\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SU(3)_C\times SU(2)_L\times U(1)_Y\left( \times U(1)_X \right) $$\end{document}SU(3)C×SU(2)L×U(1)Y×U(1)X as a symmetry of division algebraic ladder operators

We demonstrate a model which captures certain attractive features of SU(5) theory, while providing a possible escape from proton decay. In this paper we show how ladder operators arise from the division algebras \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}R, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document}C, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}$$\end{document}H, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {O}$$\end{document}O. From the SU(n) symmetry of these ladder operators, we then demonstrate a model which has much structural similarity to Georgi and Glashow’s SU(5) grand unified theory. However, in this case, the transitions leading to proton decay are expected to be blocked, given that they coincide with presumably forbidden transformations which would incorrectly mix distinct algebraic actions. As a result, we find that we are left with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{sm} = SU(3)_C\times SU(2)_L\times U(1)_Y / \mathbb {Z}_6$$\end{document}Gsm=SU(3)C×SU(2)L×U(1)Y/Z6. Finally, we point out that if U(n) ladder symmetries are used in place of SU(n), it may then be possible to find this same \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{sm}=SU(3)_C\times SU(2)_L\times U(1)_Y / \mathbb {Z}_6$$\end{document}Gsm=SU(3)C×SU(2)L×U(1)Y/Z6, together with an extra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(1)_X$$\end{document}U(1)X symmetry, related to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\!-\!L$$\end{document}B-L.


Introduction
From a wide range of possible theories, the standard model has emerged almost uniquely, after having survived decades of experimental scrutiny. And so this raises the question: What makes SU (3) C × SU (2) L × U (1) Y /Z 6 so special? Of the infinite number of imaginable gauge groups, why this gauge group?
Furthermore, as we know, a choice of gauge group does not imply a complete description of a theory. Even if we were to understand why nature's local symmetries should be given by G sm ≡ SU (3) C ×SU (2) L ×U (1) Y /Z 6 , we would still be at a loss to explain the standard model's particle content. Clearly, the requirement of anomaly cancellation alone cannot go far enough to narrow down the possibilities.
So in addition to deciphering the reasoning behind the standard model's curious gauge group, it is furthermore upon us to decipher the reasoning behind its curious list of irreducible representations. Now, in 1974, H. Georgi and S. Glashow introduced one of the first grand unified theories, based on the 24-dimensional gauge group SU (5). The group SU (5) is the smallest simple Lie group to contain G sm , admit complex representations, accommodate the standard model's particle content, and be free of anomalies [1,2]. Hence, it offers perhaps the most natural simplification of particle physics at high energies.
More importantly, SU (5) theory does reveal at least a couple of clues about the standard model's mysterious structure. For example, we find that the standard model's peculiar list of hypercharges is explained when G sm is embedded into SU (5) [3,4]. Furthermore, some modern versions of the Georgi-Glashow model have SU (5) acting on the 32-C-dimensional exterior algebra ΛC 5 (as opposed to the originally proposed 5 * and 10 irreps). The exterior algebra ΛC 5 breaks down into the 1 ⊕ 5 ⊕ 10 ⊕ 10 * ⊕ 5 * ⊕ 1 irreducible representations of SU (5). This successfully compiles a full generation of quarks and leptons into a single object, ΛC 5 , while accounting for their anti-particles, and including a right-handed neutrino. 1 However, with this being said, we note that SU (5) theory does not truly explain the origin of G sm . In the typi-cal scenario, SU (5) breaks down to G sm via spontaneous symmetry breaking, mediated by a cleverly chosen Higgs field and potential. The question: Why SU (3) C × SU (2) L × U (1) Y /Z 6 ? in this theory is then merely replaced by Why SU (5)? and Why this Higgs?
To make matters worse, the extra generators of SU (5) enable transitions which cause the proton to decay. Calculations of the proton lifetime within minimal SU (5) theory vary depending on source [5], but are generally considered to be at odds with experiment. This conflict was confirmed recently by the Super-Kamiokande collaboration [6,7], which thus far has turned up no evidence in support of the proton's decay.
As a result, minimal SU (5) theory is largely believed to be ruled out. However, given its strengths, one might be led to wonder if perhaps there could be some missing mathematical structure which ultimately saves SU (5) theory from itself.
In this paper, we begin with four special algebras: the real numbers, R, the complex numbers, C, the quaternions, H, and the octonions, O. Uniquely, these are identified as the only four normed division algebras over the real numbers. They are of dimensions 1, 2, 4, and 8 respectively.
Using only R, C, H, and O, we construct a faithful representation of the Clifford algebra Cl(10), acting on a 32-C-dimensional spinor. This 32-C-dimensional spinor is built as a minimal left ideal, which, for our purposes, can be seen to be equivalent to ΛC 5 . We then propose to identify the model's gauge symmetry with the special unitary symmetry of Cl(10) ladder operators. For Cl(10), these ladder symmetries are identified as SU (5). Consequently, we obtain a division algebraic representation of Georgi and Glashow's SU (5) model. However, upon closer inspection, we argue that SU (5) symmetry should never be fully realised in this division algebraic construction. Instead, the new underlying algebraic structure is seen to block certain transitions under the assumption that conceptually distinct algebraic actions do not mix. Incidentally, these are the transitions responsible for proton decay. In place of SU (5), we are then left with a symmetry group given by SU This work builds on an early finding [8], that the octonions break down into 1 In their paper, Günaydin and Gürsey identified the 3 and the 3 * as a triplet of quarks and anti-quarks under the colour group SU (3) C .
Since that time, a number of authors have expanded on these early results, notably [9][10][11]. The authors [9,10] were able to find significant pieces of standard model structure by considering carefully chosen tensor products of Clifford algebras, whereas [11] identified significant pieces of standard model structure starting from the algebra R ⊗ C ⊗ H ⊗ O augmented to 2 ×2 matrices. Readers are encouraged to consult the work of these earlier authors. This paper contributes to the existing literature by exposing a rather straightforward path from R, C, H, and O to the standard model's gauge group, SU (3) C × SU (2) L × U (1) Y /Z 6 , with the possibility of an extra U (1) X symmetry, related to B − L. Furthermore, we find that the stable subspaces (minimal ideals) of this division algebraic model exhibit the behaviour of one full generation of quarks and leptons, supplemented with a right-handed neutrino.

The Georgi-Glashow model
The SU (5) model is perhaps the most logical choice for a grand unified theory. With its 24 dimensions, SU (5) is the smallest acceptable Lie group in which the 12-dimensional G sm can be embedded. From SU (5)'s list of irreducible representations, Georgi and Glashow selected the 5 * and 10 so as to portray the standard model's 15 quarks and leptons.
Parenthetically, we mention that ΛC 5 can alternately split into two 16-C-dimensional irreducible representations under Spin (10), for the "SO(10)" grand unified theory. In this case, one irreducible representation corresponds to the evengraded subspace of ΛC 5 , and the other corresponds to its odd-graded subspace. Furthermore, we point out that the full 32-complex-dimensional ΛC 5 provides the only irreducible representation for the complex Clifford algebra, Cl (10).
Already at the level of SU (5) acting on ΛC 5 , one might begin to suspect that the standard model's particle content is not entirely arbitrary. It seems too much a coincidence to think that ΛC 5 should gratuitously provide the perfect space for one full generation, and that SU (5) should naturally include SU (3) C × SU (2) L × U (1) Y /Z 6 . Fig. 1 The exterior algebra ΛC 5 , broken down in terms of SU (5) irreducible representations. Here, the basis elements within a given irreducible representation appear within a single row. Starting from the bottom and moving upward, the grade-0 object, 1, forms an SU (5) singlet, the grade-1 objects, α i , form the 5, the grade-2 objects, α i α j , form the 10, the grade-3 objects, α i α j α k , form the 10 * , the grade-4 objects, α i α j α k α , form the 5 * , and the grade-5 object, α 1 α 2 α 3 α 4 α 5 , forms another singlet 2.2 Breaking SU (5) Now with this being said, SU (5) does indeed condone proton decay, a process thought never to have been observed. For this reason, it is standard practice to then propose an additional Higgs field so as to break SU (5) → G sm at high energy.
The typical representation chosen for this task is the 24dimensional adjoint Higgs [5]. Its vacuum expectation value is then made to be proportional to the generator of U (1) Y .
As for fermions, the singlet remains unchanged, while the 5 * and the 10 break as Finally, the familiar Higgs field φ, responsible for breaking G sm → SU (3) C × U (1) em , is often embedded in the 5 of SU (5). At the GUT scale, this breaks as where the 3, 1, − 1 3 describes a new triplet Higgs field, H.

SU (5) summary
In searching for a minimalistic model of particle physics, one would be hard-pressed to surpass SU (5) acting on ΛC 5 . Its particle content is concisely defined, and its simple gauge group comes as close as could be expected to that of the standard model of particle physics. However, compliance with experiment prompts the introduction of a carefully chosen 24-dimensional Higgs field and potential. This breaks SU (5) → G sm , but in the process, also compromises the simplicity of the original model.
More troublesome still, we find that SU(5) theory conflicts with experiment, even with this adaptation. For example, experimental lack of proton decay, lack of 't Hooft-Polyakov monopoles, the doublet-triplet splitting problem, and inaccurate mass predictions have all taken their toll on SU (5) theory to varying degrees [5]. However, given the virtues of SU (5) theory, one might wonder if it could be possible to construct a similar model which can bypass these hazards.
We will now build up a division algebraic representation of the Georgi-Glashow model, piece by piece. Along the way, we will encounter several group representations familiar from particle theory. We begin with the complex quaternions, C ⊗ H.

Introduction to C ⊗ H
A generic element of C ⊗ H is written c 0 0 + c 1 1 + c 2 2 + c 3 3 , where the c i ∈ C and 0 ≡ 1. The basis vectors 1 , 2 , 3 follow the associative, but non-commutative quaternionic multiplication rules from which we obtain 1 We define two notions of conjugation on an element a ∈ C ⊗ H. The complex conjugate of a, denoted a * , maps the complex i to −i. That which we will call the hermitian conjugate of a, denoted a † , maps i to −i and j to − j for j = 1, 2, 3, while reversing the order of multiplication, A well-known correspondence exists between the complex quarternions and the Pauli matrices. That is, However, readers should note that these objects behave more symmetrically than do the Pauli matrices under complex conjugation. Explicitly,

Clifford algebraic structure
It is straightforward to confirm that the left action of C⊗H on itself gives a faithful representation of the complex Clifford algebra Cl (2). Please see Fig Under anti-commutation, these operators behave as Since α and α † span the Clifford algebra's generating space, we then see that all of C ⊗ H Cl(2) may be described as sums and multiples of these ladder operators.

Right-handed Weyl spinors as minimal left ideals
The Clifford algebraic structure of C ⊗ H is important for us since it will allow us to construct Weyl spinors. That is, we will make use of the fact that spinors can be defined as minimal left ideals of Clifford algebras [45]. Given an algebra, A, a left ideal, B, is a subalgebra of A whereby ab is in B for all b in B, and for any a in A. Said another way, the subspace B is stable under left multiplication by any of the elements a ∈ A. Now, a minimal left ideal is a left ideal which contains no left ideals other than {0} and itself. (This choice of definition for spinors was motivated by the algebraic path integral model described in Chapter 2 of [46], where particles are identified as surviving subspaces of some fundamental algebra.) We will now construct spinors as minimal left ideals, largely following the procedure set out in [45] for Clifford algebras Cl(2n) with n ∈ Z > 0. In this construction, the first task is to build a particular idempotent, v = vv. Then the spinor is obtained by simply left multiplying Cl(2n) onto v, as in Ψ ≡ Cl(2n)v.
In our current case of Cl(2), our idempotent will be defined as v s ≡ αα † , where the subscript s refers to spin. For generic Cl(2n), the Clifford algebra's generating space will be split into n operators α i and n operators α † i . In this more general case, the idempotent v will be constructed as (2), readers may confirm that the resulting minimal left ideal Cl(2)v s takes the form where ψ ↑ R and ψ ↓ R are complex coefficients. Here, we have labelled the basis vector α † v s as spin-up since 1 2 We have furthermore labeled the spinor Ψ R as righthanded, which may be viewed as an arbitrary choice at this point.
Readers may notice the resemblance between Ψ R and a Fock space, where v s formally plays the role of a vacuum state.

Left-handed Weyl spinors as minimal left ideals
Incidentally, another spinor may be constructed in Cl(2) by swapping the roles of α and α † . In this case, let us then define v s ≡ α † α. When expressed in terms of C ⊗ H, it so happens that v s is given by v s = v * s . From here, we can construct a minimal left ideal, linearly independent from the first.
where ψ ↑ L and ψ ↓ L are complex coefficients. The identification of Ψ L and Ψ R as left-and right-handed Weyl spinors is justified when we take γ 5 to be represented as right multiplication by −i 3 as shown in Section 4.7 of [46]. Furthermore, in Chapter 3 of [46], Ψ L is shown to transform as does a left-handed Weyl spinor under SL(2, C), Ψ L = LΨ L , and Ψ R is shown to transform as does a right-handed Weyl spinor, Putting both subspaces together, we may define Dirac spinors as Ψ D ≡ Ψ L + Ψ R . When translated into the formalism of 2 × 2 C matrices via relations (5), this gives so that each Weyl spinor occupies a column within the 2×2 C matrices. Multiplying from the left induces rotations between spin states, while multiplication from the right induces rotation between chiralities.
In this paper, we will sometimes alternate between matrix descriptions and division algebraic descriptions of our states. Having said that, readers should take note that not all descriptions are created equal. After all, it is the division algebras, not the matrix algebras, which ultimately dictate which Clifford algebras we will consider. Also, the familiar operation of charge conjugation finds a more succinct description within this division algebraic formalism, as we will now show.

Complex conjugation and charge conjugation
To give a basis for comparison, let us first consider the Dirac spinor, as described by 2 × 2 C matrices in relation (10). Under the action of complex conjugation, i → −i, we simply conjugate the four complex coefficients, a procedure which does not lead to anything of particular significance.
In contrast, let us now apply the map i → −i to Ψ D when it is described in terms of C ⊗ H, This transformation may be recognizable to readers as ψ → ψ c = −iγ 2 ψ * from the standard formalism of quantum field theory (up to a phase). In other words, i → −i in the C ⊗ H formalism yields charge conjugation on Dirac spinors.
In this light, we may see that the role of iγ 2 in the standard formalism is to take into account the complex conjugation of basis vectors in the C ⊗ H formalism. When spinors are written in terms of C ⊗ H, the object iγ 2 need no longer be put in by hand.

Dirac algebra
The Weyl spinors Ψ L and Ψ R were each constructed as minimal ideals of C ⊗ H, under left multiplication. However, transitions between Ψ L and Ψ R can be effected via right multiplication. This right action provides another faithful representation of Cl(2).
Cl(2) spin chirality (12) Hence the combined action of both left and right multiplication gives Cl(2) ⊗ C Cl(2) Cl (4). From these actions, we may construct the C ⊗ H-equivalent of the Dirac matrices. Generators of the Dirac algebra in the Weyl basis may be described as as introduced in Section 4.7 of [46]. Here, we made use of the bar notation of [47]. By definition, the operator x|y acting on some element z, for x, y, z ∈ C ⊗ H, is given by xzy. We end this section by pointing out that C ⊗ H is capable of describing more than just Weyl and Dirac spinors. This 4-C-dimensional algebra was shown in [46] to also describe Majorana spinors, scalars, four-vectors, and the field strength tensor -each in the form of generalized ideals. By generalized ideals, we mean invariant subspaces under some action of the algebra on itself. These account for all of the Lorentz representations of the standard model.
For a recent electromagnetic model which builds on this formalism, see [48].

C ⊗ H summary
In this section, we showed that the left action of C ⊗ H on itself gives a faithful representation of the Clifford algebra Cl(2). We then identified anti-commuting ladder operators generating Cl(2), and used them to build a pair of minimal left ideals, Ψ L and Ψ R . These two Weyl spinors may then be combined as Ψ D = Ψ L +Ψ R so as to give a single irreducible representation when both the left and right actions of C ⊗ H are considered. This results in a faithful representation of Cl(2) ⊗ C Cl (2) Cl(4) C ⊗ Cl(1, 3). In short, C ⊗ H lends itself naturally to the description of those spinors familiar to 3 + 1 spacetime dimensions.

C ⊗ O: colour
We will now repeat this construction for the case of the complex octonions. In analogy to the C ⊗ H minimal left ideals Ψ L + Ψ R , we will construct C ⊗ O minimal left ideals, S u + S d . Subsequently, we will find that S u and S d mirror the behaviour of one generation of quarks and leptons under SU Alternately, readers may consult [46,49], or [50], where the given Fano plane depicts these same multiplication rules. The octonions form a non-associative algebra, meaning that the relation (ab)c = a(bc) does not always hold. Octonionic automorphisms are given by G 2 , the 14-dimensional exceptional Lie group.

Clifford algebraic structure
In parallel with the case of C ⊗ H, we will now consider the left action of C ⊗ O on itself. It can be confirmed that complex linear combinations of octonions repeatedly left multiplying f ∈ C ⊗ O may always be written in the canonical form c i jk e i (e j (e k f )) + · · · + c 123456 e 1 (e 2 (e 3 (e 4 (e 5 (e 6 f ))))), where the coefficients c 0 , c i , . . . ∈ C. Readers may note that the octonionic imaginary unit e 7 is not explicitly expressed in these maps. This is due to the fact that e 7 f = e 1 (e 2 (e 3 (e 4 (e 5 (e 6 f ))))) ∀ f ∈ C ⊗ O, thereby making e 7 redundant as a left-action map. Of course, e 7 itself holds no preferred status within the octonions, and the space of left-action maps may equivalently be described by chains built from any six of the seven imaginary units. By using the identity (16), the Eq. (15) may then be written more compactly as for i, j = 1, . . . 6, and ∀ f ∈ C⊗O, and furthermore that the left action of C ⊗ O on itself gives a faithful representation of the Clifford algebra Cl (6). Please see Fig. 3. Here, multiplication is understood to be given by the composition of left action maps, and hence is associative by definition. The Clifford algebra Cl(6) is isomorphic to the 8×8 C matrices. This scenario closely emulates our earlier example where the left action of C ⊗ H on itself gave a faithful representation of the Clifford algebra Cl(2), isomorphic to the 2 × 2 C matrices. Now, the generating space given in Fig. 3 may be rewritten in terms of a new basis, Here, we define the conjugation † to map i → −i and ← − e j → − ← − e j for j = 1 . . . 7. As with the hermitian conjugation of matrices, † also reverses the order of multiplication, that is, the order of left action maps: Under anti-commutation, these operators behave as which is simply a higher-dimensional analogue of equations (7). From this point forward, we will not be interested in the object f ∈ C ⊗ O, only the maps ← − a i and ← − a j † which act on it. Hence, we will no longer refer to f explicitly. Furthermore, in the interest of simplifying notation, we will forfeit the use of arrows on our left action maps, although their presence should be implicitly understood. Equations (20) can then be rewritten more succinctly as

Quarks and leptons as minimal left ideals
Now that we have established Clifford algebraic structure generated by C ⊗ O, we would like to build minimal left ideals analogous to Ψ L and Ψ R . Again following [45], let us define ω ≡ a 1 a 2 a 3 so that our idempotent is then given by v c ≡ ωω † . Our first minimal left ideal will be given by where V,D r , . . . E + are 8 suggestively named complex coefficients. Swapping the roles of a i and a † i gives a linearly independent ideal, S d ≡ Cl (6) whereV, D r , . . . E − are eight complex coefficients. The onegeneration labeling we have used in S u and S d will be partially justified now, and fully justified by the end of this article.

SU (3) ladder symmetry
Let us now consider an SU (3) symmetry acting on the ladder operators, a 1 , a 2 , a 3 . Taking r j ∈ R, our raising and lowering operators transform as e ir j Λ j a k e −ir j Λ j and e ir j Λ j a † k e −ir j Λ j , where the eight Λ j span su (3), and are given by This representation of SU (3) is given by the subgroup of G 2 which holds the octonionic e 7 constant. Given that our minimal left ideals are built entirely out of ladder operators, we see that transformations on a i and a † j thereby induce transformations on S u and S d . Under SU (3), S u and S d are found to transform as Extending this SU (3) symmetry to U (3) = SU (3) × U (1)/Z 3 gives an additional U (1) generator, which can be found to coincide with electric charge. This U (1) em symmetry is generated by the number operator for the system, thereby providing an unusually straightforward explanation of charge quantization. Details may be found in [50].
Finally, readers are encouraged to verify that complex conjugation, i → −i, sends particles to anti-particles, S u ↔ S d . This parallels our earlier findings for C ⊗ H where i → −i similarly gave Ψ L ↔ Ψ R .

Minimal left ideals in the matrix formalism
For those more comfortable with the language of matrices, we point out that S u and S d may be formulated as Written in this way, it becomes obvious that there are in fact eight linearly independent minimal left ideals which can be built within Cl (6). Readers interested in finding additional generations of quarks and leptons within Cl(6) should consult [49], Section 9.6 of [46] and [51].
For an interesting proposal connecting this one-generation model to braids, see [52].

Towards weak isospin
As with the example of Ψ L and Ψ R , the two minimal left ideals S u and S d may be transformed into each other under right multiplication. However, this time, it is the ladder operators ω and ω † which effect these transitions, generating another copy of Cl (2). Taking the sum, S 16 ≡ S u + S d , we then have a faithful representation of Cl(6) ⊗ C Cl (2), Cl(2) colour, etc isospin type. (28) As Cl (6) operators acting on f ∈ C ⊗ O, schematically, we have S u ω f ∼ S d f and S d ω † f ∼ S u f. These two ladder operators, ω and ω † , can be seen to induce transitions between isospin pairs, eg V and E − , U r and D r , etc. However, this description of weak isospin is clearly not complete in that there is nothing at this stage to indicate that these objects should act on only left-handed states. This will be addressed in the next section.
Before moving on, it should be noted that earlier papers have been found which have much in common with the octonionic model presented here. In 1973, Günaydin and Gürsey found the SU (3) C structure for a triplet of quarks and antiquarks using the split octonions [8]. Subsequently in 1977, Barducci et al. [9], built a one-generation model of quarks and leptons from [8], based on Cl(6) and Cl (2). The main differences between the model presented here and [9] lie in the way that particles and anti-particles are related, and in how transitions between isospin states occur. Because of our use of octonionic minimal left ideals, particles and antiparticles are related here simply by i → −i. Furthermore, as we have constructed our quark and lepton states within the space of octonionic maps, not as column vectors, we may then find objects ω and ω † which automatically have the correct electric charges, without having to implement these characteristics by hand. Finally, in the case of U (1) em , we find that electric charge is proportional to the number operator for the system, as opposed to being given by a difference between number operators of two distinct Clifford algebras.

C ⊗ O summary
In this section, we showed that the left action of C ⊗ O on itself gives a faithful representation of the Clifford algebra Cl(6). We then identified anti-commuting ladder operators generating Cl(6), and used them to build a pair of minimal left ideals, S u and S d . Under SU (3) ladder symmetry, these ideals were found to transform as do the quarks and leptons of one generation of standard model particles. When this SU (3) symmetry is further extended to U (3), we then find an additional U (1) factor, generated by electric charge [50]. Combining S u with S d then provided a faithful representation of Cl(6) ⊗ C Cl (2), where the additional Cl(2) factor enables transitions between isospin up-and down-type states.

Cl(4): weak isospin
Readers are encouraged to also see a closely related model by Woit [31], which addresses weak isospin in the context of supersymmetric quantum mechanics. For a recent review article on Cl(4) and electroweak theory, see [53].

Clifford algebraic structure
We will now draw the reader's attention to the right action on Ψ L + Ψ R , and the right action on S u + S d , which each generated a copy of Cl (2). Recall that the right action on Ψ L + Ψ R induced transitions between chiralities L and R, while the right action on S u + S d induced transitions between isospin up-and down-type states. Together, these two Cl(2) right actions form Cl(2) ⊗ C Cl(2) Cl(4). Readers should note that this Cl(4) is conceptually distinct from what we have seen before, in that it effects transitions on the space of idempotents.
Nonetheless, we will now work through the same construction with Cl(4) as we did in previous sections. Generators of this Cl(4) may be carefully chosen as where It should be noted at this point that the behaviour of our generators (29) under complex conjugation differs from that of previous sections. As before, these generators may be rewritten in a new basis given by Here, ‡ maps i → −i, j → − j for j = 1, 2, 3, and e k → −e k for k = 1, . . . 7, while reversing the order of multiplication. It is then not difficult to confirm that ∀ i = 1, 2 and j = 1, 2.

Leptons as minimal right ideals
With ladder operators defined, we may now construct an algebraic vacuum state as v w ≡ β ‡ 1 β ‡ 2 β 2 β 1 . From v w , we then obtain a minimal right ideal as L ≡ v w Cl(4), where Swapping β i ↔ β ‡ i and defining v w ≡ β 1 β 2 β ‡ 2 β ‡ 1 gives a linearly independent minimal right ideal as The bars over top of the variables here, as inL andV L , are meant only to identify anti-particles; they do not imply hermitian conjugation and multiplication by γ 0 .
It should be noted that in this particular Cl(4) construction (29), swapping β i ↔ β ‡ i provided a new minimal right ideal, γ 0L , which automatically includes a factor of γ 0 . This additional γ 0 will be familiar from QFT kinetic terms of the form ψ † γ 0 γ μ ∂ μ ψ.

SU (2) ladder symmetry
In parallel with the previous section, SU (2) symmetries may now be applied to our ladder operators as e −ir j T j β k e ir j T j and e −ir j T j β ‡ k e ir j T j .
The three T j generate SU (2), and are found to be As before, transformations on the ladder operators induce transformations on our minimal right ideals. So we then find that under SU (2), the ideals L and γ 0L transform as These transformation properties agree with the leptonic identifications we made in Eqs. (32) and (33), which will be further justified in the next section. So here we have again found standard model group representations by taking the special unitary symmetries of our ladder operators, and examining the transformations they induce on minimal one-sided ideals. In this case, we have found the behaviour of leptons under SU (2) L . It is worth emphasizing here that when the SU (2) symmetry of these ladder operators was applied to minimal right ideals, it acted automatically on states of only a single chirality. We did not need to implement a projector by hand. Please see Fig. 4.
Finally, we point out that if a U (2) ladder symmetry is taken in place of SU (2), then we find an extra U (1) charge, given that U (2) = SU (2) × U (1)/Z 2 . This U (1) is again generated by the system's number operator, and in this case, coincides with weak hypercharge [46,53]. Fig. 4 The leptonic minimal right ideal L. As with our previous examples, this ideal resembles a Fock space, with the right-handed neutrino acting as the (formal) vacuum state. It should be noted that the SU (2) symmetries of our ladder operators are found to act automatically on lepton states of only a single chirality. That is, without the need to impose a chiral projector by hand

Cl(4) summary
In this section, we focussed in on a representation of Cl(4) which induces transitions of isospin and chirality idempotents. We identified a particular set of anti-commuting ladder operators generating Cl(4), and used it to build a pair of minimal right ideals, L and γ 0L . Under the SU (2) symmetry of these ladder operators, the ideals were found to transform as do the leptons of one generation of standard model particles, together with a right-handed neutrino. Here, the group SU (2) was found to act automatically on states of only a single chirality. Finally, when a U (2) ladder symmetry was used in place of SU (2), we then found an additional U (1) generator with eigenvalues consistent with weak hypercharge.

All together: ladder symmetries to SU(3) C × SU(2) L × U(1) Y /Z 6
We have just come from finding familiar standard model particle representations by considering the action of division algebras on themselves. That is, the special unitary symmetries of division algebraic ladder operators led to SU (3) C and SU (2) L when acting on minimal one-sided ideals. Our goal now is to combine these fragments into a single model.

Clifford algebraic structure
It is well known that the Clifford algebra Cl(10) can provide the background structure for Spin (10) and SU (5) grand unified theories [4]. We will then use our division algebraic actions from previous sections to build a representation of Cl(6) ⊗ C Cl(4) Cl (10).
Making use of equations (19) and (30), let us define ten ladder operators to be where I represents the identity. Given Eqs. (21) and (31), it is trivial to confirm that these obey the usual anti-commutation relations.
Readers should note that from the perspective of the Clifford algebra alone, there is no real distinction between the A i operators and the B j operators. However the same cannot be said when the A i and B j operators are realised in terms of division algebras, as they were in this article. That is, the B j may be considered as truly distinct from the A i , in that the B j were introduced so as to effect transitions between idempotents. Said another way, the A i can be seen to map a left ideal to itself, whereas the B j were introduced so as to map one ideal to another.

One generation as minimal ideals
Using the same procedure as before, we may now construct a vacuum state as from which we build our minimal left ideal, As with γ 0L of Eq. (33), the antiparticles within S can be seen to include a factor of γ 0 automatically. The complex coefficients, V R ,D i L , . . ., are written here so as to anticipate how these states will eventually transform under SU As before, this minimal left ideal exhibits the structure of a Fock space. Readers may also notice that removing the idempotent v t from these states leaves us with the exterior algebra ΛC 5 , as described in [4].
It is straightforward to see that the special unitary transformations on Cl(10) ladder operators give SU (5). These SU (5) ladder symmetries further induce transformations on the minimal left ideal (39). Readers are encouraged to confirm that the action of SU (5) on (39) coincides exactly with the SU (5) transformations of particles and anti-particles in the Georgi-Glashow model. In total, there are 24 generators of SU (5) ladder symmetries, which split into two types. The first type of generator mixes A-and B-type ladder operators. These will be known as mixing generators, and can be written using the hermitian forms Since j runs from 1 to 3 and k runs from 1 to 2, we find 12 generators of this first type. Now, as far as the Clifford algebra Cl (10) is concerned, we have no reason to exclude these generators from consideration. However, from the perspective of our division algebraic construction, A-and B-type ladder operators are clearly algebraically distinct. We will then exclude these 12 elements from this model. Incidentally, it is precisely this first type of generator which is responsible for proton decay.
The second type of generator does not mix A-and Btype ladder operators. In total, there are 12 such generators remaining. The first eight are given by which can be seen to generate SU (3) C when applied to the minimal left ideal (39). Here, the Λ j are defined as in equation (25). The next three generators are given by which generate SU (2) L when applied to (39). Here, the T j are defined as in equation (35). Finally, the twelfth generator is realised as 1 3 and can be seen to assign charges to (39) which coincide with hypercharge, Y. Hence, we have found that it is exactly the non-mixing SU (5) ladder symmetries which generate the standard model's gauge group, SU (3) C × SU (2) L × U (1) Y /Z 6 .
Finally, as with previous cases, we may consider U (5) = SU (5) × U (1)/Z 5 ladder symmetries, as opposed to SU (5). This again introduces an extra U (1) generator, proportional to the number operator. Up to an overall phase, this number operator, is found to give the X charges from the well-known symmetry breaking pattern Spin(10) → SU (5) × U (1) X /Z 5 of [54]. Explicitly, where we are taking Y according to the weak hypercharge conventions of [5].

Summary
In this section, we combined our previous Cl(4) and Cl (6) results so as to yield a division algebraic representation of Cl (10). From Cl(10) ladder operators, we constructed a 32-C-dimensional minimal left ideal. Then, under special unitary ladder symmetries, this minimal left ideal was found to transform as do the particles and anti-particles of the SU (5) Georgi-Glashow grand unified theory.
Finally, under the requirement that A-and B-type division algebraic actions be kept distinct, we find that the SU (5) ladder symmetries then reduce immediately to SU (3) C × SU (2) L × U (1) Y /Z 6 .
Making use of the full U (5) symmetry leads us to the same result, but introduces the possibility of an extra (presumably gauged) U (1) X .

Outlook
We have come from demonstrating how four low-dimensional algebras: R (1D), C (2D), H (4D), and O (8D), can act on themselves so as to yield group representations of the Georgi-Glashow model. Here, the group SU (5) arises as symmetries of Clifford algebraic ladder operators. We point out, though, that only half of these SU (5) generators preserve the underlying algebraic structure. Perhaps unexpectedly, we find that it is precisely this subset which generates the standard model gauge group, G sm = SU (3) C × SU (2) L × U (1) Y /Z 6 .
It bears mentioning that the reduction of SU (5) → SU (3) C × SU (2) L × U (1) Y /Z 6 has not been mediated here by a Higgs boson. Instead, we emphasize that the full SU (5) symmetry should never be fully realised in this division algebraic model in the first place.
Finally, we point out one last avenue worth investigation. Readers may have noticed that with the introduction of ω ≡ a 1 a 2 a 3 , and ω † = a † 3 a † 2 a † 1 , we were able to show that sequences of complex octonions can behave as Cl (2) C ⊗ H. In other words, new algebraic behaviour can arise at different chain lengths of the original algebra (length three in this case).
This algebraic phenomenon bears resemblance to the emergence of effective theories in physics at different energy scales. We might then ask if collective algebraic behaviour might ultimately be used to address currently unexplained physical phenomena, such as colour confinement.