Octonions and \(\Esix\) in physics
Physics met these objects four times: as quantum observables, as symmetries of particles, as the shape of supersymmetry, and as the moduli of supergravity, where \(\Esix/\Ffour\) itself appears. Only the last two are established mathematical physics; attempts to derive the Standard Model are open research programmes.
SummaryWhat does physics actually use?
| object | role | status |
|---|---|---|
| \(\mathfrak h_3(\Oc)\) | the one exceptional algebra of observables (1934) | theorem |
| \(\R,\C,\Ha,\Oc\) | supersymmetry in spacetime dimensions \(3,4,6,10\) | theorem |
| \(E_{6(-26)}/\Ffour\) | scalar manifold of the octonionic magical supergravity in 5 dimensions | theorem (model) |
| cubic norm \(\det\) | black-hole entropy in 5 dimensions | derived in models |
| \(\Esix\), its 27 | grand unification | model, not confirmed |
| \(\mathfrak h_3(\Oc)\), \(\Ffour\), \(\Oc\) | Standard Model gauge group, three generations | speculative |
Status labels describe the mathematics inside each physical model, not evidence that nature uses it. As Baez put it in 2002, “there is still no proof that the octonions are useful for understanding the real world” [Baez, §1, p. 3].
1934: algebras of observablesWhy did physicists define the Albert algebra?
Observables of a quantum system are self-adjoint operators, and the product of two of them is not self-adjoint, but \(X\jo Y=\tfrac12(XY+YX)\) is. Jordan, von Neumann and Wigner took this product as the axiom and classified the finite-dimensional formally real algebras: apart from matrix algebras with \(\tfrac12(XY+YX)\), only one remains [JNW]. Albert proved that this one, \(\mathfrak h_3(\Oc)\), is genuinely exceptional [Albert]. Octonionic quantum mechanics built on it had little success in nuclear and particle physics [Baez, §1, p. 2]; the algebra survived in mathematics as the Albert algebra of these pages.
1973–1976: quarks and grand unificationDo the symmetry groups of these pages act on particles?
- Colour \(\mathrm{SU}(3)\subset\Gtwo\). Günaydin and Gürsey wrote the octonions in a split basis and reduced them along the subgroups \(\mathrm{SU}(3)\) and \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) of \(\Gtwo=\Aut(\Oc)\), relating the first to quark colour [GG].
- \(\Esix\) unification. Gürsey, Ramond and Sikivie proposed a gauge theory with group \(\Esix\), fitting quarks and leptons into two 27-dimensional representations [GRS]. The smallest nontrivial representations of \(\Esix\) are 27-dimensional [Baez, §4.4, p. 46]; on these pages one of them is \(\Jalg\) itself, on which \(\Esix\) acts by definition.
These are models; their particle assignments are not confirmed by experiment.
Supersymmetry: dimensions \(3,4,6,10\)Where does the list \(\R,\C,\Ha,\Oc\) appear in spacetime?
[Baez–Huerta, abstract and §1] Proved (cited)
The classical superstring and super-Yang–Mills Lagrangians are supersymmetric if and only if spacetime has dimension \(3,4,6\) or \(10\), that is, \(2+\dim\K\) for \(\K=\R,\C,\Ha,\Oc\).
Baez and Huerta prove the “if” direction with division algebras (§6) and credit the dimension count to earlier physics literature; Kugo and Townsend had tied spinors in these dimensions to \(\R,\C,\Ha,\Oc\) through \(\SL(2,\K)\) [KT]. The mechanism is visible on the \(\SL_2/\SO_2\) page: \(2\times2\) Hermitian matrices over \(\K\) form a space of dimension \(2+\dim\K\), and their determinant is a Minkowski form,
\[\det\begin{pmatrix}t+x&\bar w\\ w&t-x\end{pmatrix}=t^2-x^2-\Nm(w),\qquad w\in\K,\]so \(\SL(2,\K)\) acts on spacetime of dimension \(2+\dim\K\) by Lorentz transformations: \(\slf(2,\R)\cong\so(2,1)\), …, \(\slf(2,\Oc)\cong\so(9,1)\) [Baez, §1, p. 3]. For \(\K=\R\) this is exactly the hyperboloid of picture (a). The same list \(1,2,4,8\) is the multiplicity \(\mult\) of the \(A_2\) row.
Magical supergravity: \(\Esix/\Ffour\) as a space of fieldsWhere does \(\Esix/\Ffour\) itself appear?
Günaydin, Sierra and Townsend classified the 5-dimensional \(N=2\) supergravities whose scalar fields form a symmetric space, by Jordan algebras of degree 3; four exceptional, “magical”, cases come from \(\mathfrak h_3(\K)\), \(\K=\R,\C,\Ha,\Oc\) [GST 1983], [GST 1984]. In these theories the scalar fields take values in the hypersurface where the cubic norm equals one [FG, §4, pp. 9–10]:
\[\{X\in\mathfrak h_3(\K):\ \det X=1,\ X\text{ with three positive eigenvalues}\}.\]For \(\K=\Oc\) this is \(E_{6(-26)}/\Ffour\), the real, Riemannian form of the variety \(\Xe\cong\Esix/\Ffour\) of these pages (see octonions over \(\C\)). The four magical theories are the four members of the \(A_2\) row, read over \(\R\).
Black holes and orbits of \(\Esix\)What does the cubic determinant measure?
In these 5-dimensional theories a black hole carries charges \(X\) in the Jordan algebra, and its entropy is \(\sqrt{|\det X|}\) [FG, abstract and §4]. Ferrara and Günaydin classify the charge orbits by how many eigenvalues vanish, as the \(\SL_3/\SO_3\) page does for symmetric matrices.
| vanishing eigenvalues | \(\det X\) | orbit |
|---|---|---|
| none | \(\ne0\) | \(E_{6(6)}/F_{4(4)}\) |
| one | \(0\) | \(E_{6(6)}/(\mathrm O(5,4)\ltimes T_{16})\) |
| two | \(0\) | \(E_{6(6)}/(\mathrm O(5,5)\ltimes T_{16})\), dim 17 |
In the maximal (\(N=8\)) theory, from string and M-theory, the algebra is built on the split octonions and the group is \(E_{6(6)}\) [FG, §2, p. 3]; the compact octonions and \(E_{6(-26)}\) belong to the magical \(N=2\) theory. Over \(\C\) both become the \(\Esix\) and \(\Jalg\) of these pages, and the generic orbit \(\{\det\ne0\}\) is, up to scaling, \(\Esix/\Ffour\).
Open programmes: the Standard ModelCould the Albert algebra explain the particles we see?
Several current research programmes propose that it might. None is established.
- Dubois-Violette proposes the 27-dimensional Albert algebra as the internal “finite quantum space” of particle physics, relating triality to three generations [DV]; with Todorov he derives the Standard Model symmetry from subgroups of \(\Ffour\) and \(\Esix\) via Borel–de Siebenthal theory [TDV].
- Boyle relates the complexified Albert algebra to the Standard Model and its left–right symmetric extension, one generation to the tangent space of the complex octonionic projective plane, and three generations to \(\mathrm{SO}(8)\) triality [Boyle].
- Furey obtains the Standard Model gauge group from ladder operators built from \(\R,\C,\Ha,\Oc\), without the Albert algebra [Furey].
Claims in this section are taken from the abstracts of these papers. Speculative