From: baez@guitar.ucr.edu (John Baez)
Newsgroups: sci.physics.research,sci.physics,sci.math
Subject: This Week's Finds in Mathematical Physics (Week 59)
Date: 3 Aug 1995 15:13:31 GMT

This Week's Finds in Mathematical Physics (Week 59)
John Baez

   As you crack your eyes one morning your reason is assaulted by
   a strange sight.  Over your head, humming quietly, there floats
   a monitor, an ethereal otherworldly screen on which is written a
   curious message.  "I am the Screen of ultimate Truth.  I am bulging
   with information and ask nothing better than to be allowed to impart
   it."

It would be nice if more math books started with something attention-
grabbing like this.  In fact, it appears near the beginning of

1) Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions,
Complex Numbers and the Algebraic Design of Physics, Kluwer Press,
ISBN 0-7923-2880-6.

Dixon is convinced that the details of the Standard Model of
particle interactions can be understood better by taking certain
mathematical structures very seriously.   There are very few
algebras over the real numbers for which every element has a 
multiplicative inverse: if we demand associativity and 
commutativity, just the reals themselves and the complex 
numbers.  If we drop the demand for commutativity,
we also get a 4-dimensional algebra called the quaternions, 
invented by Hamilton.  If in addition we drop the demand for 
associativity, we also get an 8-dimensional algebra called 
the octonions, or Cayley numbers.  Clearly these are very 
special structures, and also clearly they play an important 
role in physics... or do they?  

Well, few people doubt that the real numbers are fundamental 
to physics (though some advocates of the discrete 
might prefer the integers), and with emergence of quantum 
theory, if not sooner, the basic role of the complex 
numbers also became clear.  Hamilton discovered the quaternions
in the 1800s, and used them to formulate
a beautiful theory of rotations in 3-dimensional space.  They
fell out of favor somewhat when the vectors of Gibbs proved
simpler for many purposes, but their deeper importance became
clear when people started studying spin: indeed,
the Pauli matrices so important in physics are closely related
to the quaternions, and it is the group of unit quaternions, SU(2), 
rather than the group of rotations in 3d space, SO(3), which
turns out to be the symmetry group whose different representations 
correspond to particles of different spin.  But what about the
octonions?  

Well, there are not too many places in physics yet
where the octonions reach out and grab one with the force 
the reals, complexes, and quaternions do.  But they are
certainly out there, they have a certain beauty to them,
and they are the natural stopping-point of a certain finite
sequence of structures, so it is natural for people
of a certain temperament to believe that they are there for a
reason.  Dixon makes a passionate case for this in the beginning
of his book.  

Suppose you were confronted with the Screen of Truth.  
What would you ask it?  Dixon, being a physicist,
naturally fantasizes asking it why the universe is the way it
is!  What kind of answer could this possibly have?  Perhaps
there is only one consistent way for things to be, and mathematics,
with its unique and beautiful structures that are pure expressions
of logical necessity, is trying to tell us something about this?  

On the one hand this seems outrageous... especially to the 
hard-nosed pragmatist or empiricist in us.  It seems 
old-fashioned, naive, and too good to be true.  On the other hand, the
universe *is* outrageous!  It's outrageous that it exists in the
first place, and doubly outrageous that it has the particular
physical laws it does and no others.  It has only been through
the old-fashioned, naive belief that we can understand it using
mathematics that we discovered what we have of its physical laws.  
So maybe eventually we *will* see that the basic structures of
mathematics determine, in some mysterious sense, all the basic
laws of physics.  Or maybe we won't.  In either case, there is
a long way yet to go.  As Dixon's Screen of Truth comments, before
it departs:

  Do you believe that were I to explain as much of what I know
  as you could comprehend that you would recognize it, that you 
  would say, oh yes, this is but an extension of what we have  
  already done, and though the mathematics is broader, the principles
  deeper, I am not surprised?  Do you think you have asked even a
  fraction of the questions you need to ask?

Anyway, it is at least worth considering all the beautiful mathematical
structures one runs into for their potential importance.  For example,
the octonions.  

In order to write this week's Finds, I needed to learn a little about
the octonions.  I wanted some good descriptions of the octonions, 
that hopefully would "explain" them or at least make them easy
to remember.  So I asked for help on sci.physics.research, and I
got some help from Greg Kuperberg, Ezra Getzler, Matthew Wiener,
and Alexander Vlasov.  After a while Geoffrey Dixon got wind of this
and referred me to his work!  I'll probably talk to him later this
summer when I go back to Cambridge Massachusetts, and hopefully
I'll learn more about octonions and the like.  

But for now let me just give a quick beginner's introduction to the
octonions.  A lot of this appears in 

2) William Fulton and Joe Harris, Representation Theory --- a First
Course, Springer Verlag, Berlin, 1991.

I should add that this book is a very good place to learn about
Lie groups, Lie algebras, and their representations... I wish I had
taken a course based on this book when I was in grad school!

Let's start with the real numbers.  Then the complex number

                    a+bi

can be thought of as a pair

                    (a,b)

of real numbers.  Addition is done component-wise, and 
multiplication goes like this:

            (a,b)(c,d) = (ac - db,da + bc)

We can also define the conjugate of a complex number by

                (a,b)* = (a,-b).

Now that we have the complex numbers, we can define the
quaternions in a similar way.  A quaternion can be thought of 
as a pair

                        (a,b)

of complex numbers.  Addition is component-wise and 
multiplication goes like this

            (a,b)(c,d) = (ac - d*b, da + bc*)

This is just like how we defined multiplication of
complex numbers, but with a couple of conjugates (*'s)
thrown in.  To emphasize how similar the two multiplications
are, we could have included the conjugates in the first 
formula, since the conjugate of a real number is just itself. 

We can also define the conjugate of a quaternion by

               (a,b)* = (a*,-b).

The game continues!  Now we can define an octonion to be a pair 
of quaternions; as before, we add these component-wise and
multiply them as follows:

           (a,b)(c,d) = (ac - d*b, da + bc*).

One can also define the conjugate of an octonion by

               (a,b)* = (a*,-b).

Why do the real numbers, complex numbers, quaternions
and octonions have multiplicative inverses?  I take it as
obvious for the real numbers.  For the complex numbers, 
you can check that

(a,b)* (a,b) = (a,b) (a,b)* = K (1,0)

where K is a real number called the "norm squared" of (a,b).
The multiplicative identity for the complex numbers is (1,0).
This means that the multiplicative inverse of (a,b) is (a,b)*/K.  
You can check that the same holds for the quaternions and 
octonions!  

Of course, all this should make you want to keep playing the 
game and develop a 16-dimensional algebra, the "hexadecanions,"
consisting of pairs of octonions equipped with the same sort
of multiplication law.  What do you get?  Why aren't there
multiplicative inverses anymore?  I haven't checked, because
this is my summer vacation!  I am learning about octonions 
just for fun, since I just finished writing some rather technical papers, 
and my idea of fun does not presently include multiplying two 
hexadecanions together to see why the norm-squared law (a,b) (a,b)* =
(a,b)* (a,b) = K (1,0) breaks down.  But I'm sure someone out there 
will enjoy doing this... and I'm sure someone else out there has already 
done it!  So they should let me know what happens.  There is something
out there called "Pfister forms", which I think might be related.

Now if we unravel the above definition of quaternions, by writing
the quaternion (a+bi,c+di) as a+bi+cj+dk, we see that the multiplication
law is 

i^2 = j^2 = k^2 = -1, 

and 

ij = -ji = k,  kj = -jk = i,  ki = -ik = j.

For more about the inner meaning of these rules, see "week5".
Similarly, we can unravel the above definition of octonions by 
writing the octonion (a+bi+cj+dk,e+fi+gj+hk) as

a + b e1 + c e2 + d e3 + e e4 + f e5 + g e6 + h e7.

Note: since mathematicians are very impersonal, they usually 
call these seven dwarves e1,...,e7 instead of Sleepy, Grumpy, 
etc. as in the Disney movie.   Any one of these 7 guys
times himself is -1.  Also, any two distinct ones anticommute; for
example, e3 e7 = -e7 e3.   There is a nice way to remember 
how to multiply them using the "Fano plane".  This is a
projective plane with 7 points, where by a "projective plane" I mean
that any two points determine an abstract sort of "line", which
in this case consists of just 3 points, and any two lines intersect
in a point.  It's a bit of a pain to draw this plane using ASCII!
The book by Fulton and Harris has a nice picture, but the best
I can do is:


                      e5


              e2              e3


                      e4 


  e7                  e1                   e6


The "lines" are the 3 edges of the big triangle, the 3 lines going
through a vertex, the center and the midpoint of the opposite
edge, and the circle including e1, e2, and e3.  All the "lines" are
cyclically ordered, and that tells you how to multiply the seven
dwarves.  For example, the line that's actually a circle goes
clockwise, so e1 e2 = e3, e2 e3 = e1, and e3 e1 = e2.  The lines
that are edges of the big triangle also point clockwise, so
for example e5 e3 = e6, and cyclic permutations thereof, and
e1 e7 = e6.  The lines that go through the center point
from the vertex to the midpoint of the opposite edge, so
for example e7 e4 = e3.  I hope that made sense; you can
work it out yourself, of course.

So those are the octonions in a nutshell.  I should say a bit about
how they relate to triality for SO(8), the exceptional Lie group
G2, the group SU(3) which is so important in the study of the
strong force, and to lattices like E8, Lambda_{16} and the Leech
lattice.  But I will postpone that; for now you can consult Fulton
and Harris, and also various papers by Dixon:

3) Geoffrey Dixon, Octonion X-product orbits, preprint available
as hep-th/9410202.

Octonion X-product and E8 lattices, preprint available as hep-th/9411063.

Octonions: E8 lattice to Lambda_{16}, preprint available as hep-th/9501007.

Octonions: invariant representation of the Leech lattice, preprint available 
as hep-th/9504040.

Octonions: invariant Leech lattice exposed, preprint available as 
hep-th/9506080.


I am not presently in a position to assess these papers or Dixon's work
relating division algebras and the Standard Model, but hopefully sometime
I will be able to say a bit more.

Let me wrap up by saying a bit about the Leech lattice.  As described
in my review of Conway and Sloane's book ("week20"), there is a wonderful
branch of mathematics that studies the densest ways of packing spheres in 
n dimensions.  Most of the results so far concern lattice packings, packings
in which the centers of the spheres form a subset of R^n closed under addition
and scalar multiplication by integers.  When n = 8, the densest packing is 
given by the so-called E8 lattice.  In "week20" I described how to
get this lattice using the quaternions and the icosahedron.  Briefly, it goes as
follows.  The group of rotational symmetries of the icosahedron (not counting
reflections) is a subgroup of the rotation group SO(3) containing 60 elements.
As mentioned above, SO(3) has as a double cover the group SU(2) of unit
quaternions.   So there is a 120-element subgroup of SU(2) consisting of
elements that map to elements of SO(3) that are symmetries of the icosahedron.
Now form all integer linear combinations of these 120 special elements of
SU(2).  We get a subring of the quaternions known as the ``icosians''.  

We can think of icosians as special quaternions, but we can also think of
them as special vectors in R^8, as follows.  Every icosian is of the form

(a+sqrt(5)b) + (c+sqrt(5)d)i + (e+sqrt(5)f)j + (g+sqrt(5)h)k

with a,b,c,d,e,f,g,h rational --- but not all rational values of a,...,h give
icosians.  The set of all vectors x = (a,b,c,d,e,f,g,h) in R^8 that correspond
to icosians in this way is the E8 lattice!

The Leech lattice is the densest lattice packing in 24 dimensions.  It has
all sorts of remarkable properties.  Here is an easy way to get ones hands
on it.  First consider triples of icosians (x,y,z).  Let L be the set of such 
triples with

x = y = z mod h

and 

x + y + z = 0 mod h*

where h is the quaternion (-sqrt(5)+i+j+k)/2.  Since we can think of an
icosian as a vector in R^8, we can think of L as a subset of R^{24}.  
It is a lattice, and in fact, it's the Leech lattice!  I have a bit more to
say about the Leech lattice in ``week20'', but the real place to go for 
information on this beast is Conway and Sloane's book.  It turns out
to be related to all sorts of other ``exceptional'' algebraic structures.
People have found uses for many of these in string theory, so if string
theory is right, maybe they are important in physics.  Personally, I
want to understand them more deeply as pure mathematics before worrying
too much about their applications to physics.  

-----------------------------------------------------------------------
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