the complexification of the exceptional Jordan algebra and applications to particle physics

Recent papers of Todorov and Dubois-Violette[4] and Krasnov[7] con-tributed revitalizing the study of the exceptional Jordan algebra h 3 ( O ) in its relations with the true Standard Model gauge group G SM . The absence of complex representations of F 4 does not allow Aut ( h 3 ( O )) to be a candidate for any Grand Uni(cid:28)ed Theory, but the group of automorphisms of the complexi(cid:28)cation of this algebra, i.e. h C 3 ( O ) , is isomorphic to the compact form of E 6 . Following Boyle in [12], it is then easy to show that the gauge group of the minimal left-right symmetric extension of the Standard Model is isomorphic to a proper subgroup of Aut (cid:0) h C 3 ( O ) (cid:1) .


the octonions
Octonions O are, along with Real numbers R, Complex numbers C and Quaternions H, one of the four normed division algebras. In this section we will dene these algebras through the Cayley-Dickson construction and present some fundamental facts on Octonions, their geometry and their analysis.

normed division algebras
An algebra is a real vector space A with a bilinear multiplication. We will assume our algebras to be unital, i.e. it exists an element 1 ∈ A such that 1a = a1 = a for all a ∈ A. If for every a, b, c ∈ A the algebra is called associative, and if ab = ba for every a, b ∈ A then the algebra is called commutative. Finally, if the subalgebra generated by any two elements of A is associative then the algebra is called alternative.
Our focus on this section it will be on normed division algebras. A normed division algebra is a unital algebra with a real valued function |·| called norm from A to [0, ∞) such that |ab| = |a| |b| .
In particular, a normed division algebra is a division algebra, since (2) implies ab = 0 if and only if a = 0 or b = 0.
The Cayley-Dickson construction. Normed division algebras are better understood from the perspective of a more general kind of algebra, namely * −algebra, that are algebras equipped with a conjugation * , i.e. a R-linear map such that a * * = a, (ab) for all a, b ∈ A. We say that the algebra is real if a * = a for all of its elements and if we also have a * + a ∈ R, and a * a = aa * > 0, we say that the algebra is nicely normed. In this case we can dene a norm on A by |a| 2 = aa * .
Moreover we can dene the inverse a −1 as We thus have (see [2]) that all nicley normed and alternative * − algebra are normed division algebras. Let A be a * −algebra, then we can dene a new * −algebra A with elements (a, b) ∈ A × A, with multiplication given by and with a new conjugation given by This procedure is called the Cayley-Dickson construction and it's easy to see that starting with the real algebra R it produces four normed division algebras, i.e.
The four normed division algebras. A well known theorem due to Hurwitz [1] states that R, C, H and O, i.e. the algebras that can be produced through the Cayley-Dickson construction, are the only four normed division algebras. More explicitly, since specic properties are lost every time we apply the Cayley-Dickson construction [2], we have the following • R is a real commutative associative nicely normed; • C is a commutative associative nicely normed; • H is an associative nicely normed; • O is an alternative nicely normed.

the octonions
The Octonions are the only non-associative normed division algebra. A pratical way to work with them is to consider their R 8 decomposition. Let x ∈ O and let {i 0 = 1, i 1 , ..., i 7 } be a basis of R 8 , then we can encode the octonian multiplication through the following relations, i.e.
where δ αβ is the Kroenecker delta and αβγ is totally antisymmetric in α, β, γ, non zero and equal to 1 in the following set which can be mnemonically encoded through the Fano plane ( Fig. 1).
We therefore have and the expression for the norm squared Following [8] we note that a suitable basis for O is and therefore writing we have a decompositions of the Octonions given by Since the algebra of Octonions is alternative, any product involving two indipendent octonions do associate, i.e.
(xy) y = xy 2 , In fact, any two indipendent octonions generate an algebra isomorphic to the quaternions H. Another consequence of alternativity are the Moufang identities, i.e.
(xyx) z = x (y (xz)) , therefore, even though octonions under multiplications are not a group since they lack of associativity, nevertheless they form a Moufang loop. The same happens to S ⊂ O, the octonionic unit sphere, i.e.
which again is a Moufang loop under multiplication.

automorphisms of the octonions
Let A be an algebra, we dene Aut (A) the grup of automorphism of A, i.e. the group whose elements are the transformation ϕ : A −→ A such that and whose multiplication is given by composition of transformation. It is straightforward to see that automorphisms of the Quaternionic algebra H is given by the group of norm-preserving and orientation-preserving transformations on the vector space of imaginary elements of H, i.e.
Similarly, we could argue that any automorphism of the Octonions O is given by a group of norm-preserving and orientation-preserving transformation on the vector space of imaginary elements of H, i.e. Aut (O) ⊆ SO (7), but, since (14) the imaginary units i 1 , i 2 and i 4 generates all the Octonionic algebra, the remaining imaginary units, i.e. i 3 , i 5 and i 7 , are already costrained by the automorphism value on i 1 , i 2 and i 4 . Following [9], Aut (O) is a Lie group of dimension 14 whose generators {A i k , G i k } k=1,...,7 are rotations in two planes that point to the same imaginary unit i k . The generators A i k are rotations that leave unchanged other two imaginary units, e.g.
A i4 : while G i k leave unchanged only i k , e. g.
G i4 : It is worth noting that Aut (O) is isomorphic to the exceptional Lie group G 2 and that {A i1 , ..., A i7 , G i7 } generate a subgroup in G 2 isomorphic to SU (3) that will be used in the last section of this text.

octonionic analysis
Let Ω be a domain in O, let S ⊂ O be the imaginary sphere dened in (24) and let ι ∈ S. Following Gentili and Struppa in [8] we will now dene a real dierentiable function f from Ω to O as regular if for every ι ∈ S it restriction f ι to the complex line L ι ∼ = R⊕ιR passing through the origin and containing 1 and ι is holomorphic on Ω ∩ L ι . Moreover we will dene the directional derivatives along ι as As in the complex case, f is regular if and only if ∂ ι f (w) = 0 for every ι ∈ S and w ∈ Ω ∩ L ι . We then dene Series of results of complex analysis are then extended to the Octonions. It can be easily shown (see [8]) that the derivative of a regular function is a regular function, that that polynomials are regular functions and nally that a regular function has its series expansion on a ball centered in the origin B (0, R) where R is the radius of convergence of the series. The Liouville theorem equivalent also applies and if f is regular and |f | has a relative maximum, then f is constant. Finally we also have the Cauchy representation formula for regular functions on Octonions, i.e. where if Im (w) = 0 and w (0, r) = x + ι w y :

the exceptional jordan algebras
Let A be a commutative non-associative algebra. If A satises the Jordan identity, i.e.
(ab) (aa) = a (b (aa)) , for all a, b ∈ A, then we say is a Jordan algebra. It can be shown [10] that A is power-associative, i.e.
A well known result from Jordan, von Neumann and Wigner [10] states that every simple nite-dimensional formally real Jordan algebra is isomorphic to one of the following Jordan algebras: • h n (R) the self adjoint real matrices with product a • b = 1 2 (ab + ba); • h n (C) the self adjoint complex matrices with product a • b = 1 2 (ab + ba); • h n (H) the self adjoint quaternion matrices with product a•b = 1 2 (ab + ba); • h 3 (O) the 3 by 3, self adjoint octonion matrices with product a • b = 1 2 (ab + ba); Obviously we say that a matrix is self-adjoint if the coecient a ij are equal to a * ji , using the conjugation dened on the division algebra used, i.e. R, C, H and O respectively.
It is worth noting that since the algebra of Octonions O is alternative we can have a Jordan algebra made by 3 by 3 matrices, but since is not associative, we cannot have h n (O) with n greater than 3. Therefore h 2 (O) and h 3 (O) are the only Jordan algebra that can be made with octonian self adjoint matrices.
The h 2 (O) algebra. The h 2 (O) algebra can be realized as the 2 by 2 hermitian matrices, i.e. matrices of the form where α, β ∈ R and x ∈ O, equipped with the Jordan product based on the usual associative product over matrices. It will be shown in next section that this algebra is isomorphic to R 2 ⊕ R equipped with (x, t) • (x , t ) = (tx + t x, x · x + tt ) and therefore doesn't appear in the classication theorem due to Jordan, von Neumann and Wigner.
where α, β, γ ∈ R, x, y, z ∈ O and x * , y * , z * are their octonionic conjugates, equipped with the Jordan product. It is worth noting that, even if octonions are nor associative nor commutative, it is possible to dene the a determinant function given by

possible applications to particle physics
The study of Octonions in particle physics have been started by Jordan back in 1933 and then by Jordan, von Neumann and Wigner in an early attempt to generalize quantum mechanics and formalize the algebra of observables [10].
In 1973 Gnaydin and Gursey proposed the use of octonions in a split basis to model the quark structure [6]; later on, Dixon used the algebra C ⊗ H ⊗ O to model of some feature of particle physics [5]. In the rst decade of the century, Dray and Manogue (see [9] and reference therein) continued to pursue the use of Octonions in particle physics in a series of articles; while Baez and Huerta [2][3] focused on the exceptional Jordan algebra h 3 (O). More recently a series of paper by Todorov and Dubois-Violette [4], Krasnov [7] and Baez advanced on the role that h 3 (O) might have in the Standard Model. One of the main issues against the use of Octonions in Grand Unication Theories is the absence of a complex representation of groups related to Octonions such as G 2 and F 4 . This issue was recently addressed by Boyle in [12] introducing the complexication of h 3 (O) which we will briey present in section 4.3.

minkowsky spacetime from jordan algebras
Let X be an element of h 2 (C) and let σ x , σ y , σ z be the Pauli matrices, i.e.
then we can consider an element (t, x, y, z) in the Minkowski space M 4 as an element of h 2 (C) given by Since Lorentz transformation are those transformations that preserve the element then we have the identication between Spin (3, 1), i.e. the double cover of the Lorentz group SO (3, 1), and where Λ † is the transpose of Λ * .
The same construction applies to all the division algebras K, therefore we can identify the Jordan algebras h 2 (K) with the Minkowski spacetime of 3, 4, 6 and 10 dimension respectively, i.e.
On the other hand, the previous construction can also be used as a way to generalize the denition of SL (2, K) and obtaining the identication SL (2, O) ∼ = Spin (9, 1) .
(54) 4.2 octonionic representation of SU (3) One of the main interest for the use of Octonions in particle physics was motivated by the realization of SU (3) as subgroup of the exceptional Lie group G 2 , as was pointed out in [6]. We have already shown that the group of automorphisms of O is the compact form of G 2 . If we choose an immaginary unit in O, the group of automorphisms of O that xes the immaginary unit is indeed isomorphic to SU (3). The Lie group is relevant for quantum chromodynamics since a gauge colour rotation is a SU (3) group element U (x) obtained through the exponential map, i.e.
where x ∈ M 4 and the λ k are generators for the Lie algebra su (3) and are called the Gell-Mann matrices, i.e.
Using the Gell-Mann matrices we can have a 14-dimensional representation fo G 2 as where we used e 1 , e 2, and e 3 to represent the standard ortonomal vectors Using the above representation of G 2 it is straightforward to identify SU (3) as the subgroup generated by T 1 , ..., T 8 . It is worth noting that the above representation of G 2 , and consequently of SU (3), is isomorphic to its complex conjugate representation. This is a necessary condition for the representation o be real and all representation of and imposing the preservation of the idempotent, automorphisms in (66) would then require an additional feature on V ∈ SU (3), i.e. being of the form where V ∈ SU (2) and ϕ ∈ U (1) and therefore the whole automorphism would be determined by To conclude it is worth noting that that the automorphism results unchanged with the following transformations U, V , ϕ −→ e 2πi/3 U, V , e 2πi/3 ϕ , and therefore the resulting subgroup of automorphisms is indeed the true gauge group, i.e.
It is well known that E 6 is the only exceptional group to have complex representation and therefore it has been a natural candidate as gauge group in Grand Unied Theories along with SU (5) and SO (10) (e.g. [14,15]).
We can therefore reproduce the previous construction on h C 3 (O). Let ι 1 ∈ S be the imaginary unit used to decompose O as in the previous section, ι 2 the imaginary unit for the complex scalar eld in C ⊗ h 3 (O) and Then, the general automorphism of h C 3 (O) that preserves the embedding of C in O is given by where z −→ z is the conjugation in respect to the scalar eld C, and U, V L , V R ∈ SU (3). Then, if we impose, as in the previous section, an idempotent to be invariant, we then have the decomposition another complex line in the remaining C 3 structure. The latter construction has the property of having a complex representation that therefore is suitable to be be a real candidate for Grand Unied Theories.