lepton line

Part IV, The Exceptional Interior · defined in Chapter XVI, The Quartet

Also lepton plane, lepton’s line

The line spanned by the octonion unit 1 and an observer’s clock $e_p$ , $\mathrm{span}\{1,e_p\}$ , on which the clock’s complex structure sends 1 to $e_p$ and $e_p$ to $-1$ . Chapter X calls it the lepton plane, the plane spanned by the shared octonion unit and the clock; on the sky it is the functions constant on the observer’s pair and on the complement, and no relabelling of the observer’s rods moves it.

Plate W.29The lepton as a zero: the eight units graded by

\F_2^3

with the clock

e_7

vertical; a relabelling fixes the gold class

\{1,e_7\}

and permutes the three colour classes, and only the spinor transport’s shift moves the zero.

Chapter XVI identifies it. It is the zero among the four classes of octonion units modulo the clock: every relabelling of the program that fixes the clock fixes it, and every other carries it to the lepton’s line of the image clock. Read through the observer’s letters as a Fock space, it is the line of the lepton of the Pati–Salam quartet. The names quark, lepton and neutrino are Furey’s reading of the Fock states, but the line itself is structural, and that the octonion product with its unit is physical is the program’s decision.

The double cover moves the line only through spin, by the shift of the spinor transport; with spin carried on the observer’s report qubit, beside the fiber, every relabelling keeps the lepton in place, and there is no symmetry between the lepton and a quark to gauge. Across a meeting of the Coxeter graph, a relabelling that carries one observer’s lepton plane onto the other’s carries clock to clock and is unique up to colour, so whatever such comparisons do around a loop is pure colour.

Theoremproved

Fix an observer’s clock, with time unit $e_p$ , and six letters $\gamma_a$ . The letters define three fermionic modes $a_j=(\gamma_{2j}+i\gamma_{2j+1})/2$ , and $\C\otimes\Oct\cong\C^8$ is their Fock space, with number operator $N$ and chirality $D=-iL_{e_p}=-(-1)^N$ . The fifteen bivectors $\gamma_a\gamma_b$ generate one $\Spin(6)\cong\SU(4)$ , under which $\C^8=\mathbf4\oplus\overline{\mathbf4}$ . Colour $\SU(3)=\mathrm{Stab}_{G_2}(e_p)$ is the subgroup of $\SU(4)$ fixing the lepton vector, and its centralizer in $\SU(4)$ is the circle generated by $B-L=\tfrac23N-1$ .

The lepton’s line $\mathrm{span}\{1,e_p\}$ is the zero among the four classes of octonion units modulo the clock; every relabelling of the program that fixes the clock fixes it, and every other carries it to the lepton’s line of the image clock.

As mathematics

Label the units by $\F_2^3$ , with $e_0=1$ ; the table is graded, $e_ae_b=\pm e_{a+b}$ . Made complex by $L_{e_c}$ , the octonions are a $\C^4$ , the orthogonal sum of the four lines $\mathrm{span}\{e_x,e_{x+c}\}$ , one for each class of $\F_2^3/\langle c\rangle\cong\Z_2\times\Z_2$ , and a product of units from two classes lies in the class of the sum. The lepton line is the class of 0. A collineation fixing $c$ is linear, so it fixes the zero class and permutes the three axis classes through all of $\GL(2,2)\cong S_3$ : the zero displacement of Chapter V, which no relabelling can move.

The planes $\mathrm{span}\{1,e_c\}$ are among the few structures that tell two transports of the table apart. Carried across a broken loop of the scale tree by the map that keeps every clock’s complex structure, the table becomes one whose unit has moved to the line of one clock $e_{c_1}$ , and for every other clock $c$ the plane it attaches to $c$ is orthogonal to $\mathrm{span}\{1,e_c\}$ .

Its name in another field	Bridge
$\mathrm{span}\{1,e_c\}$ , the zero class of the grading by $\F_2^3/\langle c\rangle$	built
the functions on the sky constant on an observer’s pair and on its complement	built
the lepton of the Pati–Salam quartet	a reading

Built from: fiberclock
In the dictionary: the seven points
In the volume: IThe Founding SentenceXThe Finite Celestial SphereXIRulial RelativityXIIThe Coxeter GraphXIVWhy OctonionsXVSpin from the Double CoverXVIThe QuartetXIXRulial InvariantsXXIIILight, Vacuum and Handedness