Part IV · The Exceptional InteriorChapter XIV

Why Octonions

Plate XIV.1Two constraints on the imaginary dimension

d

: composition, after Hurwitz, and complete equiangular networks, after Gerzon, Lemmens and Seidel.

XIV.1
XIV.2
XIV.3
XIV.4
XIV.5
XIV.6
XIV.7

Why would observers who are blind, complete and democratic force the octonions?

The octonions have entered physics several times, and always by invitation. Günaydin and Gürsey observed in 1973 that the automorphisms of the octonions fixing one imaginary unit form $\SU(3)$ and read the six remaining units as a colour triplet and its conjugate; Dixon organized a family of fermions around $\R\otimes\C\otimes\Ham\otimes\Oct$ ; Furey obtained the gauge groups of the Standard Model from ladder operators of $\C\otimes\Oct$ . In each case the algebra came first and the physics was read from it. This chapter asks whether the observers of this program choose the algebra themselves.

Part III studied the observers from outside, as points of a finite sky related by a finite group; here is the picture from inside. Every anchored observer carries an eight-dimensional interior, the fiber, and marks one of its directions as its own time, so the network has twenty-eight marked directions. Ask of it what Part I asked of one observer, and one thing more. Each observer alone is blind, reading one direction and missing the rest; together they miss nothing; none is privileged; and their directions compose. The first three are conditions on a set of lines, the fourth is the hypothesis of Hurwitz’s theorem, and together they leave two possibilities, of which the octonions are one.

The central result

Let $q_1,\dots,q_N$ be unit vectors spanning distinct lines in the imaginary part $\R^d$ , $d>1$ , of a normed division algebra. If their projectors $P_x=q_xq_x^{\mathsf T}$ span the real symmetric matrices $\mathrm{Sym}(d)$ (completeness) and all pairs of lines meet at one angle, $\lvert\langle q_x,q_y\rangle\rvert=a$ for $x\neq y$ (the equiangular half of democracy), then

N=\frac{d(d+1)}{2},\qquad a^2=\frac{1}{d+2},\qquad\sum_xP_x=\frac{d+1}{2}\,I,\qquad d\in\{3,7\}.

Both cases occur: at $d=3$ the six diagonals of the icosahedron, with $a^2=1/5$ ; at $d=7$ the twenty-eight lines of $E_7$ , with $a^2=1/9$ , which are exactly the time directions of the program’s twenty-eight anchored observers.

Status

What is proved is the dimension count, the geometry at seven, and its identity with the program’s observers. The theorem does not choose seven over three; the observer’s six letters and one time do that, since $7=6+1$ . It does not fix how the octonion product sits on the twenty-eight lines: two inequivalent placements satisfy every hypothesis. And “the octonions are an output of an observer principle, not an input” is a reading of the theorem, not a further theorem. Blindness does no work in the proof; it is the reason the question arises.

Colour as the stabilizer of time is exact group theory, and calling it colour is the identification of Günaydin, Gürsey and Furey; that the report group lifts into each observer’s colour group is exact, and it is the source of the conflict between colour and space. That the lepton line is the unit’s class, and that every relabelling carries lepton lines to lepton lines, is exact; that the octonion product, with its unit, is physical is chosen. The lift’s spinor transport, a permutation of the units followed by a shift that moves the unit, is exact and is read as the transport of spin. The complex octonions are the unique carrier whose whole multiplication is complex-linear, a selection that holds given the division-algebra programme’s thesis that physics is what multiplication generates, which is not derived from the program’s axioms. The clocks as the octavian orders, and the lines as their intersections, is an exact identification.

Four requirements on a network

Plate XIV.1Two constraints on the imaginary dimension

d

: composition, after Hurwitz, and complete equiangular networks, after Gerzon, Lemmens and Seidel.

Each clause translates an earlier commitment. Blindness is the premise of Chapter II: an observer audits a class of histories, never the interior of the class. Completeness says that nothing is hidden from all observers together. Democracy is the relativity principle among observers in its most symmetric form: every pair is equally complementary, and no one is distinguished. Composability is the Fano attachment: an observer’s two letters multiply to a third letter or to its time.

The proof uses completeness, the equiangular half of democracy, and composability; transitivity is used once, to show that the network’s symmetry acts irreducibly. A single observer that could read the whole interior would need no network, which is why blindness, though idle in the proof, poses the question.

Definition(A network of time directions)

Let $V$ be a real inner-product space of dimension $d$ . A network is a finite family of unit vectors $q_x\in V$ , one for each observer $x$ in a set $X$ of size $N$ , spanning distinct lines; observer $x$ reads a real symmetric operator $\rho$ through the single number $\operatorname{Tr}(\rho P_x)$ , with $P_x=q_xq_x^{\mathsf T}$ . The network is blind if no single reading determines $\rho$ , which holds as soon as $d>1$ ; complete if the $P_x$ span $\mathrm{Sym}(V)$ ; democratic if $\operatorname{Tr}(P_xP_y)$ takes one value $a^2$ for all $x\neq y$ and the network’s symmetries act transitively on $X$ ; and composable if $V$ is the imaginary part of a normed division algebra $\R1\oplus V$ .

The forcing theorem

Plate XIV.2The two rows meet at the trivial

d=1

and at

d=3

and

d=7

; the figure does not choose between three and seven.

The equiangular condition alone is far from settled mathematics: equality in Gerzon’s bound is known only in dimensions 2, 3, 7 and 23, and whether it holds in other dimensions with $d+2$ an odd square is partly open. Hurwitz’s theorem makes that open problem irrelevant here, because it removes every dimension except 3 and 7 before the geometry is consulted.

At each surviving dimension the line system is unique up to orthogonal maps. At three, with $\varphi=(1+\sqrt5)/2$ , the six vectors $(0,1,\pm\varphi)$ , $(1,\pm\varphi,0)$ , $(\pm\varphi,0,1)$ meet pairwise at $a^2=\varphi^2/(\varphi+2)^2=1/5$ : the icosahedron’s diagonals in the imaginary quaternions, whose product-preserving symmetry is the icosahedron’s rotation group $A_5$ . At seven, by Bannai and Sloane, the lines are the fifty-six minimal vectors of the weight lattice of $E_7$ taken up to sign. The clause $d>1$ is a premise: the literal conditions admit one observer and no angle, and a single observer is not a network.

Theorem(Complete democratic networks) proved

Let $d>1$ and let $(q_x)_{x\in X}$ be a complete network in $V=\R^d$ whose lines are pairwise at the same angle, $\lvert\langle q_x,q_y\rangle\rvert=a$ for $x\ne y$ . Then $N=d(d+1)/2$ , $a^2=1/(d+2)$ and $\sum_xP_x=\tfrac{d+1}{2}I$ . If moreover the network is composable, then $d\in\{3,7\}$ .

Proof

The Hilbert–Schmidt Gram matrix of the projectors is $(1-a^2)I+a^2J$ , positive definite since $a<1$ , so the $N$ projectors are independent in $\mathrm{Sym}(d)$ and $N\le d(d+1)/2$ , Gerzon’s bound; completeness gives equality.

Completeness also gives $I=\sum_xc_xP_x$ , and pairing with each $P_y$ shows that the constant $c=1/(1+(N-1)a^2)$ is the solution. Taking the trace, $d=N/(1+(N-1)a^2)$ ; with $N-1=(d+2)(d-1)/2$ and $d>1$ this gives $a^2=1/(d+2)$ , and then $\sum_xP_x=c^{-1}I=\tfrac{d+1}{2}I$ .

If the network is composable, $\R1\oplus V$ is a normed division algebra, of dimension 1, 2, 4 or 8 by Hurwitz’s theorem. So $d\in\{0,1,3,7\}$ , and $d>1$ leaves 3 and 7.

The twenty-eight observers are the solution at seven

Plate XIV.3Time directions as chords: those sharing an end with the gold one meet it at

+\tfrac13

, the disjoint ones at

-\tfrac13

In the global frame of Chapter X the fiber is the space of real functions on the eight sky points: the octonion unit is the constant function, which no relabelling moves, and the imaginary octonions are the functions with sum zero. An observer is an unordered pair $\{a,b\}$ of sky points, and its time direction is the centred indicator $q_{ab}=\sqrt{2/3}\,(\mathbf{1}_{\{a,b\}}-\tfrac14\mathbf{1})$ . Of the 378 pairs of observers, 168 share a sky point, and these are the promotions of Chapter XII.

The readings reconstruct every real symmetric $A$ on the imaginary octonions, $A=\sum_x[\tfrac98\operatorname{Tr}(AP_x)-\tfrac18\operatorname{Tr}A]P_x$ , and only those. Of a Hermitian operator’s forty-nine real parameters the twenty-one imaginary antisymmetric ones, its relative phases, are invisible to every observer, and distinct positive states can agree on all twenty-eight readings: the network is complete for the real part and collectively blind to its phases.

What the observers share is not only classical. With $c$ the unit, the covariant questions $P_c-P_x$ and $\lvert c\rangle\langle q_x\rvert+\lvert q_x\rangle\langle c\rvert$ anticommute and form an observer-covariant qubit on each lepton plane $\operatorname{span}\{c,q_x\}$ , the plane of the unit and the observer’s time direction, which no relabelling of the observer’s rods moves.

Proposition(The observers’ time directions) proved

The twenty-eight vectors $q_{ab}$ are unit vectors in the imaginary octonions. Two of them have inner product $+1/3$ if their pairs share one point and $-1/3$ if the pairs are disjoint. Their projectors span $\mathrm{Sym}(7)$ and sum to $4I_7$ . The four observers of one clock, whose pairs partition the eight points, have time directions summing to zero, pairwise at $-1/3$ : the vertices of a regular tetrahedron.

Proof

Each vector has two entries $\sqrt{2/3}\cdot\tfrac34$ and six entries $-\sqrt{2/3}\cdot\tfrac14$ , so its squared norm is $\tfrac23(\tfrac{18}{16}+\tfrac6{16})=1$ , and its entries sum to zero. For pairs sharing one point the inner product is $\tfrac23(\tfrac9{16}-\tfrac6{16}+\tfrac5{16})=\tfrac13$ ; for disjoint pairs it is $\tfrac23(-\tfrac{12}{16}+\tfrac4{16})=-\tfrac13$ . So the network is equiangular with $a^2=1/9$ , its projectors are independent, and twenty-eight is $\dim\mathrm{Sym}(7)$ . Over a partition of the eight points the indicators add to $\mathbf{1}$ , so a clock’s four vectors sum to zero.

Arrows, and the placement of the product

Plate XIV.4Arrows reconstruct the sky: the seven observers through the point 0, pairwise at

+\tfrac13

, recover that point.

The two products have symmetry groups of different orders, so no orthogonal map carries one pair of lines and product to the other: the principle, read on lines, fixes the geometry of the network and leaves the placement of the product undetermined. Both groups consist of automorphisms of a product, so both fix the unit and carry each observer’s lepton plane $\operatorname{span}\{1,q_x\}$ to another observer’s, and transitivity on the lines makes the seven imaginary directions an irreducible real representation of the network’s symmetry. The arrows of time, $+q_x$ rather than $-q_x$ , carry more. They reconstruct the eight sky points as the eight sets of seven arrows with pairwise inner product $+\tfrac13$ , and their stabilizer is $\PSL(2,7)$ . The Cayley placement’s stabilizer of its twenty-eight positive vectors has order only 24, so it is no counterexample to a principle stated on arrows; whether that principle selects the program’s placement uniquely is open.

Democracy has a price. Under $G_2(2)$ , or its simple derived group $U_3(3)$ , there is no covariant way to compare two distinct observers: a comparison from $x$ to $y$ must commute with every symmetry fixing both, the centralizer of that joint stabilizer has order 2 or 8, and none of its elements carries $x$ to $y$ . Once arrows are chosen, the symmetry drops to $\PSL(2,7)$ , covariant comparisons exist, and by the rulial curvature theorem each has holonomy the full stabilizer $S_3$ . At seven, observers can be compared only after each has chosen which way its time points, and then the comparison is curved. The quaternionic sibling differs: $A_5$ admits exactly two covariant reversible comparisons of the six lines, both with full dihedral holonomy of order ten.

Proposition(Symmetry of the network) computed

Give the fiber the program’s global octonion product, in which every element of $\PSL(2,7)$ acting on the sky points is an automorphism. The orthogonal maps preserving both this product and the twenty-eight lines form the group $G_2(2)$ , of order $12{,}096$ , which acts 2-transitively on the lines; the subgroup preserving the twenty-eight positive vectors $q_{ab}$ is exactly $\PSL(2,7)$ . The ordinary Cayley product, written in another orthonormal frame, is compatible with the same lines, and the maps preserving it and the lines form $2^3{\cdot}\PSL(3,2)$ , of order $1{,}344$ , again transitive on the lines.

Colour is what time cannot see

Plate XIV.5The Fano plane of the imaginary octonions on the volume’s chart, with

e_7

as time at the centre: the medians pair the letters into axes, and the incircle and sides are the four reports.

Take the octonions with the oriented triples 124, 235, 346, 457, 561, 672, 713, so that $e_1e_2=e_4$ and cyclically, and mark $e_7$ as time. The lines through 7 pair the other six points as $\{1,3\}$ , $\{2,6\}$ , $\{4,5\}$ , and the four lines missing 7 each take one point from every pair: the incidence of an anchored observer, with letters, axes and reports. Left multiplication by time turns each axis into a complex line, $e_7e_1=e_3$ , $e_7e_2=e_6$ , $e_7e_4=e_5$ : to turn a letter into its antipode, multiply by time. With $\operatorname{span}\{1,e_7\}$ these make the octonions a $\C^4$ of four complex lines, which are not the four reports: a report is a Fano line, three letters any two of which multiply to $\pm$ the third, while a complex line is spanned by two units that time exchanges up to sign.

An automorphism conjugates the clock to $gL_tg^{-1}=L_{g(t)}$ , so $\operatorname{Stab}_{G_2}(t)$ is exactly the group of symmetries of the product that leave the clock unchanged: it fixes 1 and $t$ and acts on the six directions the clock turns among themselves. The 24 collineations fixing 7 lift, four ways each, to 96 signed permutations of the units that are automorphisms fixing $e_7$ ; on $\C^3$ they form the complex reflection group $G(4,4,3)$ , monomial with entries in $\{\pm1,\pm i\}$ and determinant one, and act irreducibly. So the report group of an observer lifts into its colour group, and when the letters also build space, colour moves space; the octonions create that difficulty rather than hide it. These ninety-six preserve the product and fix the unit. Chapter XV lifts the same collineations differently: each map of the spinor transport permutes the units without the product’s signs and then multiplies on the right by a unit, a shift that in general moves 1.

Theorem(The stabilizer of time)

The automorphism group $G_2$ of the octonions acts transitively on the unit imaginary octonions. The stabilizer of one of them, $t$ , is $\SU(3)$ . It fixes 1 and $t$ and acts on $\{1,t\}^\perp\cong\C^3$ , made complex by $L_t$ , as the defining representation.

The clock’s four lines

Plate XIV.6The eight units graded by

\F_2^3

with the clock

e_7

vertical: the four vertical edges are the clock’s complex lines, and the gold one, the unit’s class, is the zero.

For the oriented table of the previous step, giving $e_7$ , $e_1$ and $e_2$ the basis vectors $c$ , $u$ , $v$ makes $e_4=u+v$ , $e_3=u+c$ , $e_6=v+c$ and $e_5=u+v+c$ , and the classes modulo the clock are $\{1,e_7\}$ , $\{e_1,e_3\}$ , $\{e_2,e_6\}$ and $\{e_4,e_5\}$ : the unit’s class and the three axes.

The four classes and the four reports are quartets of different kinds. The reports are places: none is an origin, and the report group carries any one to any other. The four lines are displacements: no step, and one step along each axis, a group whose zero every relabelling fixing the clock leaves in place. The unit’s class is the zero displacement, the line Chapter XVI identifies as the lepton’s; the identification rests on one decision of the program, that the octonion product, with its unit, is physical.

Across the network each observer has its own colour, $\SU(3)_x=\operatorname{Stab}_{G_2}(q_x)$ , which a relabelling conjugates to $\SU(3)_{\pi x}$ ; the part visible in the network’s finite symmetry has order 216, $3^{1+2}\rtimes Q_8$ , and contains a qutrit Pauli group on the colour triplet. Only the multiples of the unit are fixed by all twenty-eight colour groups, and the unit is an eigenvector of no observer’s chirality $D_x=-iL_{q_x}$ : each observer resolves it into its own pair of colour-neutral chiral lines $(1\pm iq_x)/\sqrt2$ . What the observers share is the rule that picks out the lepton. Observer $x$ ‘s lepton line is $\operatorname{span}\{1,q_x\}$ , the unit’s class with $q_x$ as clock, and every relabelling carries one observer’s lepton line to another’s and never mixes a lepton line with a colour triplet. Read this way, colour is frame data: what moves with the frame is the labelling of the three colours, and the spinor transport of Chapter XV, though not a symmetry of the product, carries each observer’s colour group to the next in the same way. That is a reading; its test is whether any record or instrument can read a colour rotation that no change of observer accompanies, and so far none can.

Proposition(The grading and the clock’s four lines)

Label the octonion units by $\F_2^3$ , the unit 1 by 0 and each imaginary unit by its point of the Fano plane, and write $e_x$ for the unit labelled $x$ . (1) The table is graded: $e_ae_b=\pm e_{a+b}$ . (2) Fix a clock $e_c$ . Made complex by $L_{e_c}$ , the octonions are the orthogonal sum of the four complex lines $\operatorname{span}\{e_x,e_{x+c}\}$ , one for each class $\{x,x+c\}$ modulo $c$ , graded by $\F_2^3/\langle c\rangle\cong\Z_2\times\Z_2$ , whose zero is the unit’s class $\operatorname{span}\{1,e_c\}$ . (3) Every collineation fixing $c$ fixes the unit’s class and permutes the three axis classes, and the twenty-four induce all of $S_3$ . There are $1{,}344$ signed permutations of the units that preserve the product, every one fixing 1; the 96 that cover a collineation fixing $c$ and commute with $L_{e_c}$ carry the four lines as their collineation carries the classes, and the four over the identity are sign changes $\operatorname{diag}(1,\pm1,\pm1,\pm1)$ lying in colour $\SU(3)$ .

Proof

(1) The lines are the triples of nonzero vectors summing to zero, so for distinct nonzero $a,b$ , $e_ae_b=\pm e_k$ with $k=a+b$ , and $e_ae_a=-1$ . (2) $L_{e_c}$ sends $e_x$ to $\pm e_{x+c}$ , so it preserves each plane, and it squares to $-1$ by alternativity; for $a\notin\{0,c\}$ the class $\{a,a+c\}$ is the line through the clock with the clock removed, an axis. (3) A collineation fixing $c$ is linear on $\F_2^3$ and induces a linear map of the quotient, which fixes its zero. An automorphism fixes the unit, and commutes with $L_{e_c}$ exactly when it fixes $e_c$ . The counts, the diagonal form, and the fact that the ninety-six contain no subgroup mapping isomorphically onto the twenty-four collineations, since every lift of a collineation of order four has order eight, were checked by direct computation; so the clock’s relabellings act on the fiber only together with the colour sign changes.

The clocks are the octavian orders

Plate XIV.7Two clocks, two octavian orders, meeting along the line through them in that line’s quaternion order.

The fiber is $\C\otimes\Oct\cong\C^8$ , not $\Oct$ . Each observer’s time direction gives a real complex structure $J_x=L_{q_x}$ , and the twenty-eight generate the whole matrix algebra $M_8(\R)$ , so no real square root of $-1$ is shared by all observers; the six letters alone already generate $M_8(\R)$ . By the commutant trichotomy, among the sixteen-dimensional candidates $\Oct\oplus\Oct$ , $\C\otimes\Oct$ and the sedenions, only $\C\otimes\Oct$ has a complex structure making every left multiplication complex-linear, and it is $\pm i\otimes1$ . The choice is the one Furey made, a choice between two named principles. The complex octonions then hold two square roots of $-1$ of opposite kinds: the scalar $i$ commutes with every letter, and the clock $L_{e_7}=L_{e_1}\cdots L_{e_6}$ , the volume element of the letters’ Clifford algebra, anticommutes with every letter; their product $iL_{e_7}$ is $(-1)^N$ on the Fock space of the three axes, the chirality of Chapter XVI.

The integral forms come next. Coxeter found that the octavians form seven maximal orders, all containing the sixteen units $\pm1,\pm e_i$ ; Kirmse’s earlier set is not closed under the product. To compare them with the lift, the units are indexed by the sky points with $e_\infty=1$ . In that labelling the unit is one sky point among eight, as it is for the spinor transport, whose shift carries 1 to every other unit up to sign; the product-preserving relabellings fix 1, and the integral statements use the labelling only as coordinates.

The orders see clocks and lines and nothing finer: a clock’s four observers give one and the same order. So $E_7$ appears as the complete democratic network, $G_2(2)$ as its symmetry, $\PSL(2,7)$ as the stabilizer of its arrows, $E_8$ once for each clock as that clock’s ring of integral octonions, and the Hurwitz order once for each line.

Proposition(Clocks and lines as integral octonions) computed

Index the octonion units by the eight sky points, with $e_\infty=1$ and $e_ie_{i+1}=e_{i+3}$ for $i$ modulo 7. (1) Each colour of the lift’s tetrahedra, with $\emptyset$ and the whole sky, is a doubly even self-dual $[8,4,4]$ code, and half of its Construction A lattice is an $E_8$ lattice, invariant under $\PSL(2,7)$ ; the associative quadruples of the table and their complements are exactly the blocks of the line-cube colour. (2) Neither colour’s 240 units is closed under the product. Of the thirty Steiner systems $S(3,4,8)$ on the sky exactly seven give rings, each an octavian order, each the line-cube colour moved by a transposition of the sky, and the four transpositions that give one of them are the four observers of one clock: the seven octavian orders are the seven clocks. (3) Two of these orders meet, in index four in each, in an order containing the 24 Hurwitz units of the line through their clocks; three share it exactly when their clocks are collinear, and the codes of all seven share only $\emptyset$ and the whole sky.

Seen from the whole volume, the chapter adds one link to each anchor. The celestial sphere carries the observers’ time directions as centred indicators of pairs of its points; $G_2/\SU(3)=S^6$ is the sphere of possible times, with colour as the stabilizer; rulial relativity appears as the price of democracy; and on the imaginary octonions what the network does not see is exactly twenty-one phase directions, the unaudited part of the interior given a definite size. In Seams the bridge from octavian orders to clocks is built, and the failure of Kirmse’s lattice to be closed under the product is an absence whose imprint is the seven orders.

Whether the principle stated on arrows, with completeness, democracy and composition, selects the program’s product placement uniquely is open; the placements of a normed product relative to fixed lines form a continuous family that no finite computation has yet exhausted. The next chapter follows the observers’ relabellings into the fiber. There the lift forces the spinor transport, which realizes their double cover; it permutes the units without the product’s signs and adds a shift that moves the unit, whereas the product-preserving relabellings of this chapter fix it.