THERE IS MORE TO IT
Throughout middle school years, the first chapter of the Mathematics textbook was about real numbers. Before they could learn about another number system, some of us bid adieu to Math forever. The rest of us went on to learn, in our last years at school, about complex numbers.
But these are just two of the many possible number systems. Some of the advanced systems were, although discovered a couple of centuries ago, remained just another chapter in the history of Mathematics until String Theory came into the picture and indeed if String theory is a correct representation of the universe, these number systems may explain why the universe has the number of dimensions it has.
THE IMAGINARY TURNS REAL
Fundamental Property of a line – The line is one dimensional because specifying a point on it requires one real number. Before the 1500s, mathematicians played with real numbers only. Enter the European Renaissance. The scientific community became competitive and to solve more complex questions, new tricks came to light. An Italian mathematician introduced one such ‘never seen before’ trick – the square root of -1. Although a debatable issue, he published this idea in his book Ars Magna. Centuries later, Descartes named this square root as imaginary, abbreviated as i.
All the same, mathematicians kept doing work on numbers of the form a + bi, where a and b are real numbers. In 1806, Jean-Robert Argand put forward the idea that complex numbers describe points on a plane – given a reference, a tells us how far left or right the point is, whereas b tells us how far up or down it is. More importantly, he also showed how fundamental operations – addition, subtraction, multiplication and division - on complex numbers are just geometric manipulations in the plane.REAL vs. COMPLEX
Adding or subtracting any real number slides the real line to the left or right.
Adding (or subtraction) any complex number a +bi to a point in the plane slides that point right (or left) by an amount a and up (or down) by an amount b.
Multiplying or dividing by any positive number stretches or squashes the line.
Multiplying by a complex number not only stretches or squashes but also rotates the complex plane. Division is the opposite of multiplication, so to divide, we just shrink instead of stretching, or vice versa, and then rotate in the opposite direction
Multiplying by -1 flips the line over.
In particular, multiplying by i rotates the plane by a quarter turn. Thus, if we multiply 1 by i twice, we rotate the plane a full half-turn from the starting point to arrive at -1.
NEED ONE MORE
It was in 1835 that an Irish mathematician William Rowan Hamilton came up with ways to perform the four fundamental operations with complex numbers in such a way that they retained their geometric meaning given by Argand in 1806.
However, Hamilton didn’t stop there and started trying to come up with methods to perform these operations in 3 dimensions (triplets). He didn’t know it then but he had self assigned an impossible task.
Now, a number system is defined by its rules of four fundamental operations – add, subtract, multiply and divide. Division is the critical part – a number system in which we can divide is called division algebra. In 1958, three mathematicians proved a decades old suspicion – any division algebra must have dimensions one (real numbers), two (complexnumbers), four (quaternions) and eight (octonions). To break through the wall, Hamilton had to tweak the game rules and he finally hit the home run on October16, 1843.
While walking with his wife along the Royal Canal towards the Royal Irish Academy, Dublin, he suddenly realized (remember that the proof of requirements ofdivision algebra came a century later in 1958) that in 3D, describing rotations, stretching and shrinking can’t be done with three numbers. He needed a fourth number, thereby generating a 4-D set called quaternions of the form a+ bi + cj + dk, where i, j and k are three different square roots of -1.
“I then and there felt the galvanic circuit of thought close; and the sparks
which fell from it were the fundamental equations between i, j and k; exactly
such as I have used them ever since.”
And in a remarkable act of mathematical vandalism, he carved these equations in the stone of Brougham Bridge.Thus, we need one more dimension, a 4-D space to describe changes in 3D space. Strange, but true.
KEEP MAKING GOLD
Hamilton’s lawyer friend John Graves, who incidentally had gotten Hamilton interested in complex numbers in the first place, asked Hamilton, after his discovery, some intriguing questions- Do these square roots of -1 really exist? Or can we make them up as per our convenience? And if yes, if we can, with your alchemy, make 3 pounds of gold, why stop here?
Hamilton was uninterested this time, Graves set his mind to work and himself discovered a new 8-D number system – octaves or octonions. However he did not publish his work, which was done by Arthur Cayley in 1845. Hence, the second name for octonions– Cayley numbers.
Why was Hamilton uninterested in octonions? For what it’s worth, he had his reasons.
1. He was too absorbed in the research of his own discovery about the quaternions.
2. Now, there are some mathematical laws about multiplication that are cherished by mathematicians worldwide. Simply put, the octonions broke those rules.
Quaternions were already deviating from some of these laws. Real and complex numbers upheld the commutative law of multiplication. But not quaternions; order of multiplication was important because unlike real and complex numbers, quaternions describe rotations in 3-D.
Check it out.
Take a book,
(i) Flip it top to bottom (so that the back cover is now visible) and then give a quarter turn clockwise. Note the final orientation.
(ii) Put the book in its original state. Now, first give it a quarter turn clockwise and then flip it top to bottom. Note the final orientation.
The two orientations are different.
3. Perhaps the most important reason was that the practical uses of octonions were not clear. For more than a century, they were just another chapter in a Mathematics textbook.
It would take thedevelopment of modern particle physics – and String Theory in particular – to see how the octonions might be useful in the real world.SYMMETRICAL STRINGS
In the 1970s and 1980s, theoretical physicists developed a very appealing idea called supersymmetry.
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 photon, the carrier of electromagnetic force) has
a twin matter particle.
Simply put, the laws of physics would remain unchanged if we exchanged all the matter particles with their corresponding force particles.
Even in the absence of experimental proof, physicists all over the world hope and expect supersymmetryto be real, such is the seductive beauty of the math involved.
Quantum Mechanics is one thing, however, that has been experimentally proven. The particle nature of waves (and vice-versa) is fact and so is the use of vectors and spinors in a 3D QM model. Now, to understand particle interactions, vectors and spinors must be combined together in a simulacrum of multiplication and although the math in use today might work, the process is not very appealing at all.
Firstly, imagine a peculiar universe with space but not time. Such a universe will need only one number to describe the wave properties of both force and matter particles –meaning spinors and vectors coincide to give one type of number – a number in division algebra, thereby giving permissible dimensions as one, two, four and eight. Supersymmetry emerges naturally.
But, in the real universe, time has to be taken into consideration which shows an intriguing effect in string theory. At any moment of time, a string is a one dimensional curve or line but traces out a 2-D surface as time passes, thereby changing the dimensions in which supersymmetry can arise by plus 2 – one for the string and one for time. Hence, instead of one, two, four or eight, we have three, four, six or ten.
However, only the 10-D version works out to be free of anomalies and consistent with string theory. Also, it has the use of octonions in its foundations. Thus, if string theory is right, the octonions are not just another mathematical concept but the basis of the answer to the question – a multi-dimensional universe?
Recently, one dimensional strings (String Theory) have given way to two dimensional membranes (M-Theory), which, as time passes, trace a 3-D volume in spacetime. Thus, its plus three to the standard one, two, four and eight dimensions giving us a new set – four, five, seven and eleven. Here too, the 11-D version turns out to be the most consistent. Octonions come in use, once again.
Bursting the bubble at this point, neither of the theories have been experimentally proved and scientists have yet to see any concrete proof of supersymmetry, although there are some odds of seeing such evidence in the LHC at CERN.
Yes, it is still uncertain whether octonions are built into the intricate fabric of our universe or not, but should it be proved true in future, it would hardly be the first time that a pure mathematical invention was exactly the tool that physicists needed.
DISCLAIMER: The above is just an attempt at precis-writing to a wonderful article written by John C. Baez and John Huerta in Scientific American, May 2011. The original document can be downloaded from the following link
The original article can be found on John Baez’s website-http://math.ucr.edu/home/baez/octonions/strangest.html