(adapted from & credit: MissMJ, Wikipedia)

I admit that I'm drawn to its mathematical elegance even though my understanding of its mathematics is limited. I find it's potential for bringing relativity and quantum mechanics together into one unified theory in physics exciting. Theoretical physicist Michio Kaku, cofounder of string field theory (easy introduction to this theory) and best-selling author has made great strides in helping the general public understand its promise.

I haven't explored in depth how the notion of forces and particles as strings came to be, why there is more than one string theory, and what the merits and weaknesses are. In this article I attempt to really sink my teeth into string theory, come join me.

All The Best Stories Seem to Start With Special Relativity . . .

Let me set the stage for you: It's 1930. Einstein's theory of special relativity is an enormous breakthrough in our understanding of space and time, and it all has to do with light. He realized that the speed of light was the same no matter how fast, in what direction, or where the light source was traveling. This means that photons of light will travel at 3 x 10

^{8}m/s, in a vacuum, whether they come from a flashlight shone out the front window of a space ship traveling at 3/4 light speed relative to an observer or if they come from a flashlight held at rest relative to an observer. The key point to this description of light is the notion of one measurement being relative to another. Any observer of light will measure light speed to be 3 x 10

^{8}m/s, in a vacuum. Physicists call light Lorentz invariant and that simply means it doesn't change.

But if the speed of light doesn't change then how does the spaceship example make sense? Does light from the flashlight know how to slow just enough to maintain light speed? No. And this simple answer has huge implications! It means that time and space must change - they are not Lorentz invariant. This is exactly the point at which Einstein transcended Newton's concept of a universe operating within an unchanging space and with time as an external parameter.

We've Got Space and Time Worked Out . . .

When an object travels close to the speed of light, time (relative to an observer at rest) slows down. This is called time dilation. In fact, time even stops for an object traveling at light speed, such as a photon of light. It stops according to an observer at rest. For the object itself time is ticking along as usual. This point here is the core concept of relativity. Space is also affected. If an observer at rest watches an object, lets say some kind of ultrafast motorbike whizzing past at close to light speed, that bike will appear squished flat perpendicular to the observer. That effect is called length contraction. Time dilation and length contraction are two logical consequences of the Lorentz invariance of light speed. Weird as they are, they are both well verified through experiment. This 9-minute animated video may help you visualize special relativity:

From special relativity we get spacetime, a four-dimensional stretchy fabric, where time and space have an inversely proportional relationship to each other. As time is "stretched," space is "squeezed." As space is stretched, time is squeezed.

. . . Except for How Subatomic Particles Act in It.

Special relativity was a fantastic leap forward in understanding but now there is a catch: Schrodinger's equation. This highly successful equation, one of the cornerstone theories of quantum mechanics (the physics of the very small), was formulated in 1925. It describes the motion of molecules, atoms and subatomic particles, much like Newton's second law of motion describes motion in classical mechanics. It very successfully describes a quantum system, and it, like special relativity, has been well verified through experiment.

The problem is that it isn't a relativistic theory; it doesn't take into account Lorentz transformations. Put another way, it can't describe particles traveling at or near the speed of light because it does not take into account how space and time behave at those speeds.

This Problem Ushers in A New Field of Study: Relativistic Quantum Field Mechanics

Out of this this impasse came some groundbreaking theoretical breakthroughs. A new field called relativistic quantum mechanics was born based on two mathematical equations that attempt to deal with time and space: the Klein-Gordon equation and the Dirac equation. These two equations bring the "relativistic" into quantum mechanics.

Electromagnetism Is Worked Out First

From this new field of study came quantum electrodynamics (QED) pioneered by Richard Feynman in the1940's. This theory was a huge breakthrough in physics. It describes how light and matter interact with each other (electromagnetism) through an exchange of virtual photons. This is the first solved relativistic quantum theory. In other words, it is the first theory where the equations of quantum mechanics and special relativity agree with each other. It perfectly predicts phenomena such as something called the anomalous magnetic moment of the electron as well as the Lamb shift of the energy levels of hydrogen atoms.

First Accidental Sighting of String Theory

Relativistic quantum mechanics works great for electromagnetism, one of the four fundamental forces in the universe. The other three fundamental forces - strong, weak and gravity - haven't been nearly so cooperative. Physicists trying to describe the strong force in a relativistic quantum manner came up with something called a dual resonance model in the 1960's. Researchers realized that the mathematical description of the strong force took the shape of a 1-dimensional string. In fact the model was a quantum theory of a relativistic vibrating string! They hoped that these strings, corresponding to massless 1- and 2-spin virtual subatomic particles, could describe interactions between hadrons inside atomic nuclei. This string-based description made many predictions that were then soundly contradicted by experimental findings. In a few years the scientific community lost interest in this so-called string theory of the strong force.

Electroweak Force and Strong Force Are Worked Out

Meanwhile, the weak force was successfully described by combining it with electromagnetism into the electroweak theory in the 1970's. The same wisdom was applied to the strong force soon after, resulting in another successful theory called quantum chromodynamics. Now, three of the four fundamental forces could be described, all of them using Feynman's famous diagrams showcasing virtual subatomic fundamental force-carrying particles. This is an example of a Feynman diagram, showing the radiation of a gluon when an electron (e

^{-}) and its antimatter twin, a positron (e

^{+}), are annihilated. They produce a gamma photon (wavy blue line) that becomes a quark/antiquark pair. One of particles of this pair radiates a gluon (green wavy light).

(credit: Joel Holdsworth, Wikipedia)

Aside: Notice, for interest's sake, how antimatter particles move backward through time? At the quantum level, time moves in both directions, an action forbidden by the second law of thermodynamics. We'll be talking about this important law later on in this article. If you are curious about time, please see my article on the subject.

Electromagnetism is mediated by the virtual photon. The electroweak force is mediated by virtual bosons. And the strong force is mediated by virtual gluons. These theories are successful relativistic quantum theories.

Now, What About Gravity?

This left pesky gravity. Many physicists have tried to formulate some kind of quantum gravity theory. None worked. As physicists would say, none were renormalizable. That means that unless you can treat all the values, which arise as infinities (a process called renormalizing) in a mathematical formulation, you're hooped. It doesn't work.

In the framework of quantum field theory, an extension of quantum mechanics, something called a first rank tensor describes the three fundamental force theories. A tensor is a geometric object that describes the density, flux of energy and momentum in spacetime. It's an all-around word that describes matter, radiation and force fields. This is what a stress tensor looks like in a 3-dimensional Cartesian coordinate system:

(credit: Sanpaz, derivative work: TimothyRias, Wikipedia)

The mathematics of quantum field theory suggests that gravitation can be described by something called a stress-energy tensor. It is a second rank tensor compared to the first rank tensor of other three fundamental forces. This tensor happens to describe a massless spin-2 particle that could mediate the force of gravity, a particle analogous to the virtual photon, bosons and gluon. This massless spin-2 particle would "act" like gravitation because it interacts with the stress-energy tensor the same way a gravitational field does. This means that if you find a massless spin-2 particle, you can rest assured you've discovered the quantum field counterpart for gravity. The problem is that gravitons would not be easy to find.

To a few physicists this spin-2 particle had an eerily familiar ring to it. Didn't that old dual resonance model say something about a massless spin-2 particle for the strong force?

So, our stage has been set, and this is where we pick up on how string theory came to be.

String Theory as a Relativistic Quantum Theory of Gravity. Or not.

Our current understanding of gravity is that it is an emergent property of spacetime. It is based on Einstein's extension of special relativity to his theory of general relativity, which describes gravity. Objects with mass curve spacetime, and we observe that curvature as gravity, visualized in this 4-minute video:

This understanding works great except at the extremes at both ends.

Singularities Don't Work

Gravity around the most massive objects in the universe, black holes, is reduced to a singularity - infinite gravity confined to a single point. Singularities such as this one are generally a sign that there is a problem with the mathematics behind the theory. The theory is not renormalizable. At the other extreme, gravity is not even a variable within quantum mechanics equations describing subatomic behaviour. Unless you are trying to describe the behaviour of collapsed atoms (where the effects of gravity are massive), leaving gravity, a relatively weak force, out of quantum descriptions is not a problem. But this also means that we can't describe the behaviours of particles moving near the speed of light, where relativistic effects (on time, length and gravity) become significant.

Black holes (composed of collapsed atoms) are a unique laboratory where both quantum effects and significant gravitational effects merge into one phenomenon. Only a relativistic quantum theory of gravity could adequately describe a black hole. Incidentally, only such a theory could describe the beginning of the Big Bang as well. Like relativity, quantum mechanics does not permit singularities either - particles cannot inhabit a space smaller than their wavelengths. Both general relativity and quantum mechanics break down at black hole (and Big Bang) singularities.

Strings Don't Have Singularities

String theory gets around the singularity problems in a mathematically elegant way. It offers a smooth two-dimensional surface that is analogous to the Feynman diagrams describing the other three fundamental forces. Tiny 1-dimensional strings are mathematical loop integrals over this surface, with their minimum length being Planck length, about 10

^{-35}m. Something called the Ployakov action describes how these strings want to contract to minimize their potential energy like springs do, but conservation of energy keeps them from disappearing.

This construct avoids the zero distance and infinite momentum problems of such integrals over particle, or point, loops (the mathematical way of saying singlularity). For strings, the relationship between distance and momentum doesn't break down at a singularity. It becomes a measure of string tension instead. Just like a guitar string, as the tension in a fundamental string is increased, the wave velocity and the normal frequency increase. In a guitar you hear a higher note. In a fundamental string you get a different particle.

How To Build A Relativistic Quantum String Theory

The Ployakov action above works for nonrelativistic strings. But what if the wave velocity approaches light speed? You need to incorporate the formulas that describe Lorentz transformation properties. This gets tricky: think about a string vibrating. Its oscillation in space and time sweeps out a two-dimensional surface in spacetime. Physicists call that surface a world sheet, and the division of space and time depends on the observer (remember?). This diagram gives you an idea of what a worldsheet is. Notice the brane shown in blue. We will be discussing branes a little later on. Keep in mind this is a three-dimensional representation of 4-dimensional spacetime (one space dimension is left out so we can visualize the structures):

(credit: Stevertigo, en.wikipedia)

When physicists plug relativity into the formula describing strings, they discover that this fundamental string no longer resembles a guitar string. As it oscillates it's no longer tied down at either end. It travels freely through spacetime instead. They also discover there are two kinds of strings - open strings like this one and closed strings. In closed strings, the boundary conditions are periodic. The mathematical solution looks like oscillations that can move around the string in one of two opposite directions. You can think of them more simply like this: the ends are floppy on open strings and closed into a loop on closed strings. On closed strings, you can have right-mover closed string mode or a left-mover closed string mode, which we'll talk about later on.

So far, we have created a model for a classical relativistic string. Now we have to incorporate quantum mechanics. We have to make the string momentum and position obey what is called quantum commutation relations. When we incorporate these equations, we get something called quantized string oscillator modes. These modes also happen to beautifully describe the quantum state of the mass and spin of particles in a relativistic quantum field theory. Particles, in string theory, are harmonic notes on fundamental 1-dimensional strings.

Oops, Didn't Mean To Make All Those Extra Dimensions

This last part sounded very slick but there is a catch when you incorporate quantum mechanics into string theory. When you use the quantum mathematics above, you get, in addition to particles, quantum states with negative norm. Physicists call these bad ghosts, and good mathematical formulations don't have them. They aren't particles at all; they are in fact unphysical states, states with negative probability. If you increase the number of spacetime dimensions from the usual 4 to 26 (one time and 25 spatial), these unphysical states disappear. This means that the quantum mechanics is only consistent if spacetime has 26 dimensions. This sounds like exchanging one problem for another one. However, because the theory can be formulated so that the 22 excess dimensions fold up into a kind of compact manifold, leaving the familiar 3 dimensions of space and 1 dimension of time visible to the ordinary phenomena we (and physicists) encounter, the theory can still make sense.

One of the particle state solutions of a closed string with two units of spin is the massless graviton.

One Theory Becomes 4 Theories

The string theory I just described is called bosonic string theory. It was developed in the late 1960's and it was the first generic string theory formulated. As you will soon see, it doesn't describe real life particles all that well, but it's a good toy theory, one that students usually learn first when they study string theory. Bosonic string theory gives rise to 4 different string theories, depending on how you choose the boundary conditions used to solve the equations of motion. Each of these closely related theories has a graviton particle in it. It also has a tachyon particle, a hypothetical particle that moves faster than the speed of light (and backward through time). The tachyon corresponds to the lowest energy ground state string. Unlike the graviton, most physicists don't believe that tachyons, which break Lorentz invariance (break the rules of special relativity), exist. Sorry Star Trek fans.

4 Theories Become Lots Of Theories

As I mentioned, bosonic string theory isn't realistic in terms of its particles. The particles it describes all have whole integer spins (0,1, 2, etc). These are called, well no surprise, bosons. They are generally force-mediating virtual particles. In contrast, particles that make up familiar matter are called fermions. They all have a half integer spin. Examples are the quarks and electrons inside atoms. Bosons (red squares) and fermions (purple and green squares) make up the elementary particles of the Standard Model in physics, shown here:

(credit: MissMJ, Wikipedia)

Adding fermions to the formulas gives you a whole new set of negative norm states or bad ghosts. To get rid of these you have to confine the number of spacetime dimensions to ten and you have to make the theory supersymmetric, so that there are equal numbers of bosons and fermions in the mix. Here too, several string theories arise as you choose various boundary conditions for the strings. This process is even more complex than it is for bosons. There is one high note though. All of these string theories contain the graviton but none contain the problematic tachyon of the bosonic theories.

Superstring Theory's Handedness Problem

Supersymmetry string (also called superstring) theory, developed in the early 1970's, has a fermionic partner for every boson particle. So, the supersymmetric partner for a graviton would be a gravitino, with a spin 3/2. Some researchers think this particle might exist as a stable fermion with mass and it might be a component of dark matter. However, at present both it and the graviton are theoretical particles.

Supersymmetric string theories come with their own set of problems. For example, Type IIA superstring theory incorporates the handedness that massless fermions exhibit. Allow me to use a particle called a neutrino to explain what handedness means: A neutrino, a real fermion particle with a spin 1/2 and zero mass, can theoretically spin in one of two directions - with the spin axis in the same direction as its angular momentum or with the spin axis in the opposite direction of its angular momentum, as shown here:

Type IIA string theory can describe this handedness in terms of string oscillations that move in opposite directions. There is no problem with any of this right up to here: The theory predicts that every fermion has a partner of opposite handedness. In real life however, this doesn't pan out. All neutrinos, fermions that travel at light speed, are left-handed (for fermions with mass, such as electrons and quarks, handedness is meaningless because these particles can travel at different speeds, all under light speed, and their handedness depends on the reference frame in which they are observed).

Type IIB supersymmetry theory gives the two superspace coordinates these theories use the same handedness. This means that it predicts massless fermions without partners of opposite handedness, as we see in real life. But it also means there is no way to add a gauge symmetry to the theory. The concept of gauge symmetry is a fundamental role in particle physics. All the fundamental forces (except gravity) are expressed in terms of gauge symmetry and that means we can't include any of them in Type IIB string theory.

Several Breakthroughs Lead to a New Unified Superstring Theory

As you have seen, none of these string theories gives us a complete description of reality, just various snippets of it. This led theorists to try something new. They separated the left and right oscillation modes of a string and treated them as two different theories. In 1984, they tried something crazy and it seemed to work: They realized a consistent theory could be made by combining a bosonic string theory moving in one direction with a supersymmetric theory with a single superspace coordinate moving in the opposite direction, yielding two theories, depending on direction. In doing so, the 26 spacetime dimensions of the bosonic theory dissolve into the 10 spacetime dimensions of the supersymmetric theory. In 1985, physicists realized they could achieve N=1 supersymmetry. This is a good, or at least simplifying, thing - it means that each particle has just one supersymmetric copy of itself.

They also formulated a way to compact the six extra dimensions down into a microscopic structure called a Calabi-Yau manifold, a 3-dimensional section of which is represented here:

(credit: Lunch, en.wikipedia (this image originally appeared in Scientific American magazine))

You don't see these dimensions and they don't interact in ordinary physics (or maybe they do, you'll see what I mean a little later). This series of discoveries held such promise it was referred to as the first superstring revolution, with a cover story in Discover magazine (November 1986) devoted to it.

Welcome M-Theory (Membranes, Mothers, Monsters and Magic)

A second superstring revolution in 1994 was ushered in by a new string theory called M-theory. Physicists found strong evidence that all the previous string theories could be considered as different limits of a single 11-dimensinal theory called M-theory. This theory requires the inclusion of higher dimensional objects called D-branes. In this new class of objects, open strings can end with something called Dirichlet boundary conditions, hence the "D." D-branes add a new level of mathematical richness to the theory, opening up the possibility of building cosmological models with it. We can think of D-branes as spatial dimensions, localizations in space in other words, so a D0-brane is a single point, a D1-brane is a line, D2-branes are planes, and so on:

Taking M-Theory Out For A First Spin, Tackling the Black Hole Entropy Puzzle

Black holes present a huge mystery to physicists. They break the second law of thermodynamics, one of the most basic and important laws of science. According to this law, all processes tend toward increasing entropy, which is a measure of disorder in a closed system. Black holes seem to proceed in the opposite direction. All the objects that are swallowed by a black hole are lost from the universe and that makes the universe, a closed system, simpler. It decreases the universe's overall entropy. Black holes violate the law in another related way as well. A quantum mechanical version of this second law states that information in a closed system cannot be lost. For atoms of matter that are sucked in, that means that the quantum states of the atoms are lost. Hawking radiation from black holes eventually releases information back into the universe but it is generic in nature; it doesn't preserve all the quantum information that went in. Physicists turned to an interesting formulation of string theory to help solve this paradox. Using this formulation, they can preserve quantum information in the universe even when it is sucked into black holes. The solution, which I'm going to try to describe next, is a complex mathematical journey, the kind of journey string theorists embark on everyday.

Analyzing the D-branes in M-theory, physicists came up with something called anti de Sitter/conformal field theory (AdS/CFT for short). Allow me to break down and explain this mouthful. You might recognize anti de Sitter space from my previous article called Holographic Universe (and that's a clue about the string theory solution you're going to follow here!). Anti de Sitter space is hyperbolic spacetime (one of several spacetime geometries physicists use) that behaves as special relativity says it should. A field describes a vector force, gravity for example, at every point in space. A conformal transformation is a little bit harder to explain. It works like this: Let's say you have a spherical globe of Earth and you want to stretch that out onto a flat surface, make a map in other words. If you don't want a bunch of bumps all over it you have to do a geometric conformal transformation. Even though landmasses may end up distorted on the map, the angles where latitude and longitude meet up will be preserved at 90°, just like on the original globe. In our case, we are dealing with something more complex - a mathematical conformal group (a bunch of transformations lumped together) that represents supersymmetry, along with another conformal group called an internal symmetry group.

The two conformal groups have to "talk" somehow with the anti de Sitter spacetime mathematics, and from all this you will get a description of this special field theory. Correspondence does this job for us. Correspondence is a math term that relates two different things, like a translator between two mathematical languages. Here, we are going to correspond between anti de Sitter space, a description of relativistic curved spacetime, and the product of the two groups of transformations. Something very cool happens when we do this: If you take any representation of anti de Sitter space with a certain number of dimensions you get a corresponding conformal field theory that always has one less dimension in it. Does this sound familiar to you? This AdS/CFT correspondence is a concrete realization of the holographic principle, which describes all the information of three-dimensional space limited to, spread across, and preserved on the two-dimensional space of a black hole's event horizon. In this way, information is not lost from the universe when stuff is sucked in and the information paradox of black holes is solved.

M-Theory Might Be THE Theory (When it Gets Older)

The mathematics of M-theory (the M generally stands for "membrane" but you can call it mother, monster, matrix, mystery or magic - I'm partial to monster considering its Frankenstein-ish origin) suggest something quite interesting, that string theory might not be about strings after all, that 1-dimensinal strings may actually be slices of a 2-dimensional membrane vibrating in 11-dimensional space. In doing so, it goes even further to bring all the string theories together into one theory. In other words, some theorists believe that all the other string theories are simply mathematical descriptions of different angled views of the same thing. M-theory potentially brings them all under one umbrella with the possibility that all phenomena in the universe could be described by one theory. Some physicists have great hope that M-theory will be the Theory of Everything they've long been looking for. One of them is physicist Michio Kaku. You can read his article on M-theory here.

M-theory is not a complete construction yet. So far, the theory has passed many rigorous tests for mathematical consistency, a good sign. But to be a truly successful theory, it must be able to predict phenomena, which can then be experimentally confirmed, a vital missing ingredient. Here, string theory is a bit odd: As a scientist you usually come up against some new data, a new behaviour, process or structure for example, and you try to formulate a theory based on that new information. Then you design experiments to test your theory. Did it predict your new results? String theory, on the other hand, was born from a series of mathematical formulas. The string, the D-brane, the manifolds, the graviton all live strictly inside math. They have only been "seen" in formulas. In a way string theory is a breach birth and now scientists are trying to fit the data to it. I wonder, as a string theorist, is it tempting to get lost in the beautiful math? Writing this article, I felt like I was drowning in it! String theory provides unequivocal proof that all aspiring physicists need to know their mathematics. At the end of the day, string theory, to be a successful scientific theory, must correlate with real phenomena. Well-known physicists such as Stephen Hawking, Edward Witten and Leonard Susskind believe that M-theory is a step in the right direction toward successfully describing nature at its most fundamental level. Other physicists, notably Richard Feynman and Sheldon Glashow, are not convinced that string theory can ever by verified at the energies we have currently available to us, and therefore they question its validity.

Attempts To Confirm Strings, So Far

Some researchers recently looked to the Large Hadron Collider (LHC) for signs of supersymmetric particles. These would not be easy to create as they are supposed to exist only at energies compatible with those that existed right after the Big Bang when the all the symmetries of the universe were still intact. Finding them wouldn't prove string theory either, but it would provide a good chunk of circumstantial evidence in its favour. Unfortunately the results so far have been disappointing. Supersymmetry, like M-theory which encompasses it, has a certain simplicity, beauty and mathematical elegance going for it, but those things don't mean it's a realistic theory. In this way, these results might prove to be a cautionary tale for string theorists.

A team of researchers from Vienna are trying to actually "see" string theory by taking a very close look at how tiny variations in gravity act on ultra-cold slow moving neutrons confined in a tiny cavity. Neutrons created in a fission reactor are slowed down to just 5 m/s in a material called a moderator. They are then shot between two plates only 25 um apart. The upper plate absorbs neutrons and the lower plate reflects them. As they go through, they trace out an arc because the only force acting on them is gravity, just like how a ball thrown sideways eventually hits the ground. When the neutrons hit the bottom plate, they are reflected off it and absorbed by the top plate. They don't reach the other end and they are not detected. If the researchers vibrate the lower plate at very specific frequencies, they find that the number of neutrons detected falls into specific resonant frequencies. Specific plate frequencies can boost the neutrons into higher energy quantum states. Using neutrons can eliminate all extraneous interactions, such as short-range electrical interactions, so that gravity's effect can theoretically be observed at the quantum level. Some researchers are looking for results that show a slight deviation from Newtonian gravity. Such a deviation could be the first direct evidence that gravity interacts with extra dimensions. It could also be evidence for axions, hypothetical particles that might make up dark matter in the universe. I have not been able to find definitive results associated with it yet.

Meanwhile, another group of physicists at the LHC came up with an ingenious way to look for the extra hidden dimensions of string theory. It rests on creating tiny Planck-sized black holes by smashing protons together. These black holes would be very unstable and immediately decay, releasing a slew of subatomic particles in the process. They figured out what kinds of particles would be created if the universe contained 10 or 11 dimensions. Even in massively energetic collisions of 4.5 TeV, no micro black holes formed. String theorists suspect that gravity should increase more rapidly with decreasing distance within dimensions higher than our usual 3 dimensions. In theory a black hole can have a minimum mass of 22 ug (Planck mass). To concentrate the same amount of equivalent energy would be beyond the capacity of our current colliders. However, in tiny region micro black hole-sized spaces, the extra-dimensional gravitational boost means that a micro black hole could form at energies as low as in the TeV range. The results are a setback for string theory, but not necessarily a knock out punch.

It is so technically daunting to test for the existence of behaviours and structures at the Planck scale, where strings and tiny Calabi-Yau manifolds of curled up extra dimensions are thought to live, that there may never be a way to directly confirm the existence of M-theory. That doesn't mean that the search for new experimental designs has slowed down at all, however.

Physicists can also approach confirmation from the opposite direction by solving the theory, reducing it down at low (everyday universe) energies into a theory of ordinary particles like electrons, protons and so on, which, if the theory is solved, should exactly match the slew of real-life particles out there. While this would not be physical confirmation in itself, a mathematically complete theory that could predict subatomic particles would be very compelling. Unfortunately, reaching complete mathematical solution is predicted to be extremely difficult if not impossible.

M-theory not only brings us the tantalizing possibility of describing quantum gravity, it provides a possible framework in which all particles and their interactions can be described. M-theory may even explain why gravity is such a weak fundamental force. Not all strings are confined to D-branes. The graviton is speculated to be a closed loop string and that means it is free to move about through spacetime. The other force-mediating particles are strings with endpoints that confine them to their D-branes. In this sense gravitons can "hide" among higher dimensions so that gravity might be a function of those extra dimensions. This might explain why it is so much weaker than the other three fundamental forces: In a 3-dimensional universe, gravitational attraction follows the inverse square law. When the distance between two objects is doubled, the gravitational attraction between them is reduced to 1/4. In 4 dimensions, the reduction is a cube, so it's 1/9th of the original attraction, and so on. This is the basic logic behind the micro black hole experiment at the LHC, and to some extent behind the slow neutron experiment as well, both of which are described above.

It might be wise not to fall too much in love with the construct of spacetime dimensions, and perhaps it's a waste of time looking for them. Just as the strings in M-theory may not actually be strings, some physicists (John Swartz and Paul Townsend as mentioned in Dr. Kaku's article M-Theory: The Mother of all Strings are now questioning the very idea of dimensions in M-theory. Dimensions emerge only as possible solutions to the mathematics, and they emerge only in a semi-classical context.

So this is where string theory leaves us, for now. M-theory is still very new Its complex mathematics is not finished and that kind of leaves us in an awkward place. We have this astoundingly beautiful mathematical construct, but of what exactly?

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