[This may need attention from BenTheMan]
Motivation and Ancient History
At the start of the twentieth century, physicists had begun to understand Nature at two different levels---Einstein had given us the 'space-time' interpretation of gravity, while a completely separate group of physicists (including Schroedinger, Heisenberg, Bohr and Dirac) had given us a way to understand the dynamics of things at a subatomic level. The former field is called General Relativity and the latter Quantum Mechanics.
Motivated by Maxwell's unification of electricity and magnetism, Einstein attempted to unify gravity and electromagnetism. Einstein along with Kaluza and Klein, did have some early success---Kaluza and Klein realized that one could write a theory of gravity down in five dimensions, compactifying one dimension on a circle, and end up with a four dimensional metric, some tensor with one index (a photon), and a scalar particle. Unfortunately, this elegant solution doesn't work, which Einstein realized---the scalar hadn't been observed, and one gets a tower of extra particles, all with masses in integer multiples of the radius, in natural units.
In the fifties, Feynman discovered how to write a quantum theory of electromagnetism, called quantum electrodynamics. Aside from being fantasically successful in describing processes such as Compton scattering and electron/positron anihilation, Feynman invented path integrals, which remain the best (but poorest understood) way to calculate the dynamics of gauge theories, and Feynman diagrams, which offer a nmeumonic for calculating scattering amplitudes. The successes of this theory are build on the concept of perturbation theory---essentially, an expansion in Feynman diagrams gives us a power series in the coupling constant. Because the coupling constant is less than one, each term in the power series gets smaller and smaller, untill the correction of adding another term becomes smaller than the precision of the measurement that one wishes to preform. The theory of QED has been checked to eight terms (reference!)---the precision at this level makes QED the most accurately tested theory in all of physics.
When Feynman (and others) tried to apply the same procedures to gravity, they (like Einstein) failed miserably. The problem is that the perturbation series does not converge, and one encounters infinities very quickly.
Until the 'string theory' came about, it was thought that matter existed as tiny little 'pointlike' units, like a grain of sand on a beach. This idea has been challenged. Superstring theory is a mathematical insert, describing all fundamental particles as tiny strings vibrating at different frequencies, and correctfully predicts the work of the forces between particles, (especially gravity and the strong force).
Strings come in two distinctive forms: Open strings and closed strings. Open strings are stuck to extended objects, called d-branes, while closed strings are allowed to propogate freely. One can use the different types of strings and branes to build several different types of string theories. For example, the two Type II string theories have d-branes of various dimensions, and closed strings. The closed strings are associated with the elusive graviton, and open strings make up the electrons, quarks, neutrinos, etc. Because open strings have two endpoints, they can inexorably join with other strings forming a closed string. Strings can split and combine, describing particles emitting or absorbing other particles.
In order to make a consistent string theory, our universe must have 10 dimensions. Because we know that we live in four dimensions, it must be that 6 of the dimensions are curled up into the smallest space possible called the 'Planck length', about 10^-33 cm. This is remaniscent of the old Kaluza-Klein theories---the scalars that we get are called 'moduli' and we have an infinite tower of states at the Planck Mass, or about 10^19 GeV. (For comparisson, the largest energies that we can currently probe are around 10^3 GeV, at FermiLab.) Recent proposals by physicists Nima Arkani-Hamed, Savas Dimopoulos and Gia Dvali have proposed that it may be possible to change what we mean by 'Planck Length'---it could be that some of the curled up dimensions aren't curled up as much as some of the other dimensions, allowing us to probe gravity at much lower energy scales than we had previously anticipated.
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We see three spatial dimensions everyday and are aware of a 4th dimension of time. We know our universe has three dimensions for sure, not only because of the apparent shape of an object, but also because of the 'inverse square law of gravity,' which allows the force between two masses to decrease as the square of the distance, represented as 'r', between them. You can imagine the distance 'r' and the gravitational field strength as being radiated through a 3-dimensional sphere enclosing a mass. The surface area of the sphere increases as the square of the distance 'r^2' and the strength of the field is distributed in proportion. Thus, in a 4-dimensional space, similarly the surface area of the 4-dimensional sphere would emit a field that gives away with the cube of the distance 'r^3'. Therefore, adding extra dimensions, as string theory does, would mean that the gravitational field would decrease with a corresponding increase in the power of 'r'.
Even though we have not detected these extra dimensions in spacetime, the recent lab results show us that we can probe space to a distance of 200 micrometers. Theory suggests a visible dimension curled up to about 100 micrometers - so you can imagine, we are half way there. The smallest surface around the mass where we can experimentally measure the gravitational field, would enclose the extra dimension searched for, and they would have no effect on gravity at larger distances. There is even a theory that there is a baby universe curled up into the sixth dimension of space, see 'the theory of hyperspace.' It is these hypothetical 'larger scale dimensions' that fits in neatly with the so-called 'brane theory,' or also known as 'membrane theory,' which is an extension to a multi-dimensional string theory. 'M-theory' stands for many expressions, such as magic, mother, mystery and of course, membrane theory. The 'm' itself however, has been attacked, by scientists calling it the 'moron' theory. M-theory added an extra dimension onto the existing dimensions of string theory - before M-theory; string theory was a '1-brane theory'. It was the realization in the mid 1990's that the string theory itself could be extended to allow higher dimensional objects.
The introduction of string theory introduces branes which are 'spatially extended objects'. The variable 'p' is for the spatial dimension of a particle; thus 0-brane means a zero dimensional particle. A 1-brane is a string and a 2-brane is a 'membrane' ect. ect. Membrane-theory brought with it an extra dimension of space, and the 'fundamental string', or 'F-string' became a 2-dimensional membrane, called a 'supermembrane.' Membrane-theory has brought other new and bizarre ideas to physics, such as the 'Holographic Principle.'
Plato, the Greek philosopher (427bc - 347bc), wrote a series of dialogues which summarized many things he had learned from his great teacher Socrates, who was executed in the year 399bc. One famous dialogue was called the 'Allegory of the Cave.' It describes a disturbing picture where people are chained to the ground inside of a cave, circling a fire, which cast their shadows on the walls of the cave. However, one escapes the prison, and went out into the light of the day and see's his true reality. When he returns to the captives inside of the cave, he tells them about the deception, but they all mock him for talking absurdities. In 1993, a Dutch theorist and physicist G. t' Hooft put forward a rather bold proposal, using Plato's Allegory of the Cave. The theory became to be called the Holographic Principle. The idea arose based on two assumptions; The first says that all the information contained in some region of space can be represented as a Hologram - it is ultimately a theory that exists on the boundary of that region of space. The second assertion is that the theory on the boundary be allowed at least one degree of 'Planck area.' The Planck area is a very small 'square' measurement which has a side length equal to that of the Planck length, which is 1.6 x 10^-33 centimeters. Moving on, the principle suggests a strange reality, where everything physical in our universe is nothing more than shadows on a wall! M-Theory predicts that our 4-dimensional continuum is just the boundary of a larger space. If we could move away from this wall, this apparent restriction of reality, we would be moving into the 5th-dimension which is curled up into a space smaller than a infinitesimal size of a superstring. Move around in the dimension and you would start to shrink to the size of superstring and then back to normal size! Even if one moved into the fifth dimension, you would end up where you had started. The theory of the fifth dimension was named after its inventor Oskar Klein, 'Kaluza-Klein Theory'. The Allegory of the Cave, one might say, was the first hypothetical assumption that reality as we know it was built up of much more unseen phenomenon, such as dimensions.
String theory has come under considerable attack by many physicists, especially within the last twenty years; prominent critics involve Philip Anderson, Sheldon Glashow, Lawrance Krauss and Peter Woit. The main problem, is that string theory is not testable, and is thus not falsifiable. It is in essence, a very safe theory: Though, if nothing experimental can arise from it within the next couple of decades, it will most probably fade into the past.