My personal motivation is the notion of mathematical resonance.
In experimental physics the presence of a new particle is often made
apparent by a resonant response - a peak of activity rising above
surrounding noise. The noise is a null result; the peak signifies
existence. In mathematics there are also "resonances", algebraic
and topological dimensions that explode with intricacy and depth
compared with dimensions immediately above and below. I share a view
with many others that the connection of mathematics to physics is
not accidental, and I share an intuition with perhaps fewer
that existence is attributable in part to mathematical
resonance - that the rich and specific structure of our unique physical reality
mirrors the rich and specific structure of the few mathematical resonances.
What follows is mathematics.
What is written above is philosophy, devoid of rigor, and easily rejected.
Rejection will not effect the mathematics, only its interpretation. This material dates from 1992.
The real division algebras, C (complex numbers), Q (quaternions),
and O (octonions), as being markers of three separate mathematical
resonances encompassing concepts much broader than the simply algebraic
concepts with which they are usually associated;
Consequently my adventures in physical mathematics were played out in part
on the algebra
T = C ⊗ Q ⊗ O
(where ⊗ is the (real) tensor product symbol),
which is just the complexification of the quaternionization of the
Problem 1: Resolve the identity of T into a set of orthogonal primitive
where the last condition ensures a consistent definition of components of X
with respect to the resolution. By the way, an idempotent is primitive if
not the sum of two other nonzero idempotents.
- Ordinarily a resolution of the identity would consist of n
algebraic elements pk, k = 1,...,n, such that
||p1 +...+ pn = 1
||pk2 = pk
||pkpm = 0 (k ≠ m)
- However, that definition assumes the underlying algebra is
associative and alternative. T is neither. We must modify the
conditions to take this into account, and to fulfill the spirit of
what resolving the identity is all about (think about square matrices and
Therefore, for all X in T we demand
||pk(pk X) = pk X
||pk(pm X) = 0 (k ≠ m)
||pk(X pm) = (pk X)pm
Solution to Problem 1: Define
where x and y are arbitrary unit quaternions with no real
parts. As usual let e7 be an octonion unit, and define
|λ0 = (1 + ix)/2,
||λ1 = (1 - ix)/2,
|λ2 = (1 + iy)/2,
||λ3 = (1 - iy)/2,
|ρ+ = (1 + ie7)/2,
||ρ- = (1 - ie7)/2.
The four Δk resolve the
identity of T, satisfying each of the conditions (1), (2'), (3') and (4),
and they are primitive idempotents. Warning: I have found two other
resolutions of the identity satisfying the restricted conditions (1), (2)
and (3), but none other satisfying conditions (1), (2'), (3') and (4) (the
reader should recognize that no generality is lost in picking e7
instead of some other unit octonion with no real part to appear in the
Δk). That is, it has been
proven only in part that the general form of this resolution is unique.
Were there another form I would be both surprised and very interested
(but be careful, there are ways of writing the
Δk that look quite different
from that given above).
Problem 2: Generalize Components. If A and B are arbitrary elements of T, then
where A* is the overall complex/quaternion/octonion conjugate
of A, and the (A* • B)m are complex numbers, components
of the inner product of A and B. Our problem is to find nice transformations
on T that can be performed on A, B and
Δm on the left side of
equation (0) that will leave the right side invariant.
|| (A Δm)*
(B Δm) =
(A* • B)m
Suppose that Γm is the
result of transforming Δm.
We impose the following consistency conditions:
- For all X in T,
(associativity ensuring a consistent definition of components);
|| (Γm* X)
(invariance of components of the identity).
Solution of Problem 2: (What follows is somewhat simplified; more details
What about SU(3)? Well, that's covered very elegantly on
page 14, as well as a different approach
to U(1). Note that
the since SU(3) leaves e7 invariant, it also leaves
Γm invariant, so this SU(3) fits
in perfectly with these ideas.
There are two important differences between the SU(2) and SU(3)
invariance groups given above:
It should be emphasized that these are mathematical results, but as a consequence,
when interpreted as the basis of a physical theory, SU(3)
is exact and nonchiral, while SU(2) is broken and chiral.
- the elements Γm are invariant
under SU(3) but not SU(2);
- using the projectors λm and
ρ± of which the
Δm are composed,
T can be decomposed only to the SU(3) multiplet level
(singlet, antisinglet, triplet, antitriplet),
but all the way to
spinor component level with respect to SU(2).
To put the icing on the cake, with respect to U(1)xSU(2)xSU(3) T transforms
exactly like the direct sum of a family and antifamily of quark and lepton
Weyl (or Pauli) spinors.
For the sake of brevity, and because I spent 2 years organizing these thoughts
book, I'll quit here. The book was republished in 2002 at a reduced price: Division Algebras: Octonions, Quaternions, Complex Numbers and the
Algebraic Design of Physics, Kluwer Academic Publishers.
(Well... I can't help but mention that the physical theory based on T
is associated with spaces of dimension 10 and 11 exactly as ordinary Dirac theory
is associated with spaces of dimension 4 and 5. It is at the very least interesting
that these dimensions are resonant dimensions in String and M-theory, about
which I am far from expert (half a kilometer distant, I'd say)).
|17 June 1997.