Spin(11, 3), particles, and octonions

Krasnov, Kirill

doi:10.1063/5.0070058

The fermionic fields of one generation of the Standard Model (SM), including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S₊ of the group Spin(11, 3). We describe an octonionic model for Spin(11, 3) in which the semi-spinor representation gets identified with $S_{+} = O \otimes \tilde{O}$ ⁠, where $O, \tilde{O}$ are the usual and split octonions, respectively. It is then well known that choosing a unit imaginary octonion $u \in I m (O)$ equips $O$ with a complex structure J. Similarly, choosing a unit imaginary split octonion $\tilde{u} \in I m (\tilde{O})$ equips $\tilde{O}$ with a complex structure $\tilde{J}$ ⁠, except that there are now two inequivalent complex structures, one parameterized by a choice of a timelike and the other of a spacelike unit $\tilde{u}$ ⁠. In either case, the identification $S_{+} = O \otimes \tilde{O}$ implies that there are two natural commuting complex structures $J, \tilde{J}$ on S₊. Our main new observation is that the subgroup of Spin(11, 3) that commutes with both $J, \tilde{J}$ on S₊ is the direct product Spin(6) × Spin(4) × Spin(1, 3) of the Pati–Salam and Lorentz groups, when $\tilde{u}$ is chosen to be timelike. The splitting of S₊ into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S₊ into eigenspaces of $\tilde{J}$ corresponds to splitting of Lorentz Dirac spinors into two different chiralities. This provides an efficient bookkeeping in which particles are identified with components of such an elegant structure as $O \otimes \tilde{O}$ ⁠. We also study the simplest possible symmetry breaking scenario with the “Higgs” field taking values in the representation that corresponds to three-forms in $R^{11, 3}$ ⁠. We show that this Higgs can be designed to transform as the bi-doublet of the left/right symmetric extension of the SM and thus breaks Spin(11, 3) down to the product of the SM, Lorentz, and U(1)_B−L groups, with the last one remaining unbroken. This three-form Higgs field also produces the Dirac mass terms for all the particles.

I. INTRODUCTION

The suggestion that octonions may provide a useful language for describing elementary particles is almost as old as the Standard Model (SM) itself; see Ref. 1 and references therein. (Note, however, that in this reference, somewhat confusingly, the usual octonion division algebra written in a complex basis is referred to as the split octonion algebra.) Thus, it was observed early on that the eight-dimensional space of octonions, after a complex structure is chosen, splits as $O = C \oplus C^{3}$ ⁠. The two factors can then be identified with those describing leptons and quarks, respectively. The subgroup of the group of automorphisms of the octonions G₂ preserving this split is SU(3), which is identified with the strong force gauge group.

Many authors have since been fascinated by a possible link between elementary particles and octonions. The literature on this topic has grown to be large but also hard to summarize because the subject has never become a communal effort. Individuals that did work on it had different backgrounds and used their own distinct language, making a review of what has been achieved difficult. There is currently a renewed interest in these topics, as is manifested, in particular, by Ref. 2. Some of the most notable past developments in relation to this topic are works by Dixon, e.g., Refs. 3–5, and works by Manogue and Dray; see, e.g., Ref. 6 and references therein. More recent works are those by Furey, e.g., Refs. 7 and 8, by Dubois-Violette and Todorov,^9–13 and by Boyle.¹⁴

While some of the papers on this subject only attempt to describe the “internal” particle degrees of freedom on which the gauge group of the SM acts, others are trying to put the Lorentz spinor degrees of freedom into the picture as well. The basic observation in this regard is that spinors of (pseudo-)orthogonal groups Spin(2n), when restricted to their subgroup Spin(2k) × Spin(2l), k + l = n, decompose as spinors with respect to both Spin(2k), Spin(2l) factors. This, in particular, means that a spinor of a (pseudo-)orthogonal group in 14 dimensions Spin(14), when restricted to Spin(10) × Spin(4), will behave as a spinor with respect to both of these groups. As far as we are aware, the first mention of the possibility of such a unification of Lorentz spinor and “internal” (isospin and color) degrees of freedom is in Ref. 15.

The number of real functions that is needed to describe one generation of the SM fermions is 64. This is the dimension of the Weyl, i.e., semi-spinor representation in 14 dimensions. However, for most possible signature combinations, the arising representation is complex 64-dimensional. The only two cases that admit real, i.e., Majorana semi-spinors in this dimension are the cases Spin(7, 7) and Spin(11, 3). Case Spin(11, 3) was studied in works.^16,17 The split case Spin(7, 7) was advocated in an earlier work of this author.¹⁸ Other papers related to the present work are a series of papers by Stoica^19–21 and a recent paper.²²

The purpose of this paper is to revisit the idea that all field components of one generation of the SM particles can be described using a real Weyl spinor of a pseudo-orthogonal group in 14 dimensions. The new twist that we add is an observation that there is a natural octonionic model for both Spin(7, 7) and Spin(11, 3). One can then consider various different complex structures on S₊ arising from the octonions, similar to what was done in Ref. 23 in the context of Spin(9). We will see that there are two natural commuting complex structures, both corresponding to a choice of a unit imaginary octonion, and their joint commutant in Spin(11, 3) is the direct product of the Pati–Salam and Lorentz groups. There is also a version of the construction that gives the product of the gauge group of the left/right symmetric extension of the SM times the Lorentz group. It is the simplicity of this characterization of the groups so relevant for physics that prompted us to write this paper. At the very least, the observations of this paper provide an embedding of all known particles into such an elegant mathematical structure as $O \otimes \tilde{O}$ and also characterize which choices need to be made to fix the embedding of the groups relevant for physics into Spin(11, 3).

It is useful to compare and contrast the construction of this paper with the more familiar context of the SO(10) Grand Unified Theory (GUT). It is well known that particles of one generation of the SM, together with the right-handed neutrino, but excluding their Lorentz spin components, can be identified with components of a single semi-spinor representation S₊ of Spin(10). This representation has complex dimension 16, i.e., $S_{+} = C^{16}$ ⁠. To describe the embedding of the SM gauge group inside Spin(10), one needs to make some choices. First, one needs to select a complex structure on $R^{10}$ and, thus, identify it with $C^{5}$ ⁠. The subgroup of Spin(10) that respects this choice is U(5). It is standard to break this further into the Georgi–Glashow gauge group SU(5). This breaking is achieved by selecting, in addition to a complex structure on $R^{10}$ ⁠, a top holomorphic form on $C^{5}$ ⁠. [Both pieces of geometric data are very elegantly described by a spinor in S₊ that is in a special Spin(10) orbit. This is the orbit with the stabilizer SU(5). See Ref. 24 for mode details.] To break Spin(10) further down to the SM gauge group, one needs an additional choice. This is encoded by singling out three out of five holomorphic directions, i.e., by splitting $C^{5} = C^{3} \oplus C^{2}$ ⁠. This gives a splitting $R^{10} = R^{6} \oplus R^{4}$ that is compatible with the complex structure chosen in breaking to U(5). Once this splitting is at hand, one can obtain a complex structure J on S₊ by taking the product of the corresponding six γ-matrices. Indeed, this product is an operator that maps S₊ to itself and squares to minus the identity, and so a complex structure. The SM gauge group is then the subgroup of the Georgi–Glashow group SU(5) that also commutes with the complex structure J on S₊. The complex structure J has a simple physical interpretation. It provides the split $C^{16} = C^{8} \oplus C^{8}$ ⁠, which is the split of particles into left- and right-handed. [For us, here, the terminology is that the left-handed particles transform as the weak SU(2) doublets, while the right-handed ones are weak SU(2) singlets.] This description gives a very convincing identification of particles of one generation of the SM with states in S₊, together with understanding of what choices need to be made to select a copy of the SM model gauge group inside Spin(10). To a very large extent, this description also provides an “explanation” of why the SM is what it is, i.e., where the intricate particle pattern of the SM comes from. Note that this description is “kinematical,” which is independent of any possible dynamical realization of the symmetry breaking mechanism. Indeed, to date, there is not a single commonly accepted Spin(10) GUT. There are several different possibilities as to which representations are used for the Higgs fields, and thus, there are several different possible dynamical models that follow the symmetry breaking pattern as outlined above. The Spin(10) model building is an area of active current research; see, e.g., Ref. 25 and references therein.

This places the observations of this paper into a context. Indeed, our main observation is that to select an embedding of the Pati–Salam times the Lorenz group inside Spin(11, 3), one needs not one like in Spin(10) but two commuting complex structures on the semi-spinor representation S₊ of Spin(11, 3). When appropriately chosen, the commutant of both of the complex structures inside Spin(11, 3) is Spin(6) × Spin(4) × Spin(1, 3). One of these complex structures continues to play the role it played in the Spin(10) context by providing the split into the left- and right-handed particles. The other complex structure we introduce in this paper splits S₊ into two subspaces consisting of Lorentz spinors of opposite chirality. This provides a precise analog of the kinematic description that is so powerful and convincing in the Spin(10) case. We do not discuss in this paper a possible analog of the breaking of Spin(10) to SU(5), which is needed together with breaking to Pati–Salam to single out the SM gauge group. It is possible that there is a similar mechanism at play in the Spin(11, 3) context, but we leave this to future work. Neither do we discuss the possible dynamical realizations of the symmetry breaking. As in the context of Spin(10), it can be expected that different proposals can be made in this regard. For instance, there is the “spontaneous soldering” proposal of Ref. 15. Our desire is to keep the clean and convincing kinematics separate from the ambiguous dynamics. We describe only the former to a large extent ignoring the latter; see, however, some comments in Sec. VIII.

Thus, in the more familiar story, the embedding of the SM gauge group inside Spin(10) largely provides an explanation of the particle charges pattern. We hope that the observations of this paper realizing Spin(6) × Spin(4) × Spin(1, 3) as sitting inside Spin(11, 3) can be similarly taken as an explanation of the particle pattern, this time including their Lorentz spin. It remains to be seen, however, if there are any convincing dynamical realizations of the kinematics described in this paper. We make further comments in this regard in Sec. VIII.

We now describe the results of this paper in more detail. There are many different ways to describe Spin(7, 7), Spin(11, 3). There is always the Killing–Cartan description with roots and weights. Unfortunately, it does become difficult to work with for groups of such high dimension. For orthogonal groups, one can, instead, proceed starting with the corresponding Clifford algebras. One standard way to obtain these is by taking tensor products of γ-matrices in lower dimensions. This is, indeed, the approach that was followed in Ref. 17, but it also becomes quite cumbersome. It is more efficient to restrict to some carefully chosen commuting subgroups and build a model for the big group based on the representation theory of the smaller groups. Thus, the semi-spinor representations of Spin(8) can be identified with octonions $O$ ⁠, similarly for the semi-spinor representations of the split Spin(4, 4) that can be identified with $\tilde{O}$ ⁠. One can then restrict Spin(11, 3) to Spin(8) × Spin(3, 3) and similarly Spin(7, 7) to Spin(4, 4) × Spin(3, 3) and then view the semi-spinor representation of Spin(11, 3) and Spin(7, 7) as the tensor product of spinors of Spin(8), Spin(3, 3) and Spin(4, 4), Spin(3, 3), respectively. The semi-spinor representation of Spin(3, 3) is a copy of $R^{4}$ ⁠. This means that the semi-spinor representation of Spin(11, 3) can be viewed as eight copies of the octonions, and that of Spin(7, 7) can be viewed as eight copies of the split octonions. This formalism is useful and can be developed further, but there is an even more efficient description.

What seems to be the most natural description of Spin(7, 7), Spin(11, 3) arises by restricting to the groups Spin(7) × Spin(7) ⊂ Spin(7, 7) and Spin(7) × Spin(4, 3) ⊂ Spin(11, 3). [In the case of Spin(7, 7), there is also the possibility Spin(3, 4) × Spin(4, 3) ⊂ Spin(7, 7).] The spinor representation of Spin(7) can be identified with octonions $O$ ⁠, and the spinor representation of Spin(4, 3) can be identified with split octonions $\tilde{O}$ ⁠. This means that a semi-spinor of Spin(7, 7) can be identified with $O \otimes O$ (or with $\tilde{O} \otimes \tilde{O}$ ⁠), and a semi-spinor of Spin(11, 3) can be identified with $O \otimes \tilde{O}$ ⁠. It is this description and the corresponding models for Spin(7, 7), Spin(11, 3) that we will develop in the main text.

The resulting model for Spin(7, 7), Spin(11, 3) can be phrased more generally, as it describes some other pseudo-orthogonal groups as well. It can be summarized in the following proposition.

Theorem 1.

Let

X, Y

be a pair of division algebras from the set

C, \tilde{C}, H, \tilde{H}, O, \tilde{O}

, where

C, \tilde{C}

are the usual and split complex numbers and

H

and

\tilde{H}

are the usual and split quaternions, respectively. The matrices taking values in

E n d ({(X \otimes Y)}^{2})

⁠,

Γ_{x} = (\begin{matrix} 0 & L_{x} \otimes I \\ L_{\bar{x}} \otimes I & 0 \end{matrix}), x \in I m (X), Γ_{y} = (\begin{matrix} 0 & - I \otimes L_{y} \\ I \otimes L_{\bar{y}} & 0 \end{matrix}), y \in I m (Y)

(1)

where L_x, L_y are the operators of left multiplication by

x \in I m (X), y \in I m (Y)

, and the bar denotes conjugation in

X, Y

, satisfy

\begin{matrix} Γ_{x}^{2} = | x |^{2} I, Γ_{y}^{2} = - | y |^{2} I, \\ Γ_{x} Γ_{y} + Γ_{y} Γ_{x} = 0 . \end{matrix}

(2)

Thus, they generate the Clifford algebra of signature that can be symbolically expressed as

| I m (X) |^{2} - | I m (Y) |^{2} .

(3)

A proof is by direct verification.

Remark 1.

The construction in Theorem 1 is very closely related to that of 2 × 2 matrix algebras in Ref. 26. Our construction is a subcase of the more general construction in this reference in the sense that larger groups in Ref. 26 can be obtained by adding to the set of Clifford generators of Theorem 1 two more generators. This adds one positive-definite and one negative-definite direction. Results in Ref. 26, see also Ref. 27, are then reproduced. We will return to this in the main text.

Remark 2.

This construction of the Clifford algebra identifies spinors with ${(X \otimes Y)}^{2}$ and, thus, a single copy of $X \otimes Y$ with semi-spinors. This immediately gives a description of the Lie algebra in terms of matrices in $E n d (X \otimes Y)$ ⁠. This will be described in the main text for the cases of interest.

Remark 3.

The case of Cliff_11,3 corresponds to $X = O, Y = \tilde{O}$ ⁠. The split case Cliff_7,7 can be described by both $X = O, Y = O$ ⁠, which corresponds to restricting to the Spin(7) × Spin(7) subgroup, and by $X = \tilde{O}, Y = \tilde{O}$ ⁠, which corresponds to restricting to Spin(3, 4) × Spin(4, 3). Both are useful descriptions and capture two different types of spinor orbits in the (7, 7) case.

Remark 4.

Note that there is no way to get the compact case Spin(14) by this construction, at least not by keeping the construction real. Indeed, the semi-spinor representations of this group are complex 64-dimensional. One would need to introduce factors of the imaginary unit in one of the two Γ-matrices to counteract the unavoidable minus sign in (3). Allowing for such factors of imaginary unit generalizes the construction of Theorem 1 to an even larger set of groups.

Further implications of the construction of Theorem 1 will be discussed in the main text. With this result at hand, we can identify the semi-spinor representation of Spin(11, 3) and, thus, components of one fermion generation of the SM with $O \otimes \tilde{O}$ ⁠. The Lie algebra of Spin(11, 3) and, thus, of its various subgroups of interest can be obtained by taking the commutators of the Γ-matrices (1). This gives an explicit and useful description, as we will further illustrate in the main text.

This gives an ultimate “kinematic” unification of all known to us components of fermions of one generation in such an elegant structure as $O \otimes \tilde{O}$ ⁠. However, it does not answer the crucial question of any such unification approach: What breaks the big symmetry group such as Spin(11, 3) to the product of groups we know act on elementary particles in nature? The important part of this question is what singles out the Lorentz group Spin(1, 3) that we know plays a very different role from the gauge groups appearing in the standard model. Our main observation is then that a Lorentz subgroup Spin(1, 3) is selected inside Spin(11, 3) in a very much the same way as subgroups Spin(6), Spin(4) are selected inside Spin(10) in the more familiar GUT story.

We have already discussed the breaking of Spin(10) to the Pati–Salam gauge group above. It will now be useful to rephrase this discussion in octonionic terms; see, e.g., Ref. 24 for a description of the octonionic model of Spin(10). The octonionic model of Spin(10) proceeds by identifying eight of the ten directions of the Clifford algebra Cliff₁₀ with octonions, thus leaving out two of the directions as special. There is another direction that is singled out as special, which corresponds to the identity element in $O$ ⁠. A choice of a unit imaginary octonion $u \in I m (O)$ gives yet another special direction. From these data, one can easily construct a complex structure on semi-spinors whose commutant is the Pati–Salam group Spin(6) × Spin(4). Indeed, the product of any two γ-matrices in Cliff₁₀ gives an operator that squares to minus the identity and preserves the space of semi-spinors. However, the commutant of such a complex structure is Spin(8) × Spin(2), so this is not what we want. The next in the complexity operator that preserves the space of semi-spinors is given by the product of four of the γ-matrices. However, this squares to plus the identity, and so is not a complex structure. However, the next case is the product of six different γ’s. This squares to minus the identity and so gives a complex structure. It breaks Spin(10) down to Spin(6) × Spin(4). Finally, it can clearly be parameterized by a choice of a unit imaginary octonion u. Indeed, as we already discussed, together with u, we have four special directions in Cliff₁₀ and, thus, also remaining six directions. These are the directions in $O$ that are not in the space $S p a n (I, u)$ ⁠. The product of the corresponding γ-matrices is the desired complex structure that breaks Spin(10) to the Pati–Salam group.

Our main new observation in this paper is that there exist very analogous complex structures on Spin(11, 3). These complex structures can also be parameterized by a choice of unit imaginary octonions, but now in both $O, \tilde{O}$ ⁠. A choice of $u \in I m (O)$ still singles out six of the γ-matrices in Cliff_11,3, corresponding to the octonions not in $S p a n (I, u) \subset O$ ⁠. This gives the complex structure J. Its commutant in Spin(11, 3) is Spin(6) × Spin(5, 3), which is not interesting in itself.

More possibilities arise because we have the $\tilde{O}$ factor in our octonionic model for Spin(11, 3). The relevant Clifford algebra is Cliff_4,3, where the first four directions can be referred to as timelike. These are the negative-definite directions of the corresponding quadratic form. Now, a choice of a unit imaginary octonion $\tilde{u} \in I m (\tilde{O})$ is a choice of either a timelike or a spacelike direction. Choosing $\tilde{u}$ to be spacelike gives us the complex structure that is given by the product of six γ-matrices not in $S p a n (I, \tilde{u}) \subset \tilde{O}$ ⁠. The commutant of this complex structure in Spin(11, 3) is then Spin(7, 1) × Spin(4, 2), which does not seem to be interesting for physics. When $\tilde{u}$ is selected to be timelike, the product of six γ-matrices that correspond to directions not in $S p a n (I, \tilde{u}) \subset \tilde{O}$ is not a complex structure, as this operator squares to plus the identity. However, now, the product of the four γ-matrices that correspond to $\tilde{u}$ and to the three spacelike directions in Cliff_4,3 is a complex structure. We shall refer to it as $\tilde{J}$ ⁠. Its commutant in Spin(11, 3) is Spin(10) × Spin(1, 3), which is interesting.

This discussion can be summarized by the following proposition.

Theorem 2.

There exist two complex structures $J, \tilde{J}$ on the space of semi-spinors of Spin(11, 3), one parameterized by a unit imaginary octonion $u \in I m (O)$ and the other parameterized by a unit imaginary octonion $\tilde{u} \in I m (\tilde{O})$ that is timelike. Their common commutant in Spin(11, 3) is the product of the Pati–Salam Spin(6) × Spin(4) and Lorentz Spin(1, 3) groups.

While it is perhaps not very surprising that complex structures breaking Spin(11, 3) to Pati–Salam and Lorentz groups exist, we find it striking that both of these are parameterized by the same data—a unit imaginary octonion in either $O$ or $\tilde{O}$ ⁠. Furthermore, the fact that there are two such natural complex structures is explained by the existence of an $O \otimes \tilde{O}$ model for Spin(11, 3). While this does not yet give a sought mechanism of symmetry breaking from Spin(11, 3) to the groups that we see in nature, the similarity of the complex structures $J, \tilde{J}$ suggests that maybe the dynamical mechanism that selects them is one and the same. In any case, the result in Theorem 2 puts the symmetry breaking needed to single out the Lorentz group from Spin(11, 3) on a very similar footing with the symmetry breaking that is needed to select the Pati–Salam group from Spin(10) GUT group. This suggests that unconventional descriptions that “unify” the internal (color, isospin) with the Lorentz spin degrees of freedom should be taken more seriously.

The organization of the rest of this paper is as follows. We start by reviewing in Sec. II how the group Spin(7) is described using the usual octonions. The spinor representation of Spin(7) is naturally the octonions $O$ ⁠. We also explain here how a natural complex structure J on $O$ arises from a choice of a unit imaginary octonion and how the subgroups SU(3) and U(1) of Spin(7) arise in the process. We repeat similar analysis for Spin(4, 3) in Sec. III. The novelty here is that a new type of complex structure arises, one related to a timelike unit octonion. The commutant of this complex structure $\tilde{J}$ is then the product of the Lorentz Spin(1, 3) and “weak” SU(2) gauge groups. We put everything together in Sec. IV, where the octonionic model for Spin(11, 3) is described. We then interrupt our representation theoretic discussion by a review of some aspects of the left/right symmetric extension of the SM in Sec. V. This becomes a very useful starting point when we describe an explicit dictionary between particles and octonions in Sec. VI. We then apply the developed octonionic formalism to the problem of characterizing the symmetry breaking and possible fermion mass terms. We make a small step in this direction and show how the three-form field in $Λ^{3} (R^{11, 3})$ can be used to break the symmetry down to the product of the SM gauge group, Lorentz group, and U(1)_B−L, which remains unbroken. This Higgs also produces the Dirac mass terms for all the particles. We conclude with a discussion.

II. Spin(7) AND OCTONIONS

A. Octonions

Octonions, see, e.g., Ref. 28, are objects that can be represented as linear combinations of the unit octonions 1, e¹, …, e⁷,

x = x_{0} + x_{1} e^{1} + \dots + x_{7} e^{7}, x_{0}, x_{1}, \dots, x_{7} \in R .

(4)

The conjugation is the operation that flips the signs of all the imaginary coefficients,

\bar{x} = x_{0} - x_{1} e^{1} - \dots - x_{7} e^{7},

(5)

and we have

| x |^{2} = x \bar{x} = {(x_{0})}^{2} + {(x_{1})}^{2} + \dots + {(x_{7})}^{2} \equiv | x |^{2} .

(6)

For later purposes, we note that if we represent an octonion as an eight-component column with entries x₀, x₁, …, x₇, denoted by the same symbol x, we can write the norm as |x|² = x^Tx.

Octonions $O$ form a normed division algebra that satisfies the composition property |xy|² = |x|²|y|². The cross-products of the imaginary octonions e¹, …, e⁷ are most conveniently encoded into a three-form in $R^{7}$ that arises as

C (x, y, z) = ⟨ x y, z ⟩, x, y, z \in I m (O),

(7)

where the inner product ⟨·, ·⟩ in $O$ comes by polarizing the squared norm

⟨ x, y ⟩ = R e (x \bar{y}), x, y \in O .

(8)

One convenient form of C is

C = e^{567} + e^{5} \land (e^{41} - e^{23}) + e^{6} \land (e^{42} - e^{31}) + e^{7} \land (e^{43} - e^{12}),

(9)

where the notation is e^ijk = eⁱ ∧ e^j ∧ e^k. It is then easy to read off the products of distinct imaginary octonions from (9). For example, e⁵e⁶ = e⁷, which is captured by the first term in (9). The particular form (9) of the cross-product on $I m (O)$ is convenient because it manifests the Spin(3) × Spin(4) subgroups of Spin(7) and also one of the two maximal subgroups Spin(4) of the group of automorphisms of the octonions G₂ ⊂ Spin(7) that preserves (9).

Octonions are non-commutative and non-associative, but alternative. The last property is equivalent to saying that any two imaginary octonions (as well as the identity) generate a subalgebra that is associative and is a copy of the quaternion algebra $H$ ⁠.

B. Clifford algebra Cliff₇

Octonions give rise to a very convenient model for the Clifford algebra Cliff₇. Indeed, we can identify

{C l i ff}_{7} = I m (O) .

(10)

To realize this identification, we consider the endomorphisms of $O$ given by the operators

E^{i} ≔ L_{e^{i}}

(11)

of left multiplication by an imaginary octonion. These operators anti-commute and square to minus the identity

E^{i} E^{j} + E^{j} E^{i} = - 2 δ^{i j} .

(12)

Thus, this generates the Clifford algebra Cliff₇.

The Clifford generators can be described very explicitly as 8 × 8 anti-symmetric matrices acting on eight-dimensional columns that represent elements of $O$ ⁠. Using the octonion product encoded in (9), we get

\begin{matrix} E^{1} & = - E_{12} + E_{38} - E_{47} + E_{56}, \\ E^{2} & = - E_{13} - E_{28} + E_{46} + E_{57}, \\ E^{3} & = - E_{14} + E_{27} - E_{36} + E_{58}, \\ E^{4} & = - E_{15} - E_{26} - E_{37} - E_{48}, \\ E^{5} & = - E_{16} + E_{25} + E_{34} - E_{78}, \\ E^{6} & = - E_{17} - E_{24} + E_{35} + E_{68}, \\ E^{7} & = - E_{18} + E_{23} + E_{45} - E_{67} . \end{matrix}

(13)

These are all 8 × 8 real (anti-symmetric) matrices, and the octonions in the column are ordered as $I, e^{1}, \dots, e^{7}$ ⁠. They also have the property $E^{i} I = e^{i}$ ⁠, where both $I, e^{i}$ are eight-component columns.

C. Lie algebra $s o (7)$

With 8 × 8 matrices Eⁱ giving the γ-matrices in seven dimensions, the Lie algebra $s o (7)$ is generated by the commutators of γ-matrices or simply by the products of pairs of distinct Eⁱ,

X_{s o (7)} = \sum_{i < j} ω_{i j} E^{i} E^{j} .

(14)

The matrix $X_{s o (7)}$ is also anti-symmetric. This, in particular, implies that under the Lie algebra action x → Xx, the norm squared x^Tx is preserved.

Given the explicit form (13) of the Clifford generators, it is easy to produce an explicit 8 × 8 matrix representing a general Lie algebra element. It is most convenient to do such explicit calculations by matrix manipulations in Mathematica.

D. Complex structure

Any of the Clifford generators Eⁱ squares to minus the identity, and so is a complex structure on $O$ ⁠. Thus, a choice of a unit $u \in I m (O)$ gives rise to the complex structure J ≔ L_u. Our octonion multiplication encoded in (9) is such that e⁴ is treated as a preferred element, and so we choose u = e⁴ and J = E⁴.

It is interesting to describe the subalgebra of $s o (7)$ that commutes with J. It is clear that all products EⁱE^j that do not contain E⁴ commute with it. Thus, the subalgebra of $s o (7)$ that commutes with J is $s o (6)$ that corresponds to rotations in the e^1,2,3,5,6,7 plane.

Given that this $s o (6)$ commutes with a complex structure, it admits a complex description. In this description, it is the Lie algebra of the group that acts by unitary transformations in, e.g., the (1, 0) eigenspace of J. Indeed, the complex structure allows us to identify $O = C^{4}$ ⁠, and transformations commuting with the complex structure preserve the (1, 0) and (0, 1) eigenspaces of J. In our conventions, the (1, 0) eigenspace is the one that corresponds to the eigenvalue −i. All in all, we get $s o (6) = s u (4)$ ⁠.

E. A different complex structure

There is a different choice of the complex structure on $O$ ⁠, which is also parameterized by a unit imaginary octonion. This leads to a different but very interesting commutant in $s o (7)$ ⁠. This has been described in Ref. 23.

Instead of using the operator L_u of left multiplication by a unit imaginary octonion, one can use the complex structure J′ ≔ R_u given by the operator of the right multiplication R_u. This gives a complex structure, but one different from L_u. The operator R_u agrees with L_u on the copy of $C$ spanned by $I, u$ but acts with the opposite sign on the rest of $O$ ⁠. It is easy to check that the commutant of R_u in $s o (7)$ is the subalgebra $s u (3) \times u (1) \subset s u (4)$ ⁠. Thus, this complex structure breaks $s o (7)$ even further and produces the very relevant for physics algebra $s u (3)$ of the strong force and the “hypercharge” $u (1)$ ⁠. [When we describe the left/right symmetric model below, it will become clear that the right interpretation of this $u (1)$ is that corresponding to U(1)_B−L symmetry.]

We can also describe the R_u complex structure in a way that strongly mimics what happens in the $s o (4, 3)$ setting of Sec. III D. Let us introduce the operator ρ that acts as the identity on $S p a n (I, u)$ and changes the sign of the six-dimensional orthogonal to $I, u$ space. In other words, let ρ be the reflection in the $I, u$ plane. This operator squares to $+ I$ and commutes with L_u. Their product is precisely R_u. Thus, we consider

J^{'} ≔ R_{u} = E^{4} ρ .

(15)

As already mentioned, the commutant of J′ in $s o (7)$ is the direct sum $s u (3) \oplus u (1)$ ⁠.

F. Parameterization

For later purposes, we now give a parameterization of an octonion by its (1, 0) and (0, 1) coordinates. The (1, 0) coordinates of eigenvalue of E⁴ of −i are given by

L ≔ x_{0} - i x_{4}, Q_{i} = x_{i} - i x_{i + 4},

(16)

where we introduced suggestive names. An octonion x with coordinates x₀, x₁, …, x₇ is then parameterized as

x_{0} = \frac{1}{2} (L + \bar{L}), x_{4} = \frac{i}{2} (L - \bar{L}), x_{i} = \frac{1}{2} (Q_{i} + {\bar{Q}}_{i}), x_{i + 4} = \frac{i}{2} (Q_{i} - {\bar{Q}}_{i}) .

(17)

Here, the bar is both a part of the name and the complex conjugation that maps between the eigenspaces (1, 0), (0, 1) of the complex structure J. Another complex structure and its related complex conjugation will appear later when we consider Spin(4, 3). It will be important to distinguish them.

For later purposes, we give formulas for various useful pairings in this parameterization,

\begin{matrix} ⟨ x, x ⟩ = \bar{L} L + {\bar{Q}}^{i} Q^{i}, \\ ⟨ x, (E^{1} - i E^{5}) (E^{2} - i E^{6}) (E^{3} - i E^{7}) x ⟩ = - 4 i L^{2}, \\ ⟨ x, (E^{1} E^{5} + E^{2} E^{6} + E^{3} E^{7}) E^{4} x ⟩ = - 3 \bar{L} L + {\bar{Q}}^{i} Q^{i}, \end{matrix}

(18)

where the summation convention is implied. We will also need versions of these formulas with two different octonions paired. We have

\begin{matrix} ⟨ x_{1}, x_{2} ⟩ = \frac{1}{2} ({\bar{L}}_{1} L_{2} + {\bar{L}}_{2} L_{1} + {\bar{Q}}_{1}^{i} Q_{2}^{i} + {\bar{Q}}_{2}^{i} Q_{1}^{i}), \\ ⟨ x_{1}, E^{4} x_{2} ⟩ = \frac{i}{2} (- {\bar{L}}_{1} L_{2} + {\bar{L}}_{2} L_{1} - {\bar{Q}}_{1}^{i} Q_{2}^{i} + {\bar{Q}}_{2}^{i} Q_{1}^{i}), \\ ⟨ x_{1}, (E^{1} E^{5} + E^{2} E^{6} + E^{3} E^{7}) x_{2} ⟩ = \frac{i}{2} (- 3 {\bar{L}}_{1} L_{2} + 3 {\bar{L}}_{2} L_{1} + {\bar{Q}}_{1}^{i} Q_{2}^{i} - {\bar{Q}}_{2}^{i} Q_{1}^{i}) . \end{matrix}

(19)

We will need these results when we compute the Yukawa mass terms.

G. The group of automorphisms of the octonions

The group G₂ of automorphisms of the octonions is the subgroup of SO(7) that preserves the three-form in (9) and, thus, the cross-product. The three-form (9) can be written as

C = - \sum_{i < j < k} ⟨ I, E^{i} E^{j} E^{j} I ⟩ e^{i j k},

(20)

where the notation is that e^ijk = eⁱ ∧ e^j ∧ e^k. This shows that the stabilizer of C in $s o (7)$ is given by the transformations that fix the spinor corresponding to the identity octonion $I$ ⁠. An explicit calculation shows that this is the subalgebra of $s o (7)$ satisfying

\begin{matrix} - w_{27} + w_{36} - w_{45} = 0, w_{17} - w_{35} - w_{46} = 0, - w_{16} + w_{25} - w_{47} = 0, \\ - w_{14} - w_{23} + w_{67} = 0, w_{13} - w_{24} - w_{57} = 0, - w_{12} - w_{34} + w_{56} = 0, \end{matrix}

(21)

and w₁₅ + w₂₆ + w₃₇ = 0. This gives an explicit description of the Lie algebra $g (2)$ ⁠. The dimension of this Lie algebra is 14.

H. The group SU(3)

The special unitary group in three dimensions can be seen to arise in this context in many different ways. First, we can see it arising as the subgroup of Spin(7) that preserves two orthogonal spinors. If we take these to be $I, u = e^{4}$ ⁠, the subgroup that preserves $I$ is G₂. Imposing the condition that u is stabilized imposes six additional conditions w_i4 = 0 as well and produces the Lie algebra $s u (3)$ ⁠.

A different, but equivalent way of seeing SU(3) arising is as the intersection of the group of automorphisms of the octonions G₂ with the group Spin(6) = SU(4) that commutes with J. Indeed, the conditions reducing to $s o (6) = s u (4)$ are the six conditions w_i4 = 0. Intersected with (21), this gives us $s u (3)$ ⁠.

Yet another way to see SU(3) is to note that while J allows us to identify $O = C^{4}$ ⁠, the unit imaginary octonion u that was selected to produce this complex structure also gives us a preferred copy of the complex plane in $C^{4}$ ⁠, the one spanned by $I, u$ ⁠. The transformations from SU(4) do not preserve this copy of the complex plane, mixing all four directions in $C^{4}$ ⁠. The subgroup of SU(4) that preserves $C = S p a n (I, u)$ is U(3). This U(3) splits as U(3) = SU(3) × U(1), and we will refer to the last factor as U(1)_B−L. The subgroup of this U(3) that fits into the group of automorphisms of the octonions is SU(3).

The group U(1)_B−L ⊂ SU(4) ⊂ Spin(7) does not survive the intersection with the group of automorphisms of the octonions G₂. Indeed, its generator is precisely w₁₅ + w₂₆ + w₃₇, which is set to zero by the $g (2)$ conditions. This suggests that the dynamical mechanism that is to break the Pati–Salam SU(4) to SU(3) × U(1)_B−L is not one corresponding to taking the intersection with G₂. This remark will be important in Sec. III when we discuss the split analogs of all these statements.

III. Spin(4, 3) AND THE SPLIT OCTONIONS

A. Split octonions

Split octonions $\tilde{O}$ are similarly generated by the identity and seven imaginary octonions,

y = y_{0} + y_{1} {\tilde{e}}^{1} + \dots + y_{7} {\tilde{e}}^{7} .

(22)

To distinguish the split from the usual octonions, we will denote them either with a letter with a tilde or use letter y (with an index if necessary) rather than x. The unit split octonions satisfy

{({\tilde{e}}^{5, 6, 7})}^{2} = - 1, {({\tilde{e}}^{1, 2, 3, 4})}^{2} = 1 .

(23)

The split octonion conjugation again changes the sign of all the imaginary units. This means that the split octonion quadratic form

| y |_{s}^{2} ≔ y \bar{y} = {(y_{0})}^{2} - {(y_{1})}^{2} - {(y_{2})}^{2} - {(y_{3})}^{2} - {(y_{4})}^{2} + {(y_{5})}^{2} + {(y_{6})}^{2} + {(y_{7})}^{2}

(24)

is of the split signature (4, 4). It will be convenient to rewrite this quadratic form in a matrix notation. We introduce a diagonal matrix τ = diag(−1, 1, 1, 1, 1, −1, −1, −1), (negative of) which describes a reflection in the 1, 2, 3, 4 plane. If we represent a split octonion by an eight-dimensional column with entries y₀, y₁, …, y₇, the quadratic form is

| y |^{2} = - y^{T} τ y .

(25)

This form of writing will be useful later.

The split octonion cross-product is again most conveniently encoded by a three-form, which we choose to be

C = {\tilde{e}}^{567} - {\tilde{e}}^{5} \land ({\tilde{e}}^{41} - {\tilde{e}}^{23}) - {\tilde{e}}^{6} \land ({\tilde{e}}^{42} - {\tilde{e}}^{31}) - {\tilde{e}}^{7} \land ({\tilde{e}}^{43} - {\tilde{e}}^{12}) .

(26)

Note that the only change as compared to (9) is that the signs in front of the last three terms changed. This means that the cross-product in the ${\tilde{e}}^{5, 6, 7}$ plane is unchanged to the case of usual octonions. This plane, together with $I$ ⁠, generates a copy of the quaternions $H$ ⁠. The signs of all other cross-products get reversed as compared to $O$ ⁠.

B. The Clifford algebra Cliff_4,3

Operators ${\tilde{E}}^{i} ≔ L_{{\tilde{e}}^{i}}$ of left multiplication by a unit split imaginary octonion generate the Clifford algebra Cliff_4,3,

{C l i ff}_{4, 3} = I m (\tilde{O}) .

(27)

We can represent them explicitly as the following 8 × 8 matrices:

\begin{matrix} {\tilde{E}}^{1} & = S_{12} + S_{38} - S_{47} + S_{56}, \\ {\tilde{E}}^{2} & = S_{13} - S_{28} + S_{46} + S_{57}, \\ {\tilde{E}}^{3} & = S_{14} + S_{27} - S_{36} + S_{58}, \\ {\tilde{E}}^{4} & = S_{15} - S_{26} - S_{37} - S_{48}, \\ {\tilde{E}}^{5} & = - E_{16} + E_{25} + E_{34} - E_{78}, \\ {\tilde{E}}^{6} & = - E_{17} - E_{24} + E_{35} + E_{68}, \\ {\tilde{E}}^{7} & = - E_{18} + E_{23} + E_{45} - E_{67} . \end{matrix}

(28)

These satisfy the following Clifford algebra relations:

{\tilde{E}}^{i} {\tilde{E}}^{j} + {\tilde{E}}^{j} {\tilde{E}}^{i} = 2 η^{i j},

(29)

where η = diag(1, 1, 1, 1, −1, −1, −1) is the metric of signature (4, 3). Note that we have reversed the signs here as compared to (12), and this will be convenient later when we put $O$ and $\tilde{O}$ together. This extra sign is the same as the one required in (3). The matrices S_ij are symmetric matrices with the identity in ith row and jth column and in jth row and ith column. It is then clear that the matrices ${\tilde{E}}^{1, 2, 3, 4}$ are symmetric, while ${\tilde{E}}^{5, 6, 7}$ are anti-symmetric. Note that as matrices ${\tilde{E}}^{5, 6, 7} = E^{5, 6, 7}$ ⁠. Again, we have the property that ${\tilde{E}}^{i} \tilde{I} = {\tilde{e}}^{i}$ ⁠, where both $\tilde{I}, {\tilde{e}}^{i}$ are viewed as eight-component columns.

C. Lie algebra $s o (4, 3)$

Forming the products of pairs of distinct Clifford generators, we get the representation of a general Lie algebra in terms of 8 × 8 matrices,

X_{s o (4, 3)} = \sum_{i < j} {\tilde{ω}}_{i j} {\tilde{E}}^{i} {\tilde{E}}^{j} .

(30)

These matrices no longer have a definite symmetry. Instead, the invariance of the quadratic form (25) can be written as

X_{s o (4, 3)}^{T} τ + τ X_{s o (4, 3)} = 0 .

(31)

Here, τ is the matrix of the quadratic form (25).

D. Complex structures

We now enter into a less familiar part of the discussion. It is clear that any of the three Clifford generators ${\tilde{E}}^{5, 6, 7}$ (or any unit vector constructed as their linear combination) can serve as a complex structure on $\tilde{O}$ ⁠. This complex structure identifies $\tilde{O} = C^{4}$ ⁠. What commutes with it in $s o (4, 3)$ is the subalgebra $s o (4, 2)$ ⁠. However, this is not the complex structure that is of interest to us in relation to physics.

Any of the generators ${\tilde{E}}^{1, 2, 3, 4}$ squares to plus the identity, and so is not a complex structure. However, we also have the object

τ = {\tilde{E}}^{5} {\tilde{E}}^{6} {\tilde{E}}^{7} .

(32)

This is an operator that reverses the signs of the positive-definite directions $\tilde{I}, {\tilde{e}}^{5, 6, 7}$ of the split quadratic form (24) and leaves the negative-definite directions ${\tilde{e}}^{1, 2, 3, 4}$ intact. As a matrix, this is the already encountered matrix of the quadratic form (25). This operator squares to plus the identity $τ^{2} = I$ ⁠, but anti-commutes with any of the Clifford generators ${\tilde{E}}^{1, 2, 3, 4}$ (and commutes with all the ${\tilde{E}}^{5, 6, 7}$ generators). Thus, the product of τ with any one of ${\tilde{E}}^{1, 2, 3, 4}$ (or with a unit linear combination constructed from them) is a complex structure. We choose $\tilde{u} = {\tilde{e}}^{4}$ and define

\tilde{J} ≔ {\tilde{E}}^{4} τ .

(33)

Note the strong analogy with (15). This is the complex structure that is of our main interest. Indeed, being given by the product of four Clifford generators, its commutant in $s o (4, 3)$ is the algebra $s o (3) \times s o (1, 3)$ ⁠. Both groups are very interesting because the first factor reminds us of the SU(2) acting on isospin of particles, while the second factor is the Lorentz group. The first factor is the one describing rotations in the ${\tilde{e}}^{1, 2, 3}$ plane, while the second one mixes the directions ${\tilde{e}}^{4, 5, 6, 7}$ ⁠. The direction ${\tilde{e}}^{4}$ is a timelike direction with respect to the split octonion quadratic form (24), and we will refer to it as such in what follows. For later purposes, we note that we can rewrite the complex structure $\tilde{J}$ as

\tilde{J} ≔ - {\tilde{E}}^{1} {\tilde{E}}^{2} {\tilde{E}}^{3},

(34)

i.e., as the product of the timelike Clifford generators that are distinct from the chosen ${\tilde{E}}^{4}$ ⁠.

E. Decomposition of $\tilde{O}$ under $s o (3) \times s o (1, 3)$

The split octonions $\tilde{O}$ form the spinor representation of Spin(4, 3). When we restrict to Spin(3) × Spin(1, 3), we expect the spinor to transform as spinor with respect to both of the factors. It is, nevertheless, very interesting to see how this happens explicitly.

The (1, 0) coordinates on $\tilde{O}$ for the complex structure $\tilde{J}$ are given by

w_{0} ≔ \frac{1}{2} (y_{0} + i y_{4}), w_{i} ≔ \frac{1}{2} (y_{i} + i y_{i + 4}),

(35)

where the index i = 1, 2, 3 and we introduced the factors of 1/2 for future convenience. The transformations from $s o (3) \times s o (1, 3)$ commute with $\tilde{J}$ and thus preserve the (1, 0) subspace. Therefore, they can be described as 4 × 4 complex matrices acting on the four-columns (w₀, w₁, w₂, w₃). Explicitly, we have

X_{s o (3)} = (\begin{matrix} 0 & i x & i y & i z \\ i x & 0 & z & - y \\ i y & - z & 0 & x \\ i z & y & - x & 0 \end{matrix}),

(36)

where

x = {\tilde{ω}}_{23}, y = - {\tilde{ω}}_{13}, z = {\tilde{ω}}_{12}

(37)

are three real parameters. We also have

X_{s o (1, 3)} = (\begin{matrix} 0 & a & b & c \\ a & 0 & i c & - i b \\ b & - i c & 0 & i a \\ c & i b & - i a & 0 \end{matrix}),

(38)

where

a = - {\tilde{ω}}_{45} + i {\tilde{ω}}_{67}, b = - {\tilde{ω}}_{46} - i {\tilde{ω}}_{57}, c = - {\tilde{ω}}_{47} + i {\tilde{ω}}_{56}

(39)

are complex parameters. It can be verified that the matrices $X_{s o (3)}$ and $X_{s o (1, 3)}$ commute.

It is then easy to check that the two-component columns

u = (\begin{matrix} u_{1} \\ u_{2} \end{matrix}) ≔ (\begin{matrix} w_{0} + w_{3} \\ w_{1} - i w_{2} \end{matrix}), d = (\begin{matrix} d_{1} \\ d_{2} \end{matrix}) ≔ (\begin{matrix} w_{1} + i w_{2} \\ w_{0} - w_{3} \end{matrix})

(40)

transform under $X_{s o (1, 3)}$ transformations as two-component spinors of the same Lorentz chirality, that is,

u \to A u, d \to A d, A = (\begin{matrix} c & a + i b \\ a - i b & - c \end{matrix}) .

(41)

On the other hand, the two-component column with u, d as entries transforms under $X_{s o (3)}$ as

(\begin{matrix} u \\ d \end{matrix}) \to (\begin{matrix} i z & y + i x \\ - y + i x & - i z \end{matrix}) (\begin{matrix} u \\ d \end{matrix}) .

(42)

Summarizing, by introducing the complex structure $\tilde{J}$ ⁠, the space of split octonions $\tilde{O} = R^{8}$ splits into its (1, 0) and (0, 1) eigenspaces. The (1, 0) eigenspace $C^{4}$ transforms as the spinor representation of Spin(3) and the two-component spinor representation of the Lorentz group Spin(1, 3),

C_{(1, 0)}^{4} = (2, 2, 1) .

(43)

The complex conjugate (0, 1) eigenspace transforms as the spinor of Spin(3) and the complex conjugate two-component spinor representation of Lorentz,

C_{(0, 1)}^{4} = (2, 1, 2) .

(44)

Thus, a single copy of $\tilde{O}$ gives us a two-component Lorentz spinor that is at the same time a spinor of what can be identified as “weak” SU(2). [When we embed everything into Spin(11, 3), below, we will see that this SU(2) corresponds to the diagonal subgroup in SU(2)_L × SU(2)_R = SO(4) ⊂ SO(10).] Thus, the “weak” SU(2) and the Lorentz groups are very naturally unified within Spin(4, 3), similar to how the “strong” SU(3) gets unified with U(1) within SO(7).

F. Parameterization

It will help if we develop the notation a bit further. We have seen that a Spin(4, 3) spinor $y \in \tilde{O}$ splits into its (1, 0), (0, 1) components,

y = {\tilde{P}}_{+} y + {\tilde{P}}_{-} y ≔ y^{+} + y^{-},

(45)

where

{\tilde{P}}_{\pm} = \frac{1}{2} (I \pm i \tilde{J})

(46)

are the projectors. The split octonion y⁺ transforms as a two-component Lorentz and SU(2) spinors, and y⁻ transforms as a two-component Lorentz spinor of the opposite chirality and again as a SU(2) spinor. Note that the two complex octonions y^± are related by the complex conjugation $y^{-} = {(y^{+})}^{*}$ ⁠.

It will be helpful to write down the explicit parameterization of the octonions y_± by the components of the Lorentz and SU(2) spinors. Using (40), we have

w_{0} = \frac{1}{2} (u_{1} + d_{2}), w_{3} = \frac{1}{2} (u_{1} - d_{2}), w_{1} = \frac{1}{2} (u_{2} + d_{1}), w_{2} = \frac{i}{2} (u_{2} - d_{1}) .

(47)

We also have for the real coordinates on $\tilde{O}$ ⁠,

y_{0} = (w_{0} + w_{0}^{*}), y_{4} = - i (w_{0} - w_{0}^{*}), y_{i} = (w_{i} + w_{i}^{*}), y_{i + 4} = - i (w_{i} - w_{i}^{*}),

(48)

with i = 1, 2, 3, and where the star denotes the complex conjugation. This means that we have

y^{+} = \frac{1}{2} (\begin{matrix} u_{1} + d_{2} \\ u_{2} + d_{1} \\ i (u_{2} - d_{1}) \\ u_{1} - d_{2} \\ - i (u_{1} + d_{2}) \\ - i (u_{2} + d_{1}) \\ (u_{2} - d_{1}) \\ - i (u_{1} - d_{2}) \end{matrix}) .

(49)

If we introduce an octonion y parameterized in this way, a computation gives the expected Lorentz and SU(2) invariant pairing,

⟨ y, y ⟩ = ⟨ y^{+}, y^{+} ⟩ + ⟨ y^{-}, y^{-} ⟩ = (\begin{matrix} u & d \end{matrix}) (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) (\begin{matrix} u \\ d \end{matrix}) + c . c .,

(50)

where c.c. stands for “complex conjugate,” and we have used the index-free two-component spinor notation for the pairing of two-component spinors,

u d = - d u ≔ u^{T} ϵ d, u = (\begin{matrix} u_{1} \\ u_{2} \end{matrix}), d = (\begin{matrix} d_{1} \\ d_{2} \end{matrix}), ϵ ≔ (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) .

(51)

From now on, we will never spell out the Lorentz pairing of two-component spinors in terms of matrices, always using the index-free (and matrix-free) notation. However, in order to avoid confusion, it will be useful to explicitly write the SU(2) invariant pairing in matrix terms. In addition, from now on, the “complex conjugate” will always refer to the operation that maps between the (1, 0), (0, 1) eigenspaces of the complex structure $\tilde{J}$ ⁠. When we consider Spin(11, 3), there will be another complex structure floating around, but in order to avoid confusion, we will always spell out the complex conjugate with respect to this other complex structure explicitly.

We can also write a more general formula for the pairing of two different Spin(4, 3) spinors. We have

⟨ y_{1}, y_{2} ⟩ = ⟨ y_{1}^{+}, y_{2}^{+} ⟩ + ⟨ y_{1}^{-}, y_{2}^{-} ⟩ = {(\begin{matrix} u & d \end{matrix})}_{1} (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) {(\begin{matrix} u \\ d \end{matrix})}_{2} + c . c .

(52)

The first term is a Lorentz and SU(2) invariant pairing of two different two-component Lorentz and SU(2) spinors.

G. Some useful formulas

Here, we list some useful pairings that can be computed using the particle-friendly parameterization developed in Subsection III F. We will give them in their version analogous to (52), with two different spinors.

First, we have

- 〈 y_{1}, {\tilde{E}}^{1} {\tilde{E}}^{2} {\tilde{E}}^{3} y_{2} 〉 = 〈 y_{1}, \tilde{J} y_{2} 〉 = - i {(\begin{matrix} u & d \end{matrix})}_{1} (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) {(\begin{matrix} u \\ d \end{matrix})}_{2} + c . c .

(53)

This result follows from the fact that $\tilde{J}$ acts as the operator of multiplication by −i on (1, 0) eigenspace where $y_{1, 2}^{+}$ take values.

We will also need

\begin{matrix} 〈 y_{1}, (ϕ^{i} {\tilde{E}}^{i}) y_{2} 〉 = {(\begin{matrix} u & d \end{matrix})}_{1} (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) (\begin{matrix} ϕ^{3} & ϕ^{1} - i ϕ^{2} \\ ϕ^{1} + i ϕ^{2} & - ϕ^{3} \end{matrix}) {(\begin{matrix} u \\ d \end{matrix})}_{2} + c . c ., \\ 〈 y_{1}, \frac{1}{2} (ϵ^{i j k} ϕ^{i} {\tilde{E}}^{j} {\tilde{E}}^{k}) y_{2} 〉 = i {(\begin{matrix} u & d \end{matrix})}_{1} (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) (\begin{matrix} ϕ^{3} & ϕ^{1} - i ϕ^{2} \\ ϕ^{1} + i ϕ^{2} & - ϕ^{3} \end{matrix}) {(\begin{matrix} u \\ d \end{matrix})}_{2} + c . c . \end{matrix}

(54)

H. The split ${\tilde{G}}_{2}$

The group of automorphisms of the split octonions ${\tilde{G}}_{2}$ can be defined as the subgroup of SO(4, 3) that preserves the three-form (26). Equivalently, it is the subgroup of Spin(4, 3) that preserves the spinor $I$ ⁠.

It can be checked that the intersection of the Lie algebras $\tilde{g} (2)$ and $s o (3) \times s o (1, 3)$ is the diagonal $s o (3)$ that describes simultaneous rotations in the ${\tilde{e}}^{1, 2, 3}$ and ${\tilde{e}}^{5, 6, 7}$ planes. In particular, there are no boosts in this intersection.

While this may sound as bad news for physics, we remind the reader that the analogous intersection in the $O$ case is that between $g (2)$ and $s o (6)$ ⁠. In addition, as discussed in Sec. II H, this intersection is $s u (3)$ ⁠, but not $s u (3) \times u {(1)}_{B - L}$ that would be desirable for physics. Hence, U(1)_B−L does not survive the intersection with G₂ in the case of Spin(7). What we see in the present case of Spin(4, 3) is that this group naturally contains the weak SU(2) together with the Lorentz group, but neither of this survives the intersection with ${\tilde{G}}_{2}$ ⁠. Only the diagonal SU(2) consisting of the simultaneous weak SU(2) transformations and spatial rotations does survive as a subgroup of ${\tilde{G}}_{2}$ ⁠. Therefore, this situation is analogous to what happens for Spin(7). This just means that the to-be-found mechanism that will select the “physical” unbroken subgroups from Spin(11, 3) is not the one where intersections with $G_{2}, {\tilde{G}}_{2}$ will play role. However, there is clearly something important to be understood here.

IV. THE OCTONIONIC MODEL FOR Spin(11, 3)

We now put the $O, \tilde{O}$ building blocks together and construct an octonionic model for the group Spin(11, 3). The construction we are to describe applies to a more general set of pseudo-orthogonal groups, and so we give it in full generality.

A. Yet another magic square

There is an octonionic model for the exceptional Lie groups, which is obtained by considering 3 × 3 matrices with entries in $X \otimes Y$ ⁠, where $X, Y$ are division algebras; see Ref. 26. This leads to what in the literature is known as the “magic square” construction. These authors also describe the 2 × 2 matrix version of the magic square. This gives a set of octonionic models for groups the largest of which is Spin(12, 4). We describe a very closely related construction, which effectively forgets two of the corresponding Clifford generators, to produce models with the largest covered group being Spin(11, 3). The difference between our construction and that in Ref. 26, see also Ref. 27, is that in our case, the semi-spinor is the space $X \otimes Y$ ⁠, while in the former case, the semi-spinor is twice that ${(X \otimes Y)}^{2}$ ⁠.

We now work out a table of cases that are covered by the construction of Theorem 1. There are two tables that can be produced. One corresponds to taking $X$ to be a division algebra and $Y$ to be a split (i.e., just composition) algebra. This produces the most interesting “magic square.” The largest group covered by this construction is Spin(11, 3),

(55)

What is indicated in these tables is the signature of the arising pseudo-orthogonal group.

The second table is the case when $X, Y$ are either both division or both split. These two possibilities give the same groups. The diagonal in this case gives the split signature pseudo-orthogonal groups Spin(1, 1), Spin(3, 3), and Spin(7, 7). The only other case that is not covered by the first table is that of Spin(7, 1).

It is clear that adding (1, 1) to every entry of the first table reproduces the 2 × 2 tables in Refs. 26 and 27. However, it is not so easy to go from the construction of Theorem 1 to that in Refs. 26 and 27 because these references describe a model for the Lie algebra of the relevant groups as the algebra of 2 × 2 matrices with entries in $X \otimes Y$ ⁠. The corresponding γ-matrices are then 4 × 4 such matrices. Hence, giving a model for the Clifford algebra requires one higher level of complexity. This can be done, but is not relevant for our purposes. We refer to Ref. 27 for details.

B. Lie algebra

Taking the commutators of the Γ-matrices (1) or simply the products of distinct Γ-matrices, we get the following description of the Lie algebra of the groups that our construction covers. The Lie algebra is given by the following endomorphisms of $X \otimes Y$ ⁠:

s o (I m (X)) \otimes I + I \otimes s o (I m (Y)) + L_{x} \otimes L_{y}, x \in I m (X), y \in I m (Y) .

(56)

For the case of $s o (11, 3)$ that is of most interest for us, we can give a more explicit description,

X_{s o (11, 3)} = \sum_{i < j} ω_{i j} E^{i} E^{j} \otimes I + I \otimes \sum_{i < j} {\tilde{ω}}_{i j} {\tilde{E}}^{i} {\tilde{E}}^{j} + \sum_{i, j} a_{i j} E^{i} \otimes {\tilde{E}}^{j} .

(57)

Thus, there is the Lie algebra $s o (7)$ acting on the $O$ factor in $O \otimes \tilde{O}$ ⁠, the algebra $s o (4, 3)$ acting on the $\tilde{O}$ factor, and the terms that mix the first and second sets of directions in (11, 3) = (7, 0) + (4, 3). All the blocks here are constructed from 8 × 8 matrices $E^{i}, {\tilde{E}}^{i}$ ⁠; see (13) and (28). This gives a very explicit description of the Lie algebra.

C. Invariant pairing

There is no $s o (11, 3)$ invariant pairing on the space of semi-spinors S₊. This is because the matrices appearing in (57) have different symmetry properties. The first two terms are anti-symmetric (in appropriate sense),

{(ω_{i j} E^{i} E^{j})}^{T} = ω_{i j} E^{j} E^{i} = - ω_{i j} E^{i} E^{j},

(58)

and using ${({\tilde{E}}^{i})}^{T} τ = - τ {\tilde{E}}^{i}$ ⁠,

{({\tilde{ω}}_{i j} {\tilde{E}}^{i} {\tilde{E}}^{j})}^{T} τ = τ {\tilde{ω}}_{i j} {\tilde{E}}^{j} {\tilde{E}}^{i} = - τ {\tilde{ω}}_{i j} {\tilde{E}}^{i} {\tilde{E}}^{j} .

(59)

The last term is, however, symmetric,

{(a_{i j} E^{i} \otimes {\tilde{E}}^{j})}^{T} (I \otimes τ) = (I \otimes τ) a_{i j} E^{i} \otimes {\tilde{E}}^{j} .

(60)

Thus, if we build a paring on $O \otimes \tilde{O}$ using the octonion pairings on $O, \tilde{O}$ ⁠, it will be invariant under $s o (7), s o (4, 3)$ ⁠, but not under $s o (11, 3)$ ⁠.

Instead, there exists an invariant pairing between the spaces of different types of semi-spinors S_±. The space S₋ can also be identified with $O \otimes \tilde{O}$ ⁠. From expression (1) for the Γ-matrices, it is easy to see that the action of the Lie algebra on S₋ is given by

X_{s o (11, 3)}^{'} = \sum_{i < j} ω_{i j} E^{i} E^{j} \otimes I + I \otimes \sum_{i < j} {\tilde{ω}}_{i j} {\tilde{E}}^{i} {\tilde{E}}^{j} - \sum_{i, j} a_{i j} E^{i} \otimes {\tilde{E}}^{j} .

(61)

Note the different sign in front of the last term as compared to (57). If we denote elements of $S_{+} = O \otimes \tilde{O}$ by Ψ and those of $S_{-} = O \otimes \tilde{O}$ by Ψ′, we have the following invariant pairing:

⟨ ⟨ Ψ^{'}, Ψ ⟩ ⟩,

(62)

where the double angle brackets denote the composition of the pairings on $O$ and $\tilde{O}$ ⁠. The $s o (11, 3)$ invariance is the easily checked property,

⟨ ⟨ X_{s o (11, 3)}^{'} Ψ^{'}, Ψ ⟩ ⟩ + ⟨ ⟨ Ψ^{'}, X_{s o (11, 3)} Ψ ⟩ ⟩ = 0 .

(63)

V. THE LEFT/RIGHT SYMMETRIC MODELS

We now interrupt our representation theoretic discussion to review the left/right symmetric extensions of the SM, as these are most closely related to our formalism. We follow Ref. 25; see, in particular, Sec. 1.3.1.

A. Spinor fields

It is convenient to organize the fermionic fields into the following multiplets:

Q^{i} = (\begin{matrix} u^{i} \\ d^{i} \end{matrix}), L = (\begin{matrix} ν \\ e \end{matrix}), {\bar{Q}}^{i} = (\begin{matrix} {\bar{d}}^{i} \\ - {\bar{u}}^{i} \end{matrix}), \bar{L} = (\begin{matrix} \bar{e} \\ - \bar{ν} \end{matrix}) .

(64)

These fields transform under the left/right symmetric gauge group

G_{LR} = S U (3) \times S U {(2)}_{L} \times S U {(2)}_{R} \times U {(1)}_{B - L}

(65)

in the following representations:

Q = (3, 2, 1, + \frac{1}{3}), L = (1, 2, 1, - 1), \bar{Q} = (\bar{3}, 1, 2, - \frac{1}{3}), \bar{L} = (1, 1, 2, + 1) .

(66)

Specifically, the transformation properties under SU(2)_L × SU(2)_R are

Q \to U_{L} Q, L \to U_{L} L, \bar{Q} \to U_{R} \bar{Q}, \bar{L} \to U_{R} \bar{L} .

(67)

B. Bi-doublet Higgs field

Most (but not all; see below) versions of the left/right symmetric model require a Higgs field in the bi-doublet representation,

Φ = (1, 2, 2, 0), Φ \to U_{L} Φ U_{R}^{†} .

(68)

Out of the components of Φ* of the complex conjugate matrix, one can construct another matrix

\tilde{Φ} = ϵ Φ^{*} ϵ^{T}, ϵ ≔ (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix})

(69)

that has the same transformation properties $\tilde{Φ} \to U_{L} \tilde{Φ} U_{R}^{†}$ as Φ. The vacuum expectation value (VEV) of this Higgs field

Φ = (\begin{matrix} v_{1} & 0 \\ 0 & v_{2} \end{matrix}), \tilde{Φ} = (\begin{matrix} v_{2}^{*} & 0 \\ 0 & v_{1}^{*} \end{matrix})

(70)

breaks the symmetry down to the diagonal U(1) ⊂ SU(2)_L × SU(2)_R. U(1)_B−L remains unbroken. To break the remaining symmetries to the electromagnetic U(1), one needs to introduce further Higgs fields; see below.

It is very convenient to have the Higgs field Φ because it allows construction of Yukawa mass terms. The G_LR invariant (lepton) mass term that can be constructed from $Φ, \tilde{Φ}$ is

Y_{L} L^{T} ϵ^{T} Φ \bar{L} + {\tilde{Y}}_{L} L^{T} ϵ^{T} \tilde{Φ} \bar{L},

(71)

where $Y_{L}, {\tilde{Y}}_{L}$ are mass parameters (matrices in the case there is more than one generation). Evaluating these terms on the VEVs of $Φ, \tilde{Φ}$ ⁠, we get

(Y_{L} v_{1} + {\tilde{Y}}_{L} v_{2}^{*}) e \bar{e} + (Y_{L} v_{2} + {\tilde{Y}}_{L} v_{1}^{*}) ν \bar{ν},

(72)

which are the Dirac mass terms for the electron and the neutrino. Note that it is sufficient to take v₁ ≠ 0, v₂ = 0 to obtain the Dirac masses.

C. Two Higgs in the adjoint

As already said, the Higgs field Φ cannot break the U(1)_B−L symmetry. One thus needs additional Higgs fields. The option that currently seems preferred in the literature goes under the name of the minimal left/right symmetric model; see, e.g., Ref. 29. It introduces two more Higgs fields in the adjoint of each SU(2)_L,R. It also requires neutrinos to be Majorana particles and provides a natural room for the seesaw mechanism. The adjoint Higgs fields are (B − L) charged and transform in the following representations:

Δ_{L} = (1, 3, 1, + 2), Δ_{R} = (1, 1, 3, + 2),

(73)

with SU(2)_L × SU(2)_R transformation properties

Δ_{L} \to U_{L} Δ_{L} U_{L}^{†}, Δ_{R} \to U_{R} Δ_{R} U_{R}^{†} .

(74)

The required VEVs for these fields are

Δ_{L, R} = (\begin{matrix} 0 & 0 \\ v_{L, R} & 0 \end{matrix}) .

(75)

The field Δ_L leaves unbroken the diagonal subgroup in U(1)_L × U(1)_B−L, analogously for Δ_R. This means that there is a single electromagnetic U(1)_EM unbroken. The electric charge of all the fields is worked out via

Q = T_{L}^{3} + T_{R}^{3} + \frac{B - L}{2} .

(76)

Thus, the hypercharge $Y = T_{R}^{3} + (B - L) / 2$ ⁠.

The adjoint Higgs fields allow for the construction of Yukawa mass terms. The lepton sector mass terms that can be written using Δ_L,R are

Y_{Δ} (L^{T} ϵ Δ_{L} L + {\bar{L}}^{T} Δ_{R}^{*} ϵ \bar{L}),

(77)

where Y_Δ is a mass parameter (a mass matrix in the case when there is more than one generation). This term is designed to be invariant under the discrete $Z_{2}$ symmetry,

L \leftrightarrow \bar{L}, Δ_{L} \leftrightarrow Δ_{R}^{†} .

(78)

Evaluating everything on the VEV, we get the following contribution to leptonic mass terms:

Y_{Δ} (v_{L} ν ν + v_{R}^{*} \bar{ν} \bar{ν}) .

(79)

These are the neutrino Majorana mass terms giving rise to the seesaw mechanism.

D. Two Higgs fields in the fundamental

Another option is to add two Higgs fields in the fundamental representations of SU(2)_L,R. This is the original left/right symmetric model studied in Ref. 30. However, there is no natural realization of the seesaw mechanism.

Let us introduce the two Higgs fields χ_L,R transforming as under G_LR as

χ_{L} = (2, 1, + 1), χ_{R} = (1, 2, + 1) .

(80)

There exists a vacuum configuration in which Φ takes the form (70) and

χ_{L} = (\begin{matrix} v_{L} \\ 0 \end{matrix}), χ_{R} = (\begin{matrix} 0 \\ v_{R} \end{matrix}) .

(81)

Each one of these vacuum configurations breaks SU(2)_L,R × U(1)_B−L to a U(1) subgroup. If v_R ≫ v_L, the left/right symmetry is spontaneously broken, and we get an explanation of why only the left sector manifests itself at low energies.

E. A model with only fundamental Higgs fields

There is also a version of the model with Higgs in the fundamental representations that does not introduce the bi-fundamental Higgs Φ; see, e.g., Ref. 31. Indeed, the two VEVs (81) are sufficient to break SU(2)_L × SU(2)_R × U(1)_B−L down to the electromagnetic U(1)_EM. The drawback of this model is that it does not allow for Yukawa-type mass terms. However, such terms can be produced using higher dimensional operators with two copies of the Higgs; see Ref. 31.

VI. DICTIONARY

We are now ready to provide an explicit dictionary between elementary particles, organized as in (64), and elements of $O \times \tilde{O}$ ⁠. Note that identification with particles only becomes possible after the two complex structures $J, \tilde{J}$ are chosen.

A. Two commuting complex structures on $O \times \tilde{O}$

We have already introduced the complex structures $J, \tilde{J}$ on $O, \tilde{O}$ in Secs. II and III. They extend to two commuting complex structures on $O \times \tilde{O}$ ⁠.

Let us describe the commutant of $J, \tilde{J}$ acting on $O \times \tilde{O}$ in $s o (11, 3)$ ⁠. A part of the commutant lives in $s o (7) \oplus s o (4, 3)$ subalgebra. This is the already described $s o (6) \subset s o (7)$ and $s o (3) \oplus s o (1, 3) \subset s o (4, 3)$ ⁠. However, there are more transformations in the commutant. Indeed, it is easy to see that the generators $E^{4} \otimes {\tilde{E}}^{i}, i = 1, 2, 3$ commute with both J = E⁴ and $\tilde{J} = - {\tilde{E}}^{1} {\tilde{E}}^{2} {\tilde{E}}^{3}$ ⁠. These three additional generator extend $s o (3)$ into $s o (4)$ ⁠, rotating the directions $e^{4}, e^{\tilde{1}}, e^{\tilde{2}}, e^{\tilde{3}}$ ⁠.

All in all, we see that the commutant of $J, \tilde{J}$ in Spin(11, 3) is

G_{J, \tilde{J}} = S U (4) \times S U {(2)}_{L} \times S U {(2)}_{R} \times S p i n (1, 3),

(82)

i.e., the product of the Pati–Salam and Lorentz groups. We are not careful here about the possible discrete subgroups that are in the kernel of the embedding of this direct product into Spin(11, 3), as this is of no interest to us in the present paper.

An even more interesting case is that of the commutant of J′ = R_u and $\tilde{J}$ ⁠. The complex structure J′ breaks SU(4) to SU(3) × U(1)_B−L. Thus, we get

G_{J^{'}, \tilde{J}} = S U (3) \times U {(1)}_{B - L} \times S U {(2)}_{L} \times S U {(2)}_{R} \times S p i n (1, 3),

(83)

which is the product of the left/right symmetric (65) and Lorentz groups.

B. Identification with particles

The complex structure J splits $O$ into its (1, 0), (0, 1) subspaces, each of them being a copy of $C^{4}$ ⁠. Moreover, there is a preferred copy of $C$ ⁠, arising as the one spanned by $I, u \in O$ ⁠. This is where leptons will live. Similarly, the complex structure $\tilde{J}$ splits $\tilde{O}$ into its (1, 0), (0, 1) subspaces. We know that each of these describes Lorentz two-component spinors of the same chirality. As in all GUT discussions, we can concentrate on only one of these spaces, thus describing the particle content using Lorentz spinors of the same chirality.

Upon choosing $\tilde{J}$ ⁠, the space $O \times \tilde{O}$ of real dimension 64 becomes identified with

O \times \tilde{O} = C^{32} .

(84)

Moreover, if we concentrate on the (1, 0) eigenspace of $\tilde{J}$ ⁠, this splits as a collection of two-component irreducible representations of the Lorentz group SO(1, 3) ⊂ SO(4, 3). There are precisely 16 such two-component spinors in (84). The subgroup SU(3) ⊂ SU(4) acts on these 16 Lorentz spinors and preserves the complex structure J. This gives the following further decomposition:

O \times \tilde{O} = L \oplus Q \oplus \bar{L} \oplus \bar{Q} .

(85)

The first two terms here are in the (1, 0) subspace of J, and the second two are in the (0, 1) eigenspace. Each of these spaces is also a fundamental representation of SU(2) ⊂ Spin(4, 3) and a Lorentz spinor. The projection of every term in (85) onto the (1, 0) eigenspace of $\tilde{J}$ gives a fundamental representation of SU(2) that is also a two-component Lorentz spinor.

It is clear that the $Q, \bar{Q}$ factors in (85) transform as the fundamental and anti-fundamental representation of SU(3) ⊂ Spin(6) ⊂ Spin(7) and that all factors in (85) are charged with respect to the U(1)_B−L, with the assignment of charges being as in (66). It remains to understand the transformation properties of the factors with respect to SU(2)_L ×SU(2)_R.

The general element of the Lie algebra $s o (4)$ is given by

\frac{1}{2} a^{i} ϵ^{i j k} {\tilde{E}}^{j} {\tilde{E}}^{k} + b^{i} E^{4} \otimes {\tilde{E}}^{i} .

(86)

This splits into two commuting $s u (2)$ subalgebras as

\frac{1}{2} (a^{i} - b^{i}) (\frac{1}{2} ϵ^{i j k} {\tilde{E}}^{j} {\tilde{E}}^{k} - E^{4} \otimes {\tilde{E}}^{i}) + \frac{1}{2} (a^{i} + b^{i}) (\frac{1}{2} ϵ^{i j k} {\tilde{E}}^{j} {\tilde{E}}^{k} + E^{4} \otimes {\tilde{E}}^{i}) .

(87)

Let us name the first of these two factors as $s u {(2)}_{L}$ and the second as $s u {(2)}_{R}$ ⁠. On the (1, 0) eigenspace of $\tilde{J} = - {\tilde{E}}^{1} {\tilde{E}}^{2} {\tilde{E}}^{3}$ ⁠, we have ${\tilde{E}}^{1} {\tilde{E}}^{2} = i {\tilde{E}}^{3}$ ⁠, and so on this subspace, the splitting into the two $s u (2)$ factors becomes

\frac{1}{2} (a^{i} - b^{i}) (i {\tilde{E}}^{i} - E^{4} \otimes {\tilde{E}}^{i}) + \frac{1}{2} (a^{i} + b^{i}) (i {\tilde{E}}^{i} + E^{4} \otimes {\tilde{E}}^{i}) .

(88)

Note that the signs in brackets are reversed onto the (0, 1) subspace of $\tilde{J}$ ⁠. It is now clear that the two eigenspaces of J = E⁴ transform as (2, 1) and (1, 2) representations of SU(2)_L × SU(2)_R. Indeed, on the (1, 0) eigenspace of J, we can replace E⁴ by its eigenvalue −i. This makes it clear that this subspace is a singlet with respect to SU(2)_R. Similarly, the operator E⁴ has eigenvalue +i on the (0, 1) eigenspace, which makes it clear that it is a singlet with respect to SU(2)_L. All in all, we get precisely the assignment of representations as in (66).

It remains to understand the assignment of particles into doublets of the two different SU(2). The choice of such an assignment for the sector that transforms as the doublet of SU(2)_L is a convention. Therefore, we name the components of doublets of SU(2)_L as in (64). These transform as two-component columns χ_L → U_Lχ_L, U_L ∈ SU(2)_L. The correct assignment for the right doublets can only be motivated by the Dirac mass terms that couple the left and right sectors. The choices made in (68) and (71) motivate the assignments as in (64).

VII. MASS TERMS

We now put the formalism developed to some use. We search for a “Higgs” field that can break the Spin(11, 3) symmetry and also be used to write mass terms for the semi-spinor $S_{+} = O \otimes \tilde{O}$ ⁠. We thus need a field that lives in the tensor product of the representation S₊ with itself. We have the following decomposition of this tensor product:

S_{+} \times S_{+} = Λ^{1} (R^{11, 3}) \oplus Λ^{3} (R^{11, 3}) \oplus Λ^{5} (R^{11, 3}) \oplus Λ_{s d}^{7} (R^{11, 3}) .

(89)

Here, $Λ^{k} (R^{11, 3})$ is the space of anti-symmetric tensors of rank k in $R^{11, 3}$ ⁠. The space of seven-forms is not an irreducible representation, but the space $Λ_{s d}^{7}$ of self-dual seven-forms is. The first factor in this sum is anti-symmetric, i.e., projection on it does not vanish only if we assume that the two S₊ factors anti-commute. The second term is symmetric. We are interested in the mass terms arising by projecting onto this representation. Thus, we are interested in the symmetric bilinear forms on S₊ that can be written as

C^{μ ν ρ} ⟨ ⟨ Ψ, Γ_{μ} Γ_{ν} Γ_{ρ} Ψ ⟩ ⟩ .

(90)

A. Projection onto $Λ^{3} R^{11, 3}$

Three copies of Γ-matrices can be inserted between two S₊ states and produce a $s o (11, 3)$ invariant expression. The different arising components are

\begin{matrix} ⟨ ⟨ Ψ, Γ_{x_{1} x_{2} x_{3}} Ψ ⟩ ⟩ = ⟨ ⟨ Ψ, L_{x_{1}} L_{x_{2}} L_{x_{3}} \otimes I Ψ ⟩ ⟩, ⟨ ⟨ Ψ, Γ_{y_{1} y_{2} y_{3}} Ψ ⟩ ⟩ = - ⟨ ⟨ Ψ, I \otimes L_{y_{1}} L_{y_{2}} L_{y_{3}} Ψ ⟩ ⟩, \\ ⟨ ⟨ Ψ, Γ_{x_{1} x_{2} y_{1}} Ψ ⟩ ⟩ = ⟨ ⟨ Ψ, L_{x_{1}} L_{x_{2}} \otimes L_{y_{1}} Ψ ⟩ ⟩, ⟨ ⟨ Ψ, Γ_{x_{1} y_{1} y_{2}} Ψ ⟩ ⟩ = - ⟨ ⟨ Ψ, L_{x_{1}} \otimes L_{y_{1}} L_{y_{2}} Ψ ⟩ ⟩ . \end{matrix}

(91)

This means that we have the following expression for the projection $S_{+} \times S_{+} \to Λ^{3} R^{11, 3}$ ⁠:

\begin{matrix} ⟨ ⟨ Ψ, Γ Γ Γ Ψ ⟩ ⟩ = ⟨ ⟨ Ψ, E^{i} E^{j} E^{k} \otimes I Ψ ⟩ ⟩ e^{i j k} + ⟨ ⟨ Ψ, E^{i} E^{j} \otimes {\tilde{E}}^{k} Ψ ⟩ ⟩ e^{i j} {\tilde{e}}^{k} \\ - ⟨ ⟨ Ψ, E^{i} \otimes {\tilde{E}}^{j} {\tilde{E}}^{k} Ψ ⟩ ⟩ e^{i} {\tilde{e}}^{j} {\tilde{e}}^{k} - ⟨ ⟨ Ψ, I \otimes {\tilde{E}}^{i} {\tilde{E}}^{j} {\tilde{E}}^{k} Ψ ⟩ ⟩ {\tilde{e}}^{i} {\tilde{e}}^{j} {\tilde{e}}^{k} . \end{matrix}

(92)

B. SU(3) × U(1) and Lorentz invariant pairing

We now select among the 364 terms in (92) those that are SU(3), U(1)_EM, and Lorentz invariant.

We have the SU(3) invariant three-forms in $R^{6} \subset I m (O)$ ⁠. These arise as the real and imaginary parts of an SU(3) invariant (3, 0) form,

Ω ≔ (e^{1} - i e^{5}) (e^{2} - i e^{6}) (e^{3} - i e^{7}) .

(93)

Note, however, that Ω is a singlet with respect to SU(2)_L × SU(2)_R, but transforms non-trivially under the U(1)_B−L. This means that Ω is not a singlet with respect to U(1)_EM and cannot be used. There is then an SU(3) × U(1)_B−L invariant real two-form,

ω = e^{15} + e^{26} + e^{37} .

(94)

The last building block we are allowed to use in the $R^{7}$ factor is the one-form e⁴.

For objects in $I m (\tilde{O})$ ⁠, we are only allowed to use the one-, two- and three-forms built from ${\tilde{e}}^{1, 2, 3}$ ⁠. These one- and two-forms transform as vectors under the weak SU(2), while the three-form is invariant. We can use objects that transform non-trivially under the SU(2) because we do not intend to preserve this symmetry. In other words, objects that transform non-trivially under the SU(2) are components of a “Higgs” field that we know is present in the mass terms of the SM.

This leads us to consider the following “mass” quadratic form in S₊:

\begin{matrix} {⟨ ⟨ Ψ, Ψ ⟩ ⟩}_{M} = m_{1} ⟨ ⟨ Ψ, (E^{1} E^{5} + E^{2} E^{6} + E^{3} E^{7}) E^{4} \otimes I Ψ ⟩ ⟩ + m_{2} ⟨ ⟨ Ψ, I \otimes {\tilde{E}}^{1} {\tilde{E}}^{2} {\tilde{E}}^{3} Ψ ⟩ ⟩ \\ + ψ^{i} ⟨ ⟨ Ψ, (E^{1} E^{5} + E^{2} E^{6} + E^{3} E^{7}) \otimes {\tilde{E}}^{i} Ψ ⟩ ⟩ + \frac{1}{2} ϕ^{i} ϵ^{i j k} ⟨ ⟨ Ψ, E^{4} \otimes {\tilde{E}}^{j} {\tilde{E}}^{k} Ψ ⟩ ⟩ . \end{matrix}

(95)

Here, the indices i, j, k run over the values 1, 2, 3, and summation convention is implied. Here, $ξ \in C$ is a complex parameter, $m_{1, 2} \in R$ ⁠, and ϕⁱ, ψⁱ are two real vectors.

C. Evaluating the mass terms

We now evaluate (95) on the particle states. Every element of $O$ splits into eight different states $L, \bar{L}, Q_{i}, {\bar{Q}}_{i}$ ⁠, and we have computed the relevant pairings in this basis in (18), (19), (53), and (54).

Let us concentrate on the “quark” sector first. We get

(\begin{matrix} u & d \end{matrix}) (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) Φ_{Q} (\begin{matrix} \bar{d} \\ - \bar{u} \end{matrix}) = Q^{T} ϵ Φ_{Q} \bar{Q},

(96)

where

Φ_{Q} = ((m_{1} + i m_{2}) I + (ϕ^{i} + i ψ^{i}) σ^{i})

(97)

and σⁱ are the usual Pauli matrices. This should be compared with (the quark version of) (71).

For the lepton sector, we get similarly

(\begin{matrix} ν & e \end{matrix}) (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) Φ_{L} (\begin{matrix} \bar{e} \\ - \bar{ν} \end{matrix}) = L^{T} ϵ Φ_{L} \bar{L},

(98)

where

Φ_{L} = ((- 3 m_{1} + i m_{2}) I + (ϕ^{i} - 3 i ψ^{i}) σ^{i}) .

(99)

It is clear that we get a relation between the masses of quarks and leptons, as is typical in GUT, but the particular relation that arises does not seem to be of any phenomenological interest.

The main conclusion of the calculation performed is that the bi-doublet Higgs field Φ of the left/right symmetric version of the SM that appears in the Yukawa mass terms of this model (and thus contains the familiar Higgs field) can be identified with a general SU(3) × U(1)_EM and Lorentz invariant three-form in $R^{11, 3}$ ⁠. It is interesting that this three-form Higgs field gives masses to both leptons and quarks and breaks the symmetry between them. Note, however, that while the three-form (92), with values of the parameters chosen to align with (70), breaks Spin(11, 3) to the product of SU(3) × U(1)_B−L and the Lorentz groups, there is also the diagonal U(1) ⊂ SU(2)_L × SU(2)_R that survives, similar to the left/right symmetric model. Thus, the three-form Higgs field by itself is not sufficient to break Spin(11, 3) to the product of the unbroken part of the SM gauge and Lorentz groups, as U(1)_B−L survives. As in the left/right symmetric model, other Higgs fields are required to complete the breaking. It is possible that the seven-form Higgs can be used for this purpose, but we leave this to future study.

VIII. DISCUSSION

We have developed an efficient formalism that describes the semi-spinor representation of Spin(11, 3) as the direct product of the usual $O$ and split $\tilde{O}$ octonions. This is likely the most efficient description of this group, as is, in particular, manifested by the very simple description (57) of the Lie algebra $s o (11, 3)$ ⁠. This formalism, together with its other applications covered by the tables in (55), likely has applications other from the considered here particle physics context.

The considered here application of our construction is based on the fact that components of the fermionic fields of one generation of the SM can be identified with the components of the semi-spinor representation S₊ of Spin(11, 3). We then introduced two commuting complex structures $J, \tilde{J}$ on S₊ whose commutant is the product (83) of the left/right symmetric and Lorentz groups. The fact that there is naturally two commuting complex structures is simply explained by the product structure $S_{+} = O \otimes \tilde{O}$ because $J, \tilde{J}$ arise as complex structures on $O, \tilde{O}$ ⁠, respectively. The space S₊ then splits into its (1, 0), (0, 1) eigenspaces with respect to both of the complex structures, and we saw that this splitting for J corresponds to the splitting of fermionic states into particles and anti-particles, denoted by unbarred and barred letters in (85). The splitting for $\tilde{J}$ ⁠, on the other hand, corresponds to the splitting of Lorentz four-component spinors into two different types of two-component spinors or into two different chiralities. What arises is the structure of the SM states as described by the left/right symmetric version of the SM; see (64). Thus, the developed here octonionic model achieves the ultimate “kinematic” unification of all the fermionic states, including the Lorentz spin states. It is hard to imagine a description of all these states more elegant than the one provided by our octonionic model. At the very least, our construction can be thought of as an ultimate bookkeeping that encodes all known elementary particles as components of such an elegant structure as $O \otimes \tilde{O}$ ⁠. In this sense, it is similar to the bookkeeping of Spin(10) GUT that can be thought of as providing the most compelling known explanation of the particle pattern seen in nature.

The advocated, here, bookkeeping by the semi-spinor representation of Spin(11, 3) is useful not only to describe the pattern of particles. Thus, it can be noted that the whole collection of the fermion kinetic terms in four spacetime dimensions can be obtained very elegantly in our approach using the idea of dimensional reduction. This has been described very explicitly in the context of the Spin(7, 7) model in our previous work.¹⁸ This work used the formalism of polyforms to describe spinors. The main result was the observation that the Weyl–Dirac kinetic term

\int ⟨ ⟨ Ψ, D Ψ ⟩ ⟩,

(100)

where D is the chiral Dirac operator that maps S₊ → S₋ and Ψ ∈ S₊, when dimensionally reduced to 1 + 3 dimensions reproduces the collection of correct Dirac kinetic terms for all the SM fermions. The calculation was based on the embedding Spin(6) × Spin(1, 3) × Spin(4) ⊂ Spin(7, 7). This observation can be seen to apply also to the considered here case of Spin(11, 3). Thus, also in the Spin(11, 3) case, the dimensional reduction of (100) from $R^{11, 3}$ to $R^{1, 3}$ reproduces the kinetic terms for all the SM fermions, thus “unifying” them in the single term (100). To summarize, our description of all known particles as components of a semi-spinor of Spin(11, 3) not only gives an efficient bookkeeping of the particles but also gives an elegant unification of the particles’ kinetic terms, as arising from the dimensional reduction of a simple chiral Weyl Largangian (100) in 14 dimensions.

The mathematical facts described in this paper add an interesting new perspective on the particle pattern as seen in nature. There are two possible viewpoints on this. In the first of these, one does not take the considered here symmetry group Spin(11, 3) seriously, as one that is realized in nature. This is similar to an attitude one can take toward Spin(10). While the kinematics of Spin(10) is beautiful and convincingly describes one generation of fermions, one does not need to promote Spin(10) to the level of a GUT group that needs to be broken dynamically. Instead, one can consider an appropriate subgroup of it, for example, the left/right symmetric group, and build phenomenological models based on this smaller gauge group. The Spin(10) considerations are still useful for this, but one would not attempt to break the full Spin(10) dynamically. We can similarly take the viewpoint that Spin(11, 3) just gives a kinematics framework with the desired properties and only allow models based on subgroups of Spin(11, 3). With this point of view, the role of 14 dimensions would be just that of a convenient mathematical structure to organize the particle content, i.e., bookkeeping.

However, it is tempting to do more and take the idea of unification in 14 dimensions seriously. As soon as this is done, there are serious difficulties that plague the whole approach, and these cannot be hidden. First and foremost, one is immediately faced with the all important question of what breaks such a large symmetry as Spin(11, 3) to the ones we see in nature. The provided description of the needed breaking pattern that uses the two complex structures $J, \tilde{J}$ is elegant, but it brings us no closer to answering the question about a mechanism for this symmetry breaking. What is different to the situation with more standard GUT schemes is that in that context, we have the Higgs mechanism, which we are confident is the correct theoretical description of the symmetry breaking at least in the case of the electroweak theory. The case of Spin(11, 3) is very different because we can no longer resort for help to the usual Higgs mechanism with some collection of Higgs fields. Indeed, it is clear that if one takes the Spin(11, 3) symmetry seriously, one is putting together what all textbooks say one should never mix—the group of gauge transformations acting on the “internal” particle degrees of freedom and the Lorentz group that acts by spacetime rotations. There is certainly no established mechanism that can break the group mixing such different symmetries together. Some attempts at such a mechanism are known under the name of “spontaneous soldering;” see, e.g., Refs. 15 and 16, but their status is very different from the established Higgs mechanism of particle physics.

In this paper, we do not propose any new mechanism of dynamical symmetry breaking of such a symmetry as Spin(11, 3). This paper should instead be read as giving yet another hint, supporting the idea of mixing the Lorentz and internal symmetries. Indeed, our main new observation is that there is a tantalizing similarity in the two complex structures $J, \tilde{J}$ ⁠, the first of which is needed to go from Spin(10) to the Pati–Salam group and the second needed to see the Lorentz group arising. This similarity is so striking that it suggests that the mechanism selecting them may be one and the same.

As a small step toward the mechanism required, we described the symmetry breaking using the three-form representation $Λ^{3} (R^{11, 3})$ ⁠. This arises as the representation of the smallest dimension in the second symmetric power of the semi-spinor representation with itself. We have seen that a particular three-form (92) is a singlet with respect to SU(3) × U(1)_B−L and Lorentz groups and transforms as the bi-doublet representation of SU(2)_L × SU(2)_R. Selecting a VEV for this field as in the left/right symmetric version of the SM, see (70), we can break Spin(11, 3) to the product of SU(3) × U(1)_B−L, Lorentz, and the diagonal U(1) ⊂ SU(2)_L × SU(2)_R. However, this “Higgs” field, although a step in the right direction, does not break the remaining U(1)_B−L × U(1) down to the electromagnetic U(1)_EM. Nevertheless, this construction does suggest that a field in $Λ^{3} (R^{11, 3})$ is the most natural one to attempt to produce a dynamical symmetry breaking mechanism. What is far from clear, however, is if there is dynamics that can select precisely configuration (92) for this field. In particular, it is very important that no Lorentz symmetry breaking terms get produced, and it is far from certain that this can be achieved. More work is needed to explore all these ideas.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Author(s)

Spin(11, 3), particles, and octonions

I. INTRODUCTION

II. Spin(7) AND OCTONIONS

A. Octonions

B. Clifford algebra Cliff7

C. Lie algebra so(7)

D. Complex structure

E. A different complex structure

F. Parameterization

G. The group of automorphisms of the octonions

H. The group SU(3)

III. Spin(4, 3) AND THE SPLIT OCTONIONS

A. Split octonions

B. The Clifford algebra Cliff4,3

C. Lie algebra so(4,3)

D. Complex structures

E. Decomposition of Õ under so(3)×so(1,3)

F. Parameterization

G. Some useful formulas

H. The split G̃2

IV. THE OCTONIONIC MODEL FOR Spin(11, 3)

A. Yet another magic square

B. Lie algebra

C. Invariant pairing

V. THE LEFT/RIGHT SYMMETRIC MODELS

A. Spinor fields

B. Bi-doublet Higgs field

C. Two Higgs in the adjoint

D. Two Higgs fields in the fundamental

E. A model with only fundamental Higgs fields

VI. DICTIONARY

A. Two commuting complex structures on O×Õ

B. Identification with particles

VII. MASS TERMS

A. Projection onto Λ3R11,3

B. SU(3) × U(1) and Lorentz invariant pairing

C. Evaluating the mass terms

VIII. DISCUSSION

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

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B. Clifford algebra Cliff₇

C. Lie algebra $s o (7)$

B. The Clifford algebra Cliff_4,3

C. Lie algebra $s o (4, 3)$

E. Decomposition of $\tilde{O}$ under $s o (3) \times s o (1, 3)$

H. The split ${\tilde{G}}_{2}$

A. Two commuting complex structures on $O \times \tilde{O}$

A. Projection onto $Λ^{3} R^{11, 3}$