fiber

Part IV, The Exceptional Interior · defined in Chapter XIV, Why Octonions

The eight-dimensional interior each observer carries, built on the octonions; it holds the quark–lepton quartet.

Plate W.27The fiber as the cube of occupation masks of the three modes, by level

N

: the quartet and its conjugate are its two inscribed tetrahedra, and a commit moves along an edge.

Every anchored observer carries the fiber and marks one of its directions as its own time, so the network has twenty-eight marked directions. Asked of the whole network, completeness and equiangularity of the time directions, in the imaginary part of a normed division algebra, force dimension 3 or 7, and at 7 the twenty-eight lines of $E_7$ , with $a^2=1/9$ , which are exactly the observers’ time directions. The fiber is $\C\otimes\Oct\cong\C^8$ , not $\Oct$ : the twenty-eight complex structures generate all of $M_8(\R)$ , so no real square root of $-1$ is shared by all observers, and among sixteen-dimensional real algebras only $\C\otimes\Oct$ has a complex structure making every left multiplication complex-linear.

Read through one observer’s six letters, the fiber is the Fock space of three fermionic modes $a_j=(\gamma_{2j}+i\gamma_{2j+1})/2$ . The fifteen bivectors generate one $\Spin(6)\cong\SU(4)$ , under which $\C^8=\mathbf4\oplus\overline{\mathbf4}$ , the Pati–Salam quark–lepton quartet and its conjugate; colour is the subgroup fixing the lepton vector, and no weak $\SU(2)$ is there. The names quark and lepton are a reading of the Fock states.

Theorem(The commutant trichotomy) computed

For a sixteen-dimensional real algebra $A$ , let $\mathcal L(A)$ be the associative algebra generated by its left multiplications. For $A=\Oct\oplus\Oct$ , $\mathcal L(A)=M_8(\R)\oplus M_8(\R)$ with commutant $\R\oplus\R$ , which holds no square root of $-1$ . For $A=\C\otimes\Oct$ , $\mathcal L(A)=M_8(\C)$ with commutant $\C$ , whose square roots of $-1$ are $\pm L_{i\otimes1}$ . For the sedenions, $\mathcal L(A)=M_{16}(\R)$ with commutant $\R$ , again with none.

Consequently a complex structure on $A$ that makes every left multiplication complex-linear exists only for $A=\C\otimes\Oct$ , and there it is $\pm i\otimes1$ . The same holds for right and two-sided multiplications.

As mathematics

The octonions as a real module of dimension 8, complexified. The six letter multiplications $\gamma_a=L_{e_a}$ satisfy $\gamma_a\gamma_b+\gamma_b\gamma_a=-2\delta_{ab}$ , and their 64 monomials span $M_8(\R)$ , so the fiber is the irreducible module of $\mathrm{Cl}(0,6)\cong M_8(\R)$ ; by Schur’s lemma only real scalars commute with every letter, and after complexification only $\pm i$ . At the clock $e_p$ the chirality is $D=-iL_{e_p}=-(-1)^N$ , with $N$ the number operator.

The units carry a grading by $\F_2^3$ , $e_ae_b=\pm e_{a+b}$ , and a clock $e_c$ makes the octonions a $\C^4$ , the orthogonal sum of four complex lines $\mathrm{span}\{e_x,e_{x+c}\}$ graded by $\F_2^3/\langle c\rangle$ : the unit’s class and the three axis classes. The seven clocks’ lifts generate exactly $\SL(2,7)$ on the fiber, acting as $\mathbf4\oplus\overline{\mathbf4}$ .

Its name in another field	Bridge
the irreducible module of $\mathrm{Cl}(0,6)\cong M_8(\R)$	built
the spinors $\mathbf4\oplus\overline{\mathbf4}$ of $\Spin(6)\cong\SU(4)$	built
the Fock space of three fermionic modes	built
three qubits, with the six letters as Majorana operators	a reading
the quark–lepton quartet and its conjugate	a reading