Octonion Internal Space Algebra for the Standard Model

The paper surveys recent progress in the search for an appropriate internal space algebra for the Standard Model (SM) of particle physics. As a starting point serve Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure which implements the splitting of the octonions ${\mathbb O} = {\mathbb C} \oplus {\mathbb C}^3$ reflecting the lepton-quark symmetry. Such a complex structure in $C\ell_{10}$ is generated by the $C\ell_6(\subset C\ell_8\subset C\ell_{10})$ volume form, $\omega_6 = \gamma_1 \cdots \gamma_6$, left invariant by the Pati-Salam subgroup of $Spin(10)$, $G_{\rm PS} = Spin (4) \times Spin (6) / {\mathbb Z}_2$. While the $Spin(10)$ invariant volume form $\omega_{10}=\gamma_1 ... \gamma_{10}$ is known to split the Dirac spinors of $C\ell_{10}$ into left and right chiral (semi)spinors, ${\cal P} = \frac12 (1 - i\omega_6)$ is interpreted as the projector on the 16-dimensional \textit{particle subspace} (annihilating the antiparticles). The standard model gauge group appears as the subgroup of $G_{PS}$ that preserves the sterile neutrino (identified with the Fock vacuum). The $\mathbb{Z}_2$-graded internal space algebra $\mathcal{A}$ is then included in the projected tensor product: $\mathcal{A}\subset \mathcal{P}C\ell_{10}\mathcal{P}=C\ell_4\otimes \mathcal{P} C\ell_6^0\mathcal{P}$. The Higgs field appears as the scalar term of a superconnection, an element of the odd part, $C\ell_4^1$, of the first factor. The fact that the projection of $C\ell_{10}$ only involves the even part $C\ell_6^0$ of the second factor guarantees that the colour symmetry remains unbroken. As an application we express the ratio $\frac{m_H}{m_W}$ of the Higgs to the $W$-boson masses in terms of the cosine of the {\it theoretical} Weinberg angle.


Introduction
The elaboration of the Standard Model (SM) of particle physics was completed in the early 1970's. To quote John Baez [B21] 50 "years trying to go beyond the Standard Model hasn't yet led to any clear success". The present survey belongs to an equally long albeit less fashionable effort -to clarify the algebraic (or geometric) roots of the SM, more specifically, to find a natural framework featuring its internal space properties. After discussing some old explorations we provide an updated exposition of recent developments (in particular of [T21]), clarifying on the way the meaning and the role of complex structures, concentrating on one associated with a Clifford subalgebra (in our case Cℓ 6 ) pseudoscalar.
Most ideas on the natural framework of the SM originate in the 1970's, the first decade of its existence. (Two exceptions: the Jordan algebras were introduced and classified in the 1930's [J, JvNW]; the noncommutative geometry approach was born in the late 1980's, [C, DKM, CL], and is still vigorously developed by Connes, collaborators and followers [CC, CCS, BF, CIS, NS].) First, early in 1973, the ultimate division algebra, the octonions 1 were introduced by Gürsey 2 and his student Günaydin [GG, G] for the description of quarks and their SU (3) colour symmetry. The idea was taken up and extended to incorporate all four division algebras by G. Dixon (see [D10, D14] and earlier work cited there) and is further developed by Furey [F14,F15,F16,F,F18,FH1,FH]. Dubois-Violette (D-V) [D16] emphasizes the lepton-quark correspondence and the unimodularity of the colour group, SU (3) c , as a physical motivation for introducing the octonions. They come equipped with a complex structure preserved by the subgroup SU (3) of the automorphism group G 2 of O: (1.1)

(Split) octonions as composition algebras
One can in fact provide a basis free definition of the octonions starting with the splitting (1.1). To this end one uses the skew symmetric vector product and the standard inner product on C 3 to define a noncommutative and non-associative distributive product xy on O and a real valued nondegenerate symmetric bilinear form x, y = y, x such that the quadratic norm N (x) = x, x is multiplicative: [D16,TD]). Furthermore, defining the real part of x ∈ O by Re x = x, 1 and the octonionic conjugation x → x * = 2 x, 1 − x, we shall have where 1 stands for the (real) octonion unit. A unital algebra with a nondegenerate quadratic norm obeying (1.2) is called a composition algebra.
Another basis free definition of the octonions O and of their split version O can be given in terms of quaternions by the Cayley-Dickson construction. We represent the quaternions as scalars plus vectors with the vector product U × V ∈ R 3 satisfying (1.5) The product (1.4) is clearly noncommutative but one verifies that it is associative. The Cayley-Dickson construction defines the octonions O and the split octonions O in terms of a pair of quaternions and a new "imaginary unit" l as: (1.6) We shall encounter the split octonions as generators of Cℓ(4, 2) in Sect. 3.1 below.

Jordan algebras; GUTs; Clifford algebras
Studying quantum field theory it appears natural to replace classical observables (real valued functions) by an algebra of functions on space-time with values in a finite dimensional euclidean Jordan algebra 3 . As a particularly attractive choice, which incorporates the idea of lepton-quark symmetry, D-V proposes [D16] the exceptional Jordan or Albert algebra of 3 × 3 hermitian octonionic matrices, the only irreducible one which does not admit an associative envelope [A]. Further progress was achieved in [TD,DT,T,DT20] by considering the Clifford algebra envelope of its nonexceptional subalgebra J 8 2 which fits one generation of fundamental fermions. In these papers, as well as in the present one, we are effectively working with associative algebras which should be viewed as an internal space counterpart of Haag's field algebra (see [H]).
A second development, Grand Unified Theory (GUT), anticipated again in 1973 by Pati and Salam [PS], became mainstream 4 for a time. Fundamental chiral fermions fit the complex spinor representation of Spin(10), introduced as a GUT group by Fritzsch and Minkowski and by Georgi. A preferred symmetry breaking yields the maximal rank semisimple Pati-Salam subgroup, Spin(4) × Spin(6) Z 2 ⊂ Spin(10), Spin(4) = SU (2) L × SU (2) R , Spin(6) = SU (4) . (1.8) We note that G PS is the only GUT group which does not predict a gauge triggered proton decay. It is also encountered in the noncommutative geometry approach to the SM [CCS, BF]. In general, GUTs provide a nice home for the fundamental fermions, as displayed by the two 16-dimensional complex conjugate "Weyl (semi)spinors" of Spin(10). Their other representations, however, like the 45-dimensional adjoint representation of Spin(10) are much too big, involve hypothetical particles like leptoquarks which cause difficulties. The Clifford algebra 5 Cℓ 10 , on the other hand, like the Clifford algebra of any even dimensional euclidean vector space, has a unique irreducible representation (IR); in the case of 6 Cℓ 10 ∼ = R[2 5 ] it is the 32-component real (Majorana) spinor. Viewed as a representation of Spin(10) it splits upon complexification into two 16-dimensional (complex) IRs which can be naturally associated to the left and right chiral fundamental (anti)fermions of one generation: (1.9) Clifford algebras were also applied to the SM in the 1970's -see [CG] and references therein. An essential difference in our approach is the use of the octonions with a preferred complex structure in Cℓ −6 , Cℓ 8+ν , ν = 0, 1, 2 restricting the associated gauge group. (Another new point, the use of the Z 2 grading of Cℓ 10 to define the Higgs field, will be discussed in Sect. 1.3 below.) The Pati-Salam subgroup of Spin(10) is singled out as the stabilizer of the Cℓ 6 (⊂ Cℓ 8 ⊂ Cℓ 10 ) pseudoscalar (1.10) Here L x is the operator of left multiplication in the 8-dimensional real vector space of the octonions, L x y = xy for x, y ∈ O; σ ν , ν = 0, 1, 2, 3 are the Pauli matrices (σ 0 = 1I 2 ). The action of the operators L α ∈ R[8] on the octonion units will be made explicit in Sect. 2.2 (Eq. (2.9)). The products ǫ ⊗ L α , α = 1, ..., 7 and σ 1 ⊗ 1I 8 generate the Clifford subalgebra Cℓ 8 of Cℓ 10 . All generators of Cℓ 10 are listed in Eq. (2.23) below. The group G P S (1.8) preserves, in fact, each factor in the graded tensor product representation of Cℓ 10 : Cℓ 10 = Cℓ 4⊗ Cℓ 6 (1.11) introduced earlier by Furey [F16, F] and exploited in [T21]. The complex structure J ∈ SO(10) generated by ω 6 will be displayed and the physical interpretation of the mutually orthogonal projection operators will be revealed in Sect. 2.3.
5 Aptly called geometric algebra by its inventor -see [DL]. 6 For any associative ring K, in particular, for the division rings K = R, C, H, we denote by K[m] the algebra of m × m matrices with entries in K.

Main message and organization of the paper
The present survey focuses on ongoing attempts to answer two questions: 1) Why the arbitrarily looking gauge group of the SM, and what dictates its highly reducible representation for fundamental fermions? 2) How to put together the Higgs field with the gauge bosons? Can we explain their mass ratios?
1. Most physicists accept GUT as an answer to the first question. One has then the intriguing result of Baez and Huerta, [BH], that G SM appears as the intersection of two popular GUT subgroups of Spin(10): A top down approach, however, starting with Spin(10) should involve the maximal rank subgroup U (5) instead of SU (5), in line with the philosophy of Borel -de Siebenthal, [BdS], yielding an extra U (1) factor in the intersection.
The minority trying to go further includes, besides the fans of octonions and the already cited enthusiasts of almost commutative real spectral triples, also Holger Nielsen whose more than two decades of musing over the problem are reviewed in [NB]. Our approach exploits the complex structure and the particle projector P (1.12) associated with the Clifford pseudoscalar ω 6 (1.10). It permeates the entire paper (Sects. 2.3, 3.2-4, 5.1, ...).
2. The second problem has been universally recognized (see, e.g., the popular account [M14]). We follow the superconnection approach anticipated by Ne'eman and Fairlie -for a concise review and references see Sect. 4.1. We exploit the restricted particle projector P r which annihilates the sterile neutrino (Sect. 3.3) to deform the Fermi oscillators in the lepton sector into the odd generators of the simple Lie superalgebra postulated in [N, F79]. The resulting difference between the flavour spaces of leptons and coloured quarks allows to compute the mass ratio m H /m W in agreement with experiment (Sect. 4.2).
The paper aims to be selfcontained and combines in a single narrative our contribution with review of background material. Sect. 2.1 provides a summary of the known triality realization of Spin(8). Sect. 2.2 and Appendix A spell out the relation between left and right multiplication by imaginry octonion units, applied in Sect. 3.4 to display the stabilizer of R 7 . We should like to single out two messages from Sect. 2.3: (1) the indirect connection between the Cℓ 6 pseudoscalar and the complex structure J ∈ SO(8) (2.20) (2.22); (2) the observation that the Lie subalgebra of so(8) that commutes with ω 6 and the electric charge operator Q (2.32) is the rank four subalgebra su(3) c ⊕ u(1) Q ⊕ u(1) B−L . Sect. 3.1 contains, along with a glance on the equivalence class of Clifford algebras involving Cℓ(3, 1), Cℓ −6 (= Cℓ(0, 6)), Cℓ 10 , the observation that the conformal Clifford algebra Cℓ(4, 2) of this class is generated by the split octonions and gives rise to their isometry group SO(4, 4). Sect. 3.2 contains one of the main messages of the paper: the SM gauge group (1.13) is the soubgroup of G P S (1.8) that leaves the sterile neutrino invariant (Proposition 3.1). Sect. 3.3 discusses superselection rules and the superselection of the weak hypercharge. Sect. 3.4 reviews and comments on recent work [K, FH] on complex structure associated with the right action of O on the octonion units as well as the derivation of the gauge group for the SM [TD-V] and its left-right symmetric extension [B].
The Dirac operator γ µ (∂ µ + A µ ) anticommutes with the chirality γ 5 and hence intertwines left and right fermions; so does the Higgs field which substitutes a mass term in the fermionic Lagrangian. This has inspired Connes and coworkers, [C, CL, CC], to identify the Higgs field with the internal space part of the Dirac operator. This idea finds a natural implementation in the Clifford algebra approach to the SM superconnection (reviewed in Sect. 4.1). The concise exposition in Sect. 4 emphasizes our assumptions and some delicate points, referring the reader for calculational details to the preceding publication [T21].
We recapitulate our convoluted route to Cℓ 10 in Sect. 5.1. In Sect. 5.2 we compare our solution of the fermion doubling problem with the approach of [FH1]. A summary of the main results of the paper is given in Sect. 5.3 which also cites existing (inconclusive) attempts to understand why are there exactly three generations of fundamental fermions.

The action of octonions on themselves
The group Spin(8), the double cover of the orthogonal group SO(8) = SO(O), can be defined (see [Br, Y]) as the set of triples (g 1 , g 2 , g 3 ) ∈ SO(8) × SO(8) × SO(8) such that g 2 (xy) = g 1 (x) g 3 (y) for any x, y ∈ O . (2.1) If u is a unit octonion, u * u = 1, then the left and right multiplications by u are examples of isometries of O |L u x| 2 = ux, ux = x, x , |R u x| 2 = xu, xu = x, x for u, u = 1 . (2.2) Using the Moufang identity 7 u(xy)u = (ux)(yu) for any x, y, u ∈ O , (2.3) one verifies that the triple g 1 = L u , g 2 = L u R u , g 3 = R u satisfies (2.1) and hence belongs to Spin(8). It turns out that triples of this type generate Spin(8) (see [Br] or Yokota's book [Y] for a proof).
They are expressed by each other in terms of the involution π: We find, in particular -see Appendix A: While L 78 = 4L 13 L 26 L 45 (2.11) commutes with the entire Lie algebra spin(6) = su(4) the u(1) generator (whose physical meaning is revealed by (2.32)) where the second summand is the unbroken colour Lie algebra su(3) = su(3) c .
A compact way to identify the particle states in a Clifford algebra Cℓ 2n is to introduce a complex structure which as we shall demonstrate gives rise to a fermionic Fock space in Cℓ 2n .
A complex structure in an even-dimensional real euclidean space E 2n with a positive definite symmetric scalar product (X, Y ) = (Y, X) is an orthogonal transformation J of E 2n of square -1; equivalently, a complex structure is a skewsymmetric isometry J of E 2n : For a non zero vector X and a complex structure J the vector Y = JX is orthogonal to X (and has the same norm): It follows that for each complex structure J in E 2n there exists an orthonormal basis of the form (γ 1 , ..., γn, Jγ 1 , ..., Jγn) in Cℓ 2n . Then a j = 1 2 (γ j − iJγ j ) and a * j = 1 2 (γ j + iJγ j ), j = 1, ..., n span each the image in Cℓ 2n (= Cℓ 2n (C)) of a maximal isotropic subspace of the complexification of E 2n . Together they yield a realization of the canonical anticommutation relations (CAR). Fermionic oscillators have been used in the present context in [B77, F]. The complex structure in so(2n) involves a distinguished maximal (rank n) Lie subalgebra (a notion studied in [BdS]), u(n) ⊂ so(2n), generated by the products a j a * k . It also selects two distinguished u(n) singlet states in Cℓ 2n , the vacuum, annihilated by all a j and its antipode, annihilated by the a * j . Both singlets are annihilated by the simple part su(n) of u(n). Complex structures have been studied in relation to spinors byÉlie Cartan (since 1908), Veblen (1933), Chevalley (1954. For a carefully written servey with historical highlightssee [BT]. The second reference [BT] and the concise modern exposition [D] connect simple spinors to the states in a fermionic Fock space. We have been also influenced by their use (in SO(9)) by Krasnov [K] and by relating them to Clifford pseudoscalars in [FH].
The pseudoscalar ω 6 of Cℓ 6 belongs to Cℓ 8 but only defines a complex structure through its action on the octonion units. More precisely, taking the basic relations (2.9) and the identity ǫ 2 = −σ 0 into account, we can write (2.20) Warning: due to nonassociativity of O, −L 7 e 3 = e 1 does not imply −L 7 L 3 = L 1 etc.
We shall see that each of the subgroups Spin(n) ω6 of the spin groups of Cℓ n for n = 8, 9, 10 that leaves ω 6 invariant is relevant for particle physics: The U (1) factor in the U (4) of Spin(8) ω6 and the SU (2) in Spin(9) ω6 are generated by the third, respectively by all three components of the "total weak isospin" I = I L + I R as will be made explicit in Sect. 3.2.
We shall identify the generators (of the comlexification sℓ(3, C)) of su (3) with the traceless part of the matrix (b j b * k ) whose elements belong to H (1,1) . Then the splitting (2.29) of P into the su(3) singlet ℓ and the triplet q implements the lepton-quark splitting anticipated by its image (1.1) on the octonions. We shall thus interpret the 1-dimensional projectors ℓ and q j as describing the lepton and the coloured quark states in Cℓ 6 . The states ℓ and q j are mutually orthogonal idempotents, ℓ playing the role of Fock vacuum in Cℓ 6 : Remark.
-We shall argue in Sect. 3.3 that the identification of P as a particle subspace projector (adopted in [DT20]) would be only justified if we have a clear distinction between particles and antiparticles. This can be claimed for the 30 fundamental (anti)fermions of the Cℓ 10 multiplet 32 (1.9) which have different quantum numbers with respect to the gauge Lie algebra g SM of the SM. It fails in the 2-dimensional subspace of sterile neutrinos annihilated by g SM ; ν R andν L are allowed to form a coherent superposition -a Majorana spinor. We adopt in Sects. 3.3 and 4. the restricted projector ℓ r (3.21) on the (3-rather than 4-dimensional) lepton subspace, excluding the sterile neutrino.
In order to extend the Fock space picture to Cℓ 8 we shall set where the pair (a * , a) describes another Fermi oscillator ([a, a * ] + = 1) anticommuting with b j , b * k . We shall fix the physical interpretation of [a * , a] by postulating that the electric charge operator is given by (2.32) stands for the difference between the baryon and the lepton numbers. B − L takes eigenvalues ± 1 3 for (anti)quarks and ∓1 for (anti)leptons. Demanding that the gauge Lie algebra within so(8) commutes with both ω 6 and Q we shall further reduce it from so(6) ⊕ so(2) to the rank four Lie subalgebra (2.33) The knowledge of the charges Q, B − L along with the colour Lie algebra allows to identify the primitive idempotents of Cℓ 8 , given by ℓ, q j multiplied by aa * or a * a, with the fundamental femions: (2.34) The "isotopic doublets" (ν, e) and (u j , d j ) stand for neutrino / electron and up / down coloured quarks. We see, in particular, that the Fock vacuum in Cℓ 8 associated with the complex structure (2.20) is identified with the neutrino (as it has no charge and aν = 0 = b j ν). Note that the subalgebra of g 4 which annihilates ν is the known unbroken gauge Lie algebra u(3) of the SM: This picture ignores chirality which will find its place in Cℓ 10 (Sect. 3.2).
3 The internal space subalgebra of Cℓ 10

Equivalence class of Lorentz like Clifford algebras
Nature appears to select real Clifford algebras Cℓ(s, t) of the equivalence class of Cℓ(3, 1) (with Lorentz signature in four dimensions) inÉlie Cartan's classification (which involves 9 the signs, ω 2 (s, t) and (−1) s−t ): The elements of Cℓ(s, t) are operators acting in the 2 n dimensional real vector space S of Majorana spinors. The space S admits no nontrivial real Spin(s, t) invariant subspace. If γ 1 , · · · , γ 2n is the image in Cℓ(s, t) of an orthonormal basis of the underlying vector space R s,t then the Clifford volume form 2) defines a complex structure which commutes with the action of Spin(s, t). For a (complex) chiral basis in which ω(s, t) is diagonal (and hence pure imaginary for s − t = 2(mod8)) the Dirac spinor splits into two 2 n−1 -dimensional complex Weyl (semi) spinors transforming under inequivalent complex conjugate representations of Spin(s, t). The corresponding projectors Π L and Π R on left and right spinors are given in terms of the chirality χ which coinsides with γ 5 for Cℓ(3, 1): Another interesting example of the same equivalence class (also with indefinite metric) is the conformal Clifford algebra Cℓ(4, 2) (with isometry group O(4, 2)). We shall demonstrate that just as Cℓ −6 was viewed (in Sect. 2.2) as the Clifford algebra of the octonions, Cℓ(4, 2) plays the role of the Clifford algebra of the split octonions (also appearing in bi-twistor theory, [P23]): (3.4) Indeed, defining the mapping (cf. (1.6)) we find that the missing split-octonion (originally, quaternion) imaginary unit k (= ij = −ji) can be identified with the Cℓ(4, 2) pseudoscalar: The conjugate to the split octonion x (3.4) and its norm are so that the isometry group of O is O(4, 4). (In particular, the maximal compact subalgebra so(4) ⊕ so(4) ⊂ so(4, 4) is spanned by γ jk , j, k = 1, ..., 4 and by ω 4,2 , γ α , α = −1, 0, and their commutators. The remaining 16 noncompact generators of so(4, 4) involve the square-one matrices γ j , γ α γ j , γ j ω 4,2 .) As we are interested in the geometry of the internal space of the SM, acted upon by a compact gauge group we shall work with (positive or negative) definite Clifford algebras Cℓ 2ℓ , ℓ = 1(mod 4). The algebra Cℓ −6 , considered in Sect. 2, belongs to this family (with ℓ = −3). For ℓ = 1 we obtain the Clifford algebra of 2-dimensional conformal field theory; the 1-dimensional Weyl spinors correspond to analytic and antianalytic functions. Here we shall argue that for the next allowed value, ℓ = 5, the algebra Cℓ 10 = Cℓ 4 ⊗ Cℓ 6 (1.11), fits beautifully the internal space of the SM, if we associate the two factors to colour and flavour degrees of freedom, respectively. We shall strongly restrict the physical interpretation of the generators γ ab = 1 2 [γ a , γ b ], a, b = 1, · · · , 10 of the derivations of Cℓ 10 by demanding that the splitting (1.11) of Cℓ 10 into Cℓ 4 and Cℓ 6 is preserved. This amounts to select a first step of symmetry breakings of the GUT group Spin(10) leading to the semisimple Pati-Salam group (Spin(4)×Spin(6))/Z 2 (1.8). Each summand, so(4) and so(6), of g P S , expressed in terms of Fermi creation and annihilation operators, has a distinguished Lie subalgebra, u(2) respectively u(3), which belongs to H 1,1 . We identify the leptons and quarks with u(3) singlets and triplets. This identification implements the lepton-quark symmetry alluded to by (1.1).
(2.21) in Sect. 2.3. Conversely, I L 3 , I R 3 appear as chiral projections of I 3 : The identification of the vacuum vector, a 1 a 2 b 1 b 2 b 3 (annihilated by all a α , b j ) becomes consequential if we demand that this ket-vector is a singlet with respect to the gauge group of the SM. The fact that the left and right isospin cannot vanish simultaneously (since (2(I L 3 + I R 3 )) 2 = 1) implies that the Lie algebra g SM of the SM should be chiral: (3.12) It is therefore rewarding that we can identify the Fock space vacuum in Cℓ 10 (given by ν of (2.34) for Cℓ 8 ) with the (right handed, hypothetical) sterile neutrino (in fact, ν R and its antipodeν L do not interact with the gauge bosons): 13) The role of the electric charge Q (2.33) which breaks the u(4) symmetry of ω 6 in so(8) to u(3) ⊕ u(1) Q is played by the weak hypercharge Y in so(10): (3.14) They both annihilate the respective vacuum state as well as its antipode. This is made obvious by the two forms of Y in Eq. (3.14) as sums of normal and of antinormal products. By definition Y belongs to the centre of the broken symmetry subalgebra of g P S . As pointed out in [T21] -and will be discussed below in Sect. 3.3 -it gives rise to a superselection rule in the SM. The significance of choosing the sterile neutrino as a Fock vacuum is summarized by the following Proposition 3.1 The Lie subgroup of G P S (1.8) that leaves the Fock vacuum ν R (3.13) invariant is the SM gauge group (1.13).
Proof. We shall first complete the argument that the maximal Lie subalgebra of g P S annihilating the sterile neutrino is g SM . We have already noted that the Lie subalgebra of g P S for which the vacuum transforms as a singlet is u(2)⊕u(3) (3.12). This follows from the observation that generators involving a * 1 a * 2 and b * j b * k transform ν R into a righthanded electron e R and an up quark u R , respectively. It remains to analyze the 2-dimensional centre u(1) B−L + u(1) I R 3 of this extended algebra. ν R andν L are eigenvectors of both generators with eigenvalues of opposite sign; only multiples of Y annihilate the sterile neutrino: This establishes the characterization of the Lie algebra g SM as annihilator of sterile neutrino. It will be straightforward to extend the result to the SM gauge group (1.13) after displaying the quantum numbers of the fundamental fermions in the following subsection.

Superselection rules. Restricted particle subspace
The weak hypercharge (3.14) (3.15) generates the u(1) centre of the gauge Lie algebra of the SM, hence commutes with all gauge transformations. It is conserved not only in the observed micro processes but even in hypothetical ones, like a possible proton decay (with a conserved B −L), or in the presence of a Majorana neutrino (a coherent superposition of ν R andν L ) that would break B − L by two units. It was proposed in [T21] as a superselection rule, assuming that Y commutes with all observables. The Jordan algebra of the 32-dimensional space of internal observables of one generation splits into 11 superselection sectors corresponding to the 11 different eigenvalues of Y (see Appendix to [T21]).
The superselection of the electric charge has been thoroughly discussed (see the second paper in [WWW] and the review [G07]); for more references and a historical survey addressed to philosophers -see [E]. The charge Q (2.32) is superselected by the exact symmetry of the SM (otherwise I L ± do not commute with it). SSRs are also related to measurement theory, [T11]. SSR and superselection sectors are an essential part of the Doplicher-Haag-Roberts reconstruction of quantum fields from the algebra of observables -see [H].
For all we know, the exact symmetry of the SM is given by the rank four unbroken Lie algebra (obtained from g 4 (2.33) by the substitution B − L → Y ): (3.16) The states of the fundamental (anti)fermions are given by the primitive idempotents of Cℓ 10 , represented by the 2 5 = 32 different products of the five pairs of basic projectors π j (3.8) (2.28). The 16 particles can be labeled by the eigenvalues of the pair of superselected charges (Q, Y ): (j, k, ℓ) ∈ Perm(1, 2, 3), q = q 1 + q 2 + q 3 = q 2 , tr q = 12. (As the colour is unobservable we do not bother to assign to it eigenvalues of the diagonal operators iγ 13 , iγ 26 , iγ 45 that would replace the index j.) Note that chirality in the particle subspace, Pχ = χP is determined by the hypercharge: The charges (Q, Y ) for the corresponding antiparticles have opposite sign. The spectrum of Y and of 2I L 3 = 2Q − Y together with the analysis of [BH] allow to complete the group theoretic version of Proposition 3.1.
The fact that ν R ,ν L are not distinguished by the superselected charges has a physical implication: one can consider their coherent superposition as in the now popular theory of a (hypothetical) Majorana neutrino. This suggests the introduction of a restricted 15-dimensional particle subspace, with projector (3.21) Theories whose field algebra is a tensor product of a Dirac spinor bundle on a spacetime manifold with a finite dimensional internal space usually encounter the problem of fermion doubling [GIS] (still discussed over 20 years later, [BS]). It was proposed in [DT20] as a remedy to consider the algebra PCℓ 10 P where P is the projector (2.29) on the 16 dimensional particle subspace (including the hypothetical right-handed sterile neutrino). It is important -and will be essential in the treatment of the Higgs field (Sect. 4) -that the operators a ( * ) α and b ( * ) j behave quite differently under particle projection. While a ( * ) α commute with P so that Pb j P = 0 = Pb * j P.
(3.23) Accordingly, while the generators (3.9) of the (electroweak) flavour "left-right symmetry" su(2) L ⊕ su(2) R just get multiplied by P, the particle subspace projections of the su(3) c generators take a modified form: T a = 1 2 B jk λ kj a , λ a ∈ H 3 (C), tr λ a = 0, tr λ a λ b = 2δ ab , a, b = 1, · · · , 8 , (3.24) but still obey the same CR. It makes sense to consider the gauge Lie algebra in the lepton and the quark sectors (or the factors Cℓ 4 and Cℓ 6 in Cℓ 10 ) separately just noting that P(B − L) = −1 for leptons and P(B − L) = 1 3 for quarks. It is particularly appropriate to treat the lepton sector by itself when using the restricted particle space as there the flavour oscillators a ( * ) α are also modified: The operators A ( * ) α provide a realization of the four odd generators of the smallest simple Lie superalgebra, sℓ(2|1), whose even part is su(2) L ⊕ u(1) Y . (For a detailed identification with the standard definition of sℓ(2|1) see Sect. 3 of [T21].) The nonvanishing anticommutators of A ( * ) α are: The resulting internal space algebra which leaves out the sterile neutrinos is the direct sum Here A qq is effectively the 9 dimensional associative envelope of u(3) ⊂ Cℓ 0 6 spanned by B jk and q i (3.24); A ℓq is the 3-dimensional space spanned by ℓb 1 b 2 q 3 and its cyclic permutations, A qℓ is hermitian conjugate to it. The sum of three terms multiplying Cℓ 4 in (3.27) is isomorphic to the 15-dimensional Lie algebra su(4).

Complex structure associated with R 7 : a comment
Following [K,FH,K21] we shall display and discuss the symmetry subalgebras of Cℓ n , n = 8, 9, 10, of the complex structure generated by the Clifford pseudoscalar ω R 6 corresponding to the right action of the octonions, (3.28) Written in terms of the colour projectors p j and p ′ j the hermitian pseudoscalar iω R 6 assumes the form: where we have used While the term P ′ − P (2.29) commutes with the entire derivation algebra spin(6) = su(4) of Cℓ 6 , the centralizer of B −L in su(4) is u(3) -see Proposition A2 in Appendix A. It follows that the commutant of ω R 6 in so(8) is u(3) ⊕ u(1) while its centralizer in so(9) is the gauge Lie algebra g SM = su(3) ⊕ su(2) ⊕ u(1) of the SM; finally, in so (10), ω R 6 is invariant under the left-right symmetric extension of g SM ( [FH,K21]), (3.31) Furthermore, as proven in [K], the subgroup of Spin(9) that leaves ω R 6 invariant is precisely the gauge group 10 G SM = S(U (2) × U (3)) (1.13) of the SM (with the appropriate Z 6 factored out). One is then tempted to assume that Cℓ 9 , the associative envelope of the Jordan algebra J 8 2 = H 2 (O), may play the role of the internal algebra of the SM, corresponding to one generation of fundamental fermions, with Spin(9) as a GUT group [TD, DT]. We shall demonstrate that although G SM appears as a subgroup of Spin(9) its representation, obtained by restricting the (unique) spinor IR 16 of Spin(9) to S(U (2) × U (3)) only involves SU (2) doublets, so it has no room for (e R ), (u R ), (d R ) (3.17) (3.18). We shall see how this comes about when restricting the realization (3.9) of I L and I R to Spin(9) ⊂ Cℓ 9 . It is clear from (3.9) that only the sum a 1 + a * 1 = γ 9 (not a 1 and a * 1 separately) belongs to Cℓ 9 . So the su(2) subalgebra of spin(9) corresponds to the diagonal embedding su(2) ֒→ su(2) L ⊕ su(2) R : In other words, the spinorial IR 16 of Spin (9) is an eigensubspace of the projector P 1 = (2I L 3 ) 2 . It consists of four SU (2) L particle doublets and of their right chiral antiparticles. More generally, the only simple orthogonal groups with a pair of inequivalent complex conjugate fundamental IRs, are Spin(4n + 2) (see, e.g. [CD], Proposition 5.2, p. 571). They include Spin(10) but not Spin(9).
A direct description of the IR 16 L of Spin(10) acting on CH ⊗ CO is given in [FH1]. (Here CH and CO are a short hand for the complexified quaternions and octonions: CH := C⊗ R H.) The right action of CH on elements of CH ⊗ CO which commutes with the left acting spin(10), is interpreted in [FH1] as Lorentz (SL(2, C)) transformation of (unconstrained) 2-component Weyl spinors.
The left-right symmetric extension g LR (3.30) of g SM has a long history, starting with [MP] and vividly (with an admitted bias) told in [S17]. It has been recently invigurated in [HH, DHH]. The group G LR was derived by Boyle [B] starting with the group E 6 of determinant preserving linear automorphisms of the complexified Albert algebra CJ 8 3 and following the procedure of [TD-V].
4 Particle subspace and the Higgs field

The Higgs as a scalar part of a superconnection
The space of differential forms Λ * = Λ 0 +Λ 1 +Λ 2 +... can be viewed as Z 2 graded setting Λ ev = Λ 0 +Λ 2 +..., Λ od = Λ 1 +Λ 3 +.... Let M = M 0 +M 1 be a Z 2 graded matrix algebra. A superconnection in the sense of Quillen [Q, MQ] is an element of Λ ev ⊗ M 1 + Λ od ⊗ M 0 , the odd part of the tensor product Λ * ⊗ M . A critical review of the convoluted history of this notion and its physical implications is given in Sect. IV of [T-M] . (One should also mention the neat exposition of [R] -in the context of the Weinberg-Salam model with two Higgs doublets.) Let D be the Yang-Mills connection 1-form of the SM, where Y, I L and T a are given by (3.14), (3.9) and (3.24), respectively, G a µ is the gluon field, W µ and B µ provide an orthonormal basis of electroweak gauge bosons; the normalization constant N wil be fixed in Eq. (4.13) below. Then one defines a superconnection D in [DT20] involving the chirality χ (3.12) by (We omit, for the time being, the projector P in A µ and Φ.) The last equation follows from (3.23): the projection on the particle subspace kills the odd part of Cℓ 6 thus ensuring that the quarks' colour symmetry remains unbroken. The factor χ (first introduced in this context in [T-M]) ensures the anticommutativity of Φ and χD without changing the Yang-Mills curvature D 2 = (χD) 2 . The projector P (2.29) on the 16 dimensional particle subspace that includes the hypothetical right chiral neutrino (and is implicit in (4.2)) was adopted in [DT20]. By contrast, particles are only distinguished from antiparticles in [T21] if they have different quantum numbers in the Lie algebra of the SM Thus, in [T21] P is replaced by the 15-dimensional projector P r = q + ℓ r (3.21). We have seen that the projected odd operators A α ℓ r give rise to a realization of the four odd elements of the 8-dimensional simple Lie superalgebra sℓ(2|1) whose even part is the 4-dimensional Lie algebra u(2) of the Weinberg-Salam model of the electroweak interactions. It is precisely the Lie superalgebra proposed in 1979 independently by Ne'eman and by Fairlie [N, F79] (and denoted by them su(2|1)) in their attempt to unify su(2) L with u(1) Y (and explain the spectrum of the weak hypercharge). Let us stress that the representation space of sℓ(2|1) consists of the observed left and right chiral leptons (rather than of bosons and fermions like in the popular speculative theories in which the superpartners are hypothetical). Note in passing that the trace of Y on negative chirality leptons (ν L , e L ) is equal to its eigenvalue on the unique positive chirality state (e R ) (equal to −2) so that only the supertrace of Y vanishes on the lepton (as well as on the quark) space. This observation is useful in the treatment of anomaly cancellation (cf. [T-M20]).
We shall sketch the main steps in the application of the superconnection (4.2) to the bosonic sector of the SM emphasizing specific additional hypotheses used on the way (for detailed calculations see [T21]).
The canonical curvature form satisfies the Bianchi identity equivalent to the (super) Jacobi identity of our Lie superalgebra. It is important that the Bianchi identity, needed for the consistency of the theory, still holds if we add to D 2 a constant matrix term with a similar structure. Without such a term the Higgs potential would be a multiple of Tr Φ 4 and would only have a trivial minimum at Φ = 0 yielding no symmetry breaking. The projected form of Φ (4.2) and hence the admissible constant matrix addition to Φ 2 depends on whether we use the projector P (as in [DT20]) or P r (as in [T21]). In the first case we just replace a ( * ) α with a ( * ) α P. In the second, however, the odd generators for leptons and quarks differ and we set: where ρ (like N in (4.1)) is a normalization constant that will be fixed later.
Recalling that ℓ and q are mutually orthogonal (ℓq = 0 = qℓ, ℓ + q = P) we find This sugggests defining the SM field strength (the extended curvature form) as (while m 2 = m 2 P for the 16 dimensional particle subspace of [DT20]).

Higgs potential and mass formulas
This yields the bosonic Lagrangian (setting T rX = 1 4 trX -see [T21]) where the Higgs potential V (Φ) is given by (noting that T rℓ r = 3 4 ): (4.10) Minimizing V (Φ) gives the expectation value of the square of φ = (φ 1 , φ 2 ): The superscript m indicates that φ α take constant in x values depending on the mass parameter m.) The mass spectrum of the gauge bosons is determined by the term − Tr [A µ , Φ] [A µ , Φ] of the Lagrangian (4.9) with A µ and Φ given by (4.1) and (4.6) for φ α = φ m α . The gluon field G µ does not contribute to the mass term as Cℓ 0 6 commutes with Cℓ 1 4 . The resulting quadratic form is, in general, not degenerate, so it does not yield a massless photon. It does so however if we assume that Φ m is electrically neutral (i.e. commutes with Q (3.16)): (4.12) The normalization constant N (= tg θ w ) is fixed by assuming that 2I L 3 and N Y are equally normalized: As Y (ν R ) = 0 = I L 3 (ν R ) this result for the "Weinberg angle at unification scale" is independent on whether we use P or P r . If one takes the trace over the leptonic subspace the result would have been (tg θ w ) 2 = 1 3 (⇒ sin θ w = 1 2 , [F79]) closer to the measured low energy value.
Demanding, similarly, that the leptonic contribution to Φ 2 is the same as that for a coloured quark (which gives ρ = 1 for the projector P) we find (4.14) The ratio . (4.15) The result of [T21], much closer to the observed value, can also be written in the form m 2 H = 4 cos 2 θ W m 2 W , where θ W is the theoretical Weinberg angle (4.13).

Coming to Cℓ 10
The search for an appropriate choice of a finite dimensional algebra suited to represent the internal space F of the SM is still going on. The road to the choice of Cℓ 10 , our first step to the restricted algebra A (3.27), has been convoluted.
In view of the lepton-quark correspondence which is embodied in the splitting (1.1) of the normed division algebra O of the octonions, the choice of Dubois-Violette [D16] of the exceptional Jordan algebra F = H 3 (O) (1.7) looked particularly attractive. We realized [TD, TD-V] that the simpler to work with subalgebra corresponds to the observables of one generation of fundamental fermions. The associative envelope of J 8 2 is Cℓ 9 = R[16] ⊕ R[16] with associated symmetry group Spin(9). It was proven in [TD-V] that the SM gauge group G SM (1.13) is the intersection of Spin(9) with the subgroup of the automorphism group F 4 of J 8 3 that preserves the splitting (1.1), that is the group SU(3)×SU (3) Z3 ⊂ F 4 . So we were inclined to identify Spin(9) as a most economic GUT group. As demonstrated in Sect. 3.4, however, the restriction of the spinor IR 16 of Spin(9) to its subgroup G SM gives room to only half of the fundamental fermions: the SU (2) L doublets; the right chiral singlets, e R , u R , d R , are left out. It was then recognized that the (octonionic) Clifford algebra Cℓ 10 does the job. The particle interpretation of Cℓ 10 is dictated by the choice of a (maximal) set of five commuting operators in the Pati-Salam Lie subalgebra of so(10) that leaves our complex structure invariant. This led us to presenting all chiral leptons and quarks of one generation as mutually orthogonal idempotents (3.17) (3.18).
11 The Dublin Professor of Astronomy William Rowan Hamilton (1805-1865) and the Stettin Gymnasium teacher Hermann Günter Grassmann (1809-1877) published their papers, on quaternions and on "extensive algebras", respectively, in the same year 1844. William Kingdom Clifford (1845-1879) combined the two in a "geometric algebra" in 1878, a year before his death, aged 33, referring to both of them.
The difference of the two approaches which can be labeled by the projectors P and Π L (on left and right particles and on left particles and antiparticles, respectively) has implications in the treatment of the generalized connection (including the Higgs) and anomalies. Thus, for the Π L (anti)leptons (ν L , e L ), e L , ν L we have vanishing trace of the hypercharge, tr Π L Y = 0. For P leptons, (ν L , e L ), ν R , e R , the traces of the left and right chiral hypercharge are equal: tr(PΠ L Y ) = −2 = tr(PΠ R Y ), so that, as noted in Sect. 4.1, only the supertrace vanishes in this case. The associated Lie superalgebra fits ideally Quillen's notion of super connection. A real "physical difference" only appears under the assumption that the electroweak hypercharge is superselected and P is replaced by the restricted projector P r on the 15-dimensional particle subspace (with the sterile neutrino ν R , with vanishing hypercharge, excluded). Then the leptonic (electroweak) part of the SM is governed by the Lie superalgebra sℓ(2|1), whose four odd generators are given by third degree monomials in a ( * ) α , the Cℓ 4 Fermi oscillators. The replacement of ℓ by ℓ r breaks the quark-lepton symmetry: while each coloured quark q j appears in four flavours, the colourless leptons are just three. This yields a relative normalization factor between the quark and leptonic projection of the Higgs field and allows to derive (in [T21]) the relation where θ th is the theoretical Weinberg angle, such that tg 2 θ W = 3 5 . The relation (5.4) is satisfied within 1% accuracy by the observed Higgs and W ± masses.

Summary and discussion; a challenge
After the pioneering work of Feza Gürsey and collaborators of the 1970's, Geoffrey Dixon devoted to division algebras over 30 years, followed by Cohl Furey since the 2010's. The Clifford algebra approach to unification, coupled to fermionic creation and annihilation operators, has also been pursued, since the late 1970's, by the Italian group around Roberto Caslbuoni. The notion of superconnection was anticipated and applied to the Weinberg-Salam model during the first decade of the creation of the SM, as well. Thus the basic ingredients of our endeavour have been with us for some 50 years. The pretended new features of the present survey concern certain details. Here belong: -The interpretation of the Clifford pseudoscalar ω 6 as i(P − P ′ ) where P and P ′ are the particle and antiparticle projection operators.
-The realization that the projected Clifford algebra PCℓ 10 P = Cℓ 4 ⊗ Cℓ 0 6 , (5.5) only involves the even part Cℓ 0 6 of Cℓ 6 , coupled with the assignment of the Higgs field to the odd part, Cℓ 1 4 , of the first factor, explains the symmetry breaking of the (electroweak) flavour symmetry, while preserving the colour gauge group.
-Exhibiting the role of the sterile neutrino (of the first generation of fundamental fermions) as the vacuum state of the theory. The gauge group of the SM is identified as the maximal subgoup of the Pati-Salam group G P S (1.8) that leaves ν R invariant.
-Singling out the reduced 15-dimensional particle subspace yields a relation between the Higgs and the W boson masses and the theoretical Weinberg angle satisfied within one percent accuracy.
What is missing for completing the "Algebraic Design of Physics" -to quote from the title of the 1994 book by Geoffrey Dixon -is a true understanding of the three generations of fundamental fermions. None of the attempts in this direction [F14,D16,T,B,MDW,F23] has brought a clear success. The exceptional Jordan algebra J 8 3 = H 3 (O) (1.7) with its built in triality was first proposed to this end in [D16] (continued in [DT]); in its straightforward interpretaton, however, it corresponds to the triple coupling of left and right chiral spinors with a vector in internal space, rather than to three generations of fermions. As recalled in (Sect. 5.2 of) [T] any finite-dimensional unital module over H 3 (O) has the (disappointingly unimaginative) form of a tensor product of H 3 (O) with a finite dimensional real vector space E. It was further suggested there that the dimension of E should be divisible by 3 but the idea was not pursued any further. Boyle [B] proposed to consider the complexified exceptional Jordan algebra whose group of determinant preserving linear automorpghisms is the compact form of E 6 . This led to a promising left-right symmetric extension of the gauge group of the SM but the discussion has not yet shed new light on the 3 generation problem. Yet another development, based on the study of indecomposable representations of Lie superalgebras, can be traced back from [TJGG] where only the mathematical machinery has been discussed so far.
for the associated Clifford algebras. The even subalgebra Cℓ 0 (s, t) is defined as the (closed under multiplication) span of products of an even number of γ matrices; The odd subspace Cℓ 1 (s, t) is defined as the (real) span of products of odd numbers of γ's (which is not closed under multiplication).