principle") be predicted from a TOE, they would also hold in universes with different low-level laws, subject only to some very general conditions. Therefore it is of no help, even in principle, to invoke low-level laws when discussing the behavior of complex systems. Some argue that this attitude would violate Occam's Razor if a completely valid TOE were formulated. It is not clear that there is any point at issue in these debates (e.g., between Steven Weinberg and Philip Anderson) other than the right to apply the high-status word "fundamental" to their respective subjects of interest.
Although the name "theory of everything" suggests the determinism of Laplace's quote, this gives a very misleading impression. Determinism is frustrated by the probabilistic nature of quantum mechanical predictions, by the extreme sensitivity to initial conditions that leads to mathematical chaos, and by the extreme mathematical difficulty of applying the theory. Thus, although the current standard model of particle physics "in principle" predicts all known non-gravitational phenomena, in practice only a few quantitative results have been derived from the full theory (e.g., the masses of some of the simplest hadrons), and these results (especially the particle masses which are most relevant for low-energy physics) are less accurate than existing experimental measurements. The true TOE would almost certainly be even harder to apply. The main motive for seeking a TOE, apart from the pure intellectual satisfaction of completing a centuries-long quest, is that all prior successful unifications have predicted new phenomena, some of which (e.g., electrical generators) have proved of great practical importance. As in other cases of theory reduction, the TOE would also allow us to confidently define the domain of validity and residual error of low-energy approximations to the full theory which could be used for practical calculations.
[edit] Theory of everything and philosophy
Main article: Theory of everything (philosophy)
The status of a physical TOE is open to philosophical debate. For example, if physicalism is true, a physical TOE would coincide with a philosophical theory of everything. Some philosophers (Aristotle, Plato, Hegel, Whitehead, et al) have attempted to construct all-encompassing systems. Others are highly dubious about the very possibility of such an exercise.
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