U(1)xSU(2)xSU(3)      Background summary:
  • 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.
  • 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 = CQO

    (where ⊗ is the (real) tensor product symbol), which is just the complexification of the quaternionization of the octonions.
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.
Problem 1: Resolve the identity of T into a set of orthogonal primitive idempotents;
  • Ordinarily a resolution of the identity would consist of n algebraic elements pk, k = 1,...,n, such that
    (1) p1 +...+ pn = 1 (resolve identity),
    (2) pk2 = pk (idempotents),
    (3) pkpm = 0 (k ≠ m) (orthogonal).
  • 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 column vectors). Therefore, for all X in T we demand
    (2') pk(pk X) = pk X (idempotents),
    (3') pk(pm X) = 0 (k ≠ m) (orthogonal),
    (4) pk(X pm) = (pk X)pm (associativity),
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.
Solution to Problem 1: Define
λ0 = (1 + ix)/2, λ1 = (1 - ix)/2,
λ2 = (1 + iy)/2, λ3 = (1 - iy)/2,
where x and y are arbitrary unit quaternions with no real parts. As usual let e7 be an octonion unit, and define
ρ+ = (1 + ie7)/2, ρ- = (1 - ie7)/2.
Δ0 = λ0 ρ+, Δ1 = λ1 ρ+,
Δ2 = λ2 ρ-, Δ3 = λ3 ρ-.
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
(0) (A Δm)* (B Δm) = (A* • B)m Δm,
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.

Suppose that Γm is the result of transforming Δm. We impose the following consistency conditions:
  • For all X in T,
    (1) (Γm* X) Γn = Γm* (X Γn)
    (associativity ensuring a consistent definition of components);
  • and
    (2) Γm* Γn = Δm Δn = δmn Δn
    (invariance of components of the identity).

Solution of Problem 2: (What follows is somewhat simplified; more details in the book.)
  • Condition (2) above implies the Γm take the form

    Γm = K Δm,

    K an element of T.

  • Condition (1) implies that K must be free of the octonion units e1,..., e6, those that don't appear in the Δm. Since e7 Δm = ±i Δm, the coefficient K will reduce to a complexified quaternion (dependent on m).
  • Condition (2) finally implies that each of these reduced coefficients is an element of U(1)xSU(2) (the U(1) and SU(2) actions are a bit more complicated than those arising from the unit complexes (U(1)) and unit quaternions (SU(2)), but it won't hurt to think of them that way).
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:
  • 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).
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.

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 into a 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.