One Point of the Cayley Plane

The doublet as a point

Let $J=\mathfrak h_3(\Oct)$ be the exceptional Jordan algebra and $F_4$ its automorphism group. A point of the Cayley plane $\Oct P^2$ is a primitive idempotent of $J$, and the stabilizer of a point is $\Spin(9)$. The observer’s clock picks a copy $\operatorname{span}\{1,e_c\}$ of the complex numbers in the octonions, the lepton plane of Chapter XVI, and the subgroup of $F_4$ that commutes with the centre of the clock’s colour group is $(\SU(3)\times\SU(3))/\Z_3$. The clock can be any observer’s: the spinor transport carries each observer’s colour group to every other’s (Chapter XV), so the group the point leaves is one group seen in twenty-eight frames.

Item (4) settles the orientation. The fiber’s Pati–Salam symmetry is native kinematics, so an added point that keeps it must be read inward, and then the electroweak group is what the clock leaves of that symmetry: at the level of Lie algebras, $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1)$ is the intersection of the fiber’s $\mathfrak{su}(4)\oplus\mathfrak{su}(2)_L$ with the centralizer of the clock’s colour centre. The program’s other candidate orienters do not decide it: reversing the clock conjugates each reading within itself, a commit’s two directions act alike, the spin bundle’s two forms leave every relevant operator fixed, and the table’s handedness is undone by an isomorphism of Jordan algebras.

So the second copy of the fiber that weak isospin needs is the one a point of the Cayley plane carries. The program adds the point; nothing native to one observer supplies it, since inside one fiber no structure singles out a slot of $J$, and every three outside the fiber lies across several frames, where gauge freedom does not reach. One geometric datum, kept compatible with the fiber, replaces a group, a representation and a table of charges chosen separately.

The whole family at one point

Over the real numbers the point carries only the left half of the family: the exceptional Jordan algebra holds the two left doublets, the down antiquark, a weak triplet and two singlets, but not the up antiquark or the positron. The fiber, however, is complex, and over the complex numbers the same point carries the rest. Complexify $J$ to $\C\otimes J$, a space of dimension 27. The transformations that keep its cubic norm and its Hermitian form make up the compact group $E_6$, and its points form the complexified Cayley plane, of complex dimension 16.

So the family’s right half and $\SU(2)_R$ do not lie outside the point. They are the complex half of the space the point carries, and the two-state multiplicity of Chapter XVI is that space written in coordinates. The left–right group is what the clock leaves of the point’s symmetry, as the electroweak group was over the real numbers. The step from it to the Standard Model’s group is the choice of the singlet’s direction, which the family’s hypercharge $Y=\tfrac12(B-L)+T_{R,3}$ already makes when it names $T_{R,3}$.

A breaking, not a value

Neither the point nor the singlet’s direction is a parameter. Relative to the clock, the candidate points form a single orbit of the symmetries that commute with the clock’s colour centre, and for each point the singlet’s directions form a single orbit of the left–right group. So choosing them is a breaking, not a value. The Standard Model’s group is the stabilizer of the point together with a second point on a null line through it. The only invariants of the choice are two magnitudes and one bit, which of the two weak groups survives, and reversing the clock exchanges the two. In the standard left–right models the same breaking is made by a field that carries $B-L$. One structure on the other side is native: the commit’s factor $W_a=\gamma_a\otimes I_2$, which exchanges the two weak factors, is for each of the six letters an automorphism of the real algebra $J$, inside $\Spin(8)\subset F_4$.

That bit is relative to the clock, an interior convention, not to spacetime. A vacuum that sets it breaks the symmetry between the quartets, the first of the two things the weak force’s hand needs. The second, that the surviving weak group act on fields of spacetime’s hand, no vacuum of the fiber could supply, because the lift’s mirror image leaves every datum of the fiber unchanged. Matter’s spinor supplies it: that spinor is the lift’s own, so every field of matter carries the lift’s one hand, and the left–right arrangement of the point’s sixteen states makes the surviving weak group act on fields of that hand (Chapters XV and XVIII). What remains is a naming of charge-conjugation type: which quartet the vacuum keeps as weak, relative to which is called matter.

Nothing native makes the breaking occur. Every datum the program supplies acts on the fiber and leaves the doublet index alone, and any one-particle datum that broke the symmetry this way would change $B-L$ by one unit. Relativity is consistent with the breaking but does not cause it: its lift to the algebra fixes the frame of three points for every observer at once, so the direction named by $T_{R,3}$ and the Higgs directions are left unchanged by every relabelling, yet no observer’s own symmetries single out one axis, because relativity acts on the doublet’s two copies alike.

The carrier of the breaking

The breaking has a carrier inside the family. The family fixes the point itself: the sixteen states span a space whose symmetry singles out the point. The line has a carrier too. Among all pairs of the family’s states exactly one is left unchanged by the Standard Model’s group, a pair of right-handed neutrinos, and its vacuum is the line. That pair carries $B-L=2$, and the same pairing gives the right-handed neutrino alone a mass of the Majorana kind, which with its ordinary coupling gives the neutrinos’ masses the seesaw form. Nothing in the program’s present law acts on such a pair, so what remains added is one dynamical fact, that the pair condenses. A condensate of pairs needs pairs: with the number of registers fixed, as it is natively, every pair amplitude vanishes in a state of definite register number (Chapter XVI), so the breaking presupposes the pair structure that the program owes, or a reading of the condensate as order between states of fixed number, the off-diagonal long-range order of superconductors. The pair’s alignment with the Higgs directions is a further condition on the forces that would make it condense.

Two of the program’s own processes have been tested on the pair, and neither attracts it. The colour-neutral contact of Chapter XXII acts on the letters and leaves the fiber unchanged; dressed by the steps that write its letters into the fibers it does reach the lepton line, as an exchange of $B-L$, but in its covariant form that exchange leaves the pair exactly alone, together with the left-handed pair, so it neither binds the pair nor sets it apart. The binding of composites through an intermediate that both partners touch has the sign that Adler showed favours the pair’s channel, but it acts through colour alone, and carried onto the family it leaves every pair of leptons unchanged. An attraction that reaches the pair must carry $B-L$ or right-handed weak isospin. Only $B-L$ is native, and on its own it treats the left- and right-handed lepton pairs alike; the clock’s arrow cannot choose between them, since reversing it is a symmetry of the exchange. The step that writes a record changes $B-L$, and no relabelling of the records restores it, so if the pair condensed, the boson that would otherwise be massless with it, the Majoron, would carry a small mass.

The Higgs and the masses

In the algebra the singlet’s direction is an ordered pair of points that completes $p$ to a frame of three orthogonal points, and the same two points carry the Higgs field. Restricted to the point, the cubic norm of $\C\otimes J$ couples the family’s $\mathbf{16}$ to the point’s $\mathbf{10}$, with one coefficient for all four Yukawa types. The frame’s other two points are the directions of the neutral Higgs fields: one gives masses to the up quark and the neutrino, the other to the down quark and the electron.

With three bare copies this makes the up-type and down-type mass matrices proportional, so that the quark mixing matrix is the identity, and it sets the down-type quark masses equal to the charged-lepton masses at the matching scale. Nature refutes both (Chapter XXIV). So the point’s cubic norm is not the source of the masses: they stay data, and at least two independent Yukawa structures are needed. The program now has the second one’s form. For fermions the pair terms that respect colour and matter’s number are exactly two, at a given clock: one weights leptons and quarks alike, and one weights them as one to minus a third. The second is baryon minus lepton number read through the clock’s chirality, with Georgi and Jarlskog’s factor fixed rather than fitted (Chapter XVI). Which generation carries which structure stays data, and no part of the program’s vacuum supplies the second, so the pair term must carry it as part of the law.

Three copies

No three that the program makes can serve as a family index, because each candidate is spent on something a family index must commute with. The three axes are the coordinate lines of colour. The observers’ three copies of the chiral octet are three different spin fields. Triality’s three representations carry different baryon and lepton numbers. The Cayley point’s three slots are the weak doublet and a singlet. And the lift’s pair $\mathbf3\oplus\overline{\mathbf3}$ is one representation, of dimension three, of the little group of a light direction. The forty-eight states in which three generations have been read in $\C\otimes\Oct$ need a colour action that no transformation of the fiber induces.

So the generations are added. Matter’s one-particle space is $\C^3\otimes F$, with $F$ one family and $\C^3$ a bare multiplicity on which every native symmetry acts trivially. Each copy is free of anomalies; the operators commuting with every native symmetry are one $M_3(\C)$ for each of the six species, which is the shape of the quark and lepton mixing matrices; and nothing native tells the copies apart, so their masses and mixings are data. The copies’ frames must carry the clock’s orientation, because reversing the clock exchanges left and right isospin.