A couple of months ago I flew up to Corvallis, Oregon to an AMS meeting. The AMS, in case you're unfamiliar with it, is the American Mathematical Society. They have lots of regional meetings with special sessions on various topics. One reason I went to this one is that there was a special session on octonions, organized by Tevian Dray and Corinne Manogue.
After the real numbers come the complex numbers, and after the complex numbers come the quaternions, and after the quaternions come the octonions, the most mysterious of all. The real numbers, complex numbers, and quaternions have lots of applications to physics. What about the octonions? Aren't they good for something too? This question has been bugging me for a while now.
In fact, it bugs me so much that I decided to go to Corvallis to look for clues. After all, in addition to Tevian Dray and Corinne Manogue - the former a mathematician, the latter a physicist, both deeply interested in octonions - a bunch of other octonion experts were going to be there. One was my friend Geoffrey Dixon. I told you about him in "week59". He wrote a book on the complex numbers, quaternions and octonions and their role in physics. He has a theory of physics in which these are related to electromagnetism, the weak force, and the strong force, respectively. It's a bit far out, but far from crazy! In fact, it's fascinating.
After writing about his book I got in touch with him in Cambridge, Massachusetts. I found out that his other main interest in life, besides the octonions, is the game Myst. This is probably not a coincidence. In both the main question is "What the heck is really going on here?" He has Myst all figured out, but he loves watching people play it, so he got me to play it for a while. Someday I will buy a CD-ROM drive and waste a few weeks on that game. Anyway, I got to know him back in the summer of 1995, so it was nice to see him again in Corvallis.
Another octonion expert is Tony Smith. He too has a far-out but fascinating theory of physics involving octonions! I wrote about his stuff in "week91". I had never met him before the Corvallis conference, but I instantly recognized him when I met him, because there's a picture of him wearing a cowboy hat on his homepage. It turns out he always wears that hat. He is a wonderful repository of information concerning octonions and other interesting things. He is also a very friendly and laid-back sort of guy. He lives in Atlanta, Georgia.
I also met another octonion expert I hadn't known about, Tony Sudbery, from York. (The original York, not the "new" one.) He gave a talk on "The Exceptions that Prove the Rule". The octonions are related to a host of other mathematical structures in a very spooky way. In all sorts of contexts, you can classify algebraic structures and get a nice systematic infinite list together with a finite number of exceptions. What's spooky is how the exceptions in one context turn out to be related to the exceptions in some other context. These relationships are complicated and mysterious in themselves. It's as if there were a hand underneath the water and all we see is the fingers poking out here and there. There seems to be some "unified theory of exceptions" waiting to be discovered, and the octonions must have something to do with it. I figure that to really understand what the octonions are good for, we need to understand this "unified theory of exceptions".
Let's start by recalling what the octonions are!
I presume you know the real numbers. The complex numbers are things like
a + bi
where a and b are real. We can multiply them using the rule
i2 = -1
They may seem mysterious when you first meet them, but they lose their mystery when you see they are just a nice way of keeping track of rotations in the plane.
Similarly, the quaternions are guys like
a + bi + cj + dk
which we can multiply using the rules
i2 = j2 = k2 = -1
ij = k, jk = i, ki = j
ji = -k, kj = -i, ik = -j
They aren't commutative, but they are still associative. Again they may seem mysterious at first, but they lose their mystery when you see that they are just a nice way of keeping track of rotations in 3 and 4 dimensions. Rotations in more than 2 dimensions don't commute in general, so the quaternions had better not commute. In fact, Hamilton didn't invent the quaternions to study rotations - his goal was merely to cook up a "division algebra", where you could divide by any nonzero element (see "week82"). However, after he discovered the quaternions, he used them to study rotations and angular momentum. Nowadays people tend instead to use the vector cross product, which was invented later by Gibbs. The reason is that in the late 1800s there was a big battle between the fans of quaternions and the fans of vectors, and the quaternion crowd lost. For more on the history of this stuff, see:
1) Michael J. Crowe, A History of Vector Analysis, University of Notre Dame, Notre Dame, 1967.
Octonions were invented by Cayley later on in the 1800s. For these, we start with seven square roots of -1, say e1 up to e7. To learn how multiply these, draw the following diagram:
Draw a triangle, draw a line from each vertex to the midpoint of the opposite edge, and inscribe a circle in the triangle. Label the 7 points shown with e1 through e7 - it doesn't matter how, I've just drawn my favorite way. Draw arrows on the edges of the triangle going around clockwise, draw arrows on the circle also going around clockwise, and draw arrows on the three lines pointing from each vertex of the triangle to the midpoint of the opposite edge. Come on, DO it! I'm doing all this work for you... you should do some, too.
Okay. Now you have your very own octonion multiplication table. Notice that there are six lines and a circle in your picture. Each one of these gives us a copy of the quaternions inside the octonions. For example, say you want to multiply e6 and e7. You notice that the the vertical line says "e6, e7, e2" on it as we follow the arrow down. Thus, just as for i, j, and k in the quaternions, we have
e6 e7 = e2, e7 e2 = e6, e2 e6 = e7 e7 e6 = -e2, e2 e7 = -e6, e6 e2 = -e7So in particular we have e6 e7 = e2.
In case you lose your octonion table, don't worry: you don't really need to remember the names of those 7 square roots of -1 and their positions on the chart. You just need to remember the geometry of the chart itself. Names are arbitrary and don't really matter, unless you're talking to someone else, in which case you have to agree on them.
If you want to see spiffy high-tech octonion multiplication tables, check out the following websites:
2) Tony Smith, http://galaxy.cau.edu/tsmith/TShome.html
3) Geoffrey Dixon, http://www.7stones.com (Warning: to really get into this you need to have at least Netscape 3.0 with Java and Shockwave stuff.)
What's so great about the octonions? They are not commutative, and worse, they are not even associative. What's great about them is that they form a division algebra, meaning you can divide by any nonzero octonion. Better still, they form a "normed" division algebra. Just as with the reals, complexes, and quaternions, we can define the norm of the octonion
x = a0 + a1 e1 + a2 e2 + a3 e3 + a4 e4 + a5 e5 + a6 e6 + a7 e7
|x| = sqrt(a02 + a12 + a22 + a32 + a42 + a52 + a62 + a72).
What makes them a "normed division algebra" is that
|xy| = |x||y|.
It's a wonderful fact about the world that the reals, complexes, quaternions and octonions are the only normed division algebras. That's it!
However, the octonions remain mysterious, at least to me. They are related to rotations in 7 and 8 dimensions, but not as simply as one might hope. After all, rotations in any number of dimensions are still associative. Where is this nonassociative business coming from? I don't really know. This question really bugs me.
A while ago, in "week95", I summarized a paper by John Schwarz on supersymmetric Yang-Mills theory and why it works best in dimensions 3, 4, 6, and 10. Basically, only in these dimensions can you cook up spin-1/2 particles that have as many physical degrees of freedom as massless spin-1 particles. I sort of explained why. This in turn allows a symmetry between fermions and gauge bosons. I didn't explain how this works... it seems pretty tricky to me... but anyway, it works.
So far, so good. But Schwarz wondered: is it a coincidence that the numbers 3, 4, 6, and 10 are just two more than the numbers 1, 2, 4, and 8 - the dimensions of the reals, complexes, quaternions, and octonions?
Apparently not! The following papers explain what's going on:
4) Corinne A. Manogue and Joerg Schray, Finite Lorentz transformations automorphisms, and division algebras, Jour. Math. Phys. 34 (1993), 3746-3767.
Corinne A. Manogue and Joerg Schray, Octonionic representations of Clifford algebras and triality, preprint available as hep-th/9407179.
5) Anthony Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A 17 (1984), 939-955.
Anthony Sudbery, Seven types of incongruity, handwritten notes.
Here's the basic idea. Let
R = real numbers
C = complex numbers
H = quaternions
O = octonions
Let SO(n,1) denote the Lorentz group in n+1 dimensions. Roughly speaking, this is the symmetry group of (n+1)-dimensional Minkowski spacetime. Let so(n,1) be the corresponding Lie algebra (see "week63" for a lightning introduction to Lie algebras). Then it turns out that:
sl(2,R) = so(2,1)
sl(2,C) = so(3,1)
sl(2,H) = so(5,1)
sl(2,O) = so(9,1)
This relates reals, complexes, quaternions and octonions to the Lorentz group in dimensions 3, 4, 6, and 10, and explains the "coincidence" noted by Schwarz! But it requires some explanation. Roughly speaking, if SL(2,K) is the group of 2x2 matrices with determinant 1 whose entries lie in the division algebra K = R, C, H, O, then sl(2,K) is defined to be the Lie algebra of this group. This is simple enough for R or C. However, one needs to be careful when defining the determinant of a 2x2 quaternionic matrix, since quaternions don't commute. One needs to be even more careful in the octonionic case. Since octonions aren't even associative, it's far from obvious what the group SL(2,O) would be, so defining the Lie algebra "sl(2,O)" requires a certain amount of finesse. For the details, read the papers.
As Corinne Manogue explained to me, this relation between the octonions and Lorentz transformations in 10 dimensions suggests some interesting ways to use octonions in 10-dimensional physics. As we all know, the 10th dimension is where string theorists live. There is also a nice relation to Geoffrey Dixon's theory. This theory relates the electromagnetic force to the complex numbers, the weak force to the quaternions, and the strong force to octonions. How? Well, the gauge group of electromagnetism is U(1), the unit complex numbers. The gauge group of the weak force is SU(2), the unit quaternions. The gauge group of the strong force is SU(3)....
Alas, the group SU(3) is not the unit octonions. The unit octonions do not form a group since they aren't associative. SU(3) is related to the octonions more indirectly. The group of symmetries (or technically, "automorphisms") of the octonions is the exceptional group G2, which contains SU(3). To get SU(3), we can take the subgroup of G2 that preserves a given unit imaginary octonion... say e1. This is how Dixon relates SU(3) to the octonions.
However, why should one unit imaginary octonion be different from the rest? Some sort of "symmetry breaking", presumably? It seems a bit ad hoc. However, as Manogue explained, there is a nice way to kill two birds with one stone. If we pick a particular unit imaginary octonion, we get a copy of the complex numbers sitting inside the octonions, so we get a copy of sl(2,C) sitting inside sl(2,O), so we get a copy of so(3,1) sitting inside so(9,1)! In other words, we get a particular copy of the good old 4-dimensional Lorentz group sitting inside the 10-dimensional Lorentz group. So fixing a unit imaginary octonion not only breaks the octonion symmetry group G2 down to the strong force symmetry group SU(3), it might also get us from 10-dimensional physics down to 4-dimensional physics.
Cool, no? There are obviously a lot of major issues involved in turning this into a full-fledged theory, and they might not work out. The whole idea could be completely misguided! But it takes guts to do physics, so it's good that Tevian Dray and Corinne Manogue are bravely pursuing this idea.
Upon learning that there is a deep relation between R, C, H, O and the Lorentz group in dimensions 3, 4, 6, 10, one is naturally emboldened to take seriously a few more "coincidences". For example, in "week82" I described the Clifford algebras C_n - i.e., the algebras generated by n anticommuting square roots of -1. These Clifford algebras are relevant to n-dimensional Euclidean geometry, as opposed to the Clifford algebras relevant to n-dimensional Lorentzian geometry, which appeared in "week93". They go like this:
C0 = R
C1 = C
C2 = H
C3 = H + H
C4 = H(2)
C5 = C(4)
C6 = R(8)
C7 = R(8) + R(8)
C8 = R(16)
where K(n) stands for n x n matrices with entries taken from K = R, C, or H, and "+" stands for "direct sum". Note that C8 is the same as 16 x 16 matrices with entries taken from C0. That's part of a general pattern called "Bott periodicity": in general, Cn+8 is the same as 16 x 16 matrices with entries taken from Cn.
Now consider the dimension of the smallest real representation of Cn. It's easy to work this out if you keep in mind that the smallest representation of K(n) or K(n) + K(n) is on Kn - the vector space consisting of n-tuples of elements of K. We get
The dimension of the smallest real representation:
of C0 is 1
of C1 is 2
of C2 is 4
of C3 is 4
of C4 is 8
of C5 is 8
of C6 is 8
of C7 is 8
of C8 is 16
Note that it increases at n = 1, 2, 4, and 8. These are the dimensions of R, C, H, and O. Coincidence?
No! Indeed, Cn has a representation on a k-dimensional real vector space if and only if the unit sphere in that vector space, Sk-1, admits n linearly independent smooth vector fields. So the above table implies that:
The sphere S0 admits 0 linearly independent vector fields.
The sphere S1 admits 1 linearly independent vector fields.
The sphere S3 admits 3 linearly independent vector fields.
The sphere S7 admits 7 linearly independent vector fields.
These spheres are the unit real numbers, the unit complex numbers, the unit quaternions, and the unit octonions, respectively! If you know about normed division algebras, it's obvious that these sphere admit the maximum possible number of linear independent vector fields: you can just take a basis of vectors at one point and "left translate" it to get a bunch of linearly independent vector fields.
Now - Bott periodicity has period 8, and the octonions have dimension 8. And as we've seen, both have a lot to do with Clifford algebras. So maybe there is a deep relation between the octonions and Bott periodicity. Could this be true? If so, it would be good news, because while octonions are often seen as weird exceptional creatures, Bott periodicity is bigtime, mainstream stuff!
And in fact it is true. More on Bott periodicity and the octonions coming up next Week.
Addendum: Robert Helling provided some interesting further comments on supersymmetric gauge theories and the division algebras, which I quote below. He recommends the following reference:
6) J. M. Evans, Supersymmetric Yang-Mills theories and division algebras, Nucl. Phys. B298 (1988), 92-108.
and he writes:
Let me add a technical remark that I extract from Green, Schwarz, and Witten, Vol 1, Appendix 4A.
The appearance of dimensions 3,4,6, and 10 can most easily been seen when one tries to write down a supersymmetric gauge theory in arbitrary dimension. This means we're looking for a way to throw in some spinors to the Lagrangian of a pure gauge theory:
in a way that the new Lagrangian is invariant (up to a total derivative) under some infinitesimal variations. These describe supersymmetry if their commutator is a derivative (a generator of spacetime translations). As usual, we parameterize this variation by a parameter ε, but now ε is a spinor.
From people that have been doing this for their whole life we learn that the following Ansatz is common:
δAm = i/2 ε† Γm ψ
δ ψ = -1/4 Fmn Γmn ε
Here A is the connection, F its field strength and ψ a spinor of a type to be determined. I suppressed group indices on all these fields. They are all in the adjoint representation. Γ are the generators of the Clifford algebra described by John Baez before.
[Also, ε† is my feeble subsitute for an ε with a bar over it. - JB]
For the Lagrangian we try the usual Yang-Mills term and add a minimally coupled kinetic term for the fermions:
-1/4 F2 + ig/2 ψ† Γm Dm ψ
Here Dm is the gauge covariant derivative and g is some number that we can tune to to make this vanish under the above variations. When we vary the first term we find g = 1. In fact everything cancels without considering a special dimension except for the term that is trilinear in ψ that comes from varying the connection in the covariant derivative in the fermionic term. This reads something like
fabc ε† Γm ψa ψb Γm ψc
where I put in the group indices and the structure constants fabc. This has to vanish for other reasons since there is no other trilinear term in the fermions available. And indeed, after you've written out the antisymmetry of f explicitly and take out the spinors since this should vanish for all choices of ψ and ε. We are left with an expression that is only made of gammas. And in fact, this expression exactly vanishes in dimensions 3, 4, 6, and 10 due to a Fierz identity. (Sorry, I don't have time to work this out more explicitly.)
This is related to the division algebra as follows (as explained in the papers pointed out by John Baez): Take for concreteness d = 10. Here we go to a light-cone frame by using coordinates
x+ = x0 + x9 and
x- = x0 - x1.
Then we write the Γm as block matrices where Γ+ and Γ- have the +/- unit matrix as blocks and the others have γi as blocks where γi are the SO(8) Dirac matrices (i=1,...,9). But they are intimately related to the octonions. Remember there is triality in SO(8) which means that we can treat left-handed spinors, right-handed spinors and vectors on an equal basis (see week61, week90, week91). Now I write out all three indices of γi. Because of triality I can use i,j,k for spinor, dotted spinor and vector indices. Then it is known thatγijk = cijk for i,j,k<8 δij for k=8 (and ijk permuted) 0 for more than 2 of ijk equal 8.is a representation of Cliff(8) if cijk are the structure constants of the octonions (i.e. ei ej = cijk ek for the 7 roots of -1 in the octonions).
When you plug this representation of the Γ's in the above mentioned γ expression you will will find that it vanishes due to the antisymmetry of the associator
[a,b,c] = a(bc) - (ab)c
in the division algebras. This is my understanding of the relation of supersymmetry to the divison algebras.
© 1997 John Baez