Don’t let the title fool you, this is a story about numbers. Numbers like 1, 1/2, -5, or 3.14, and even more exotic (not “real”) ones, which you may not have studied in school. This is a story about a number system that may provide a simple explanation as to why string theorists believe we live in a 10-dimensional world.
Numbers are everywhere. Remember the 1 shirt you put on this morning? Or the 2 blueberry pancakes you had for breakfast? What about the 1/2 kilo of apples you bought in the supermarket in the afternoon? If you visited an art museum recently, you may have even come across the peculiar number 1.618…, called the golden ratio, and present in many artworks. Whether you are a scientific researcher, a businessman or even someone who hates arithmetics, you deal with numbers on a daily basis in one form or another.
Unless your work involves hard-core mathematics, you probably only deal with the number system we learn in school — the real numbers. The collection of these numbers can be represented on a straight line, with each point on it corresponding to a real number and each number to a point. Because of this, the collection is said to be one-dimensional. We can also think about this idea the other way around. We could turn it on its head, as mathematicians John C. Baez (Centre for Quantum Technologies, Singapore) and John Huerta (University of California, Riverside) explain on the May 2011 issue of Scientific American, “the line is one-dimensional because specifying a point on it requires one real number.”
In a world with more than one dimension, it seems rather reductive that the numbers most people deal with can be represented on a line. In reality, there are number systems that go beyond this one-dimensional representation. The simplest example are the complex numbers.
Suppose you want to know what number multiplied by itself gives -1. In mathematical terms, you want to find x such that x*x = -1. If you restrict yourself to the collection of real numbers, you say this equation has no solutions because, when you multiply a number by itself in this system the result is always positive: there are no real square roots of negative numbers.
But you can attempt to solve the equation by imagining there is a number, let’s call it i, equal to the square root of -1. This is the sort of trick Renaissance mathematicians used to solve increasingly complex equations, so as to be able to proceed when an equation of the sort x*x = -1 arose somewhere in their calculations.
“They had no idea what the negative square root was, but then later on in the calculation you might wind up squaring again and then you’d get an ordinary negative number. The negative square root would only show up in the intermediary steps in a calculation and so they were called imaginary numbers, because you didn’t really know what they were but you could imagine that they made sense,” Baez told Plus Magazine a few years ago.
In the centuries that followed the invention of i, scientists and thinkers debated whether the square root of -1 really existed, or if it was simply a trick. Still, mathematicians kept using this “secret weapon” in their calculations. They began working with a new number system, called the complex numbers, which have the form a + b*i, with a and b being real numbers.
While each real number provides one coordinate specifying a point on the real line, complex numbers give two coordinates, (a,b). In 1806, mathematician Jean-Robert Argand published the idea that complex numbers describe points on the (two-dimensional) plane. “How does a + b*i describe a point on the plane? Simple: the number a tells us how far left or right the point is, whereas b tells us how far up or down it is,” Baez and Huerta state.
It is clear that in mathematics it is useful to have a more general number system, one that allows us to solve more complex equations. But, given that at the core of such a system is an abstract number, the imaginary i, are the complex numbers useful to describe the real world?
In fact, complex numbers are very handy to describe a multitude of physical phenomena. One example is electromagnetism, the force responsible for the interactions between charged particles. As the name suggests, electromagnetism involves two quantities, the electric field and the magnetic field. Rather than studying electromagnetic interactions using two real quantities, they can be combined into a single complex number, which simplifies the description of electromagnetic events.
If a two-dimensional number system turns out to be more useful than real numbers in solving mathematical problems and describing physical phenomena, perhaps a higher-dimensional system is even better. However, as the Irish mathematician William Rowan Hamilton discovered in the 1800s, generalizing number systems to higher dimensions is not straightforward.
The reason why complex numbers work so well is that we can do the four basic operations — addition, subtraction, multiplication, and division — with them. More, we know what those operations represent geometrically.
The same is also true of real numbers, of course. And we also know what happens to the real line when we do basic operations. For example, adding 1 to the real numbers slides the line to the right 1 unit; subtracting by a positive number slides the line to the left. Multiplying by a positive number stretches the line: all points get further away from each other. Conversely, dividing by a positive number squashes the line. And if we multiply by -1, we flip the line over: negative numbers become positive and vice versa.
In the same way real-number operations modify the real line, operations with complex numbers describe movements of the complex plane. The most interesting changes to this plane are those brought about by multiplication and division. These operations stretch and shrink the plane, and they can also rotate it. In particular, multiplying the complex numbers by the imaginary number i rotates the plane a quarter of a turn. (Rotating again gives everything pointing the opposite way we started with. Therefore, this geometrical interpretation of multiplication is consistent with the definition of the imaginary number as the number i such that i*i = -1.)
In the 1800s, Hamilton spent many years trying to find a three-dimensional number system, (a,b,c), in which he could do the four basic operations. He eventually discovered such a task was mathematically impossible. The reason is best understood geometrically: the motions in three dimensions — rotations, stretching and shrinking — can not be described with just three numbers, a fourth is needed.
“It may seem odd that we need points in a four-dimensional space to describe changes in three-dimensional space, but it is true. Three of the numbers come from describing rotations, which we can see most readily if we imagine trying to fly an airplane. To orient the plane, we need to control the pitch, or angle with the horizontal. We also may need to adjust the yaw, by turning left or right, as a car does. And finally, we need to adjust the roll: the angle of the plane’s wings. The fourth number we need is used to describe stretching or shrinking,” Baez and Huerta write.
As Hamilton concluded, while a three-dimensional number system (a,b,c) is of no use, a four-dimensional one (a,b,c,d) — the quaternions — is. Although this number system hasn’t been used much since the time of Hamilton, it still has real-world applications today. A practical use is the representation of three-dimensional rotations on a computer, with applications from video games to attitude-control system of a spacecraft.
As before, we can ask: can we go further? Can we come up with an even higher-dimensional number system where we can add, subtract, multiply, and divide? As it turns out, the next such a system doesn’t have 5, 6, nor 7 dimensions, but 8. This discovery was done by Hamilton’s friend John Graves, and shortly after by Arthur Cayley, although the latter published the result first. The eight-dimensional number system, sometimes called Cayley numbers, are know as the octonions.
But what is this eight-dimensional system good for? The quaternions already describe motions in our three-dimensional world, so perhaps there is no practical use for the octonions.
What if the world has more than three dimensions?
That may strike you as an odd question. However, theoretical physicists have been working for years on a theory describing the overall universe that carries the promise of extra (albeit “small”) dimensions: string theory.
If the universe has extra dimensions, why can’t we see them? If the flea walks to one side it eventually goes around the rope ending back where it started. So, while for the acrobat there is only one dimension, the flea has two with one of them being a small closed loop. Source: The Particle Adventure, Berkeley Lab.
String theory attempts to unify quantum mechanics, which explains the microscopic world, with general relativity, which describes the larger scales, where gravity is the dominant force. In string theory, this unification is attempted by modeling elementary particles as vibrations of tiny strings. Strangely enough, experts say that the theory is only self-consistent if the universe has 10 dimensions, 9 spacial dimensions plus time. (You may have heard that string theory works in 11 dimensions, not 10. The extra dimension comes about when you consider basic particles to be the vibrations of two-dimensional membranes rather than one-dimensional strings.)
At the core of modern versions of string theory is yet another insightful idea, one that researchers claim to be too elegant to be wrong (even though it has yet to be verified experimentally). The theory is called supersymmetry. As Baez and Huerta write in their article, supersymmetry “states that at the most fundamental levels, the universe exhibits a symmetry between matter and the forces of nature. Every matter particle (such as an electron) has a partner particle that carries a force. And every force particle (such as a photon, the carrier of the electromagnetic force) has a twin matter particle.”
Supersymmetry also states that the laws of physics would remain the same if every matter particle was traded for a force particle and vice versa. In other words, supersymmetry provides a unified description of matter and forces.
At this stage it is important to recall the basic idea of quantum mechanics. This branch of physics — for which we do have experimental evidence — states that particles are also waves. Mathematically, in the standard three-dimensional version of quantum mechanics, this means that “one type of number (called spinors) describes the wave motion of matter particles. Another type of number (called vectors) describes the wave motion of force particles,” Baez and Huerta explain. In this view, multiplication between two of these numbers describes the interaction between two particles.
In the unified description of supersymmetry, all particles use the same number system — one where we know how to describe the basic operations so that we can represent interactions. Whether they are matter or force particles, their wave motion should be represented by the same type of number. This is where the octonions can come about. In a universe with 8 spatial dimensions, this number system would be the octonions.
But why 8 dimensions and not the 10 string theory requires? What are the missing two dimensions? As it so happens, strings, the basic elements of string theory, trace out two-dimensional surfaces as they move in time. This can be visualized using a stick of chalk (representing a string) placed on a black board. As you drag the chalk (representing time going by), smudging the board, you trace out a two-dimensional surface. As explained by Baez and Huerta, “This evolution changes the dimensions in which supersymmetry arises, by adding two—one for the string and one for time.”
“So if string theory is right, the octonions are not a useless curiosity: on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions,” they conclude.
It seems the world isn’t real, nor complex. If Baez, Huerta and string theorists are right, the world is octonion.
Reference: “”http://www.scientificamerican.com/article.cfm?id=the-strangest-numbers-in-string-theory">The Strangest Numbers in String Theory", by John C. Baez and John Huerta, May 2011 issue of Scientific American (subscription required)