Three Fermion Generations with Two Unbroken Gauge Symmetries from the Complex Sedenions

Eur. Phys. J. C (2019) 79:446 https://doi.org/10.1140/epjc/s10052-019-6967-1

Regular Article - Theoretical Physics

Three fermion generations with two unbroken gauge symmetries from the complex sedenions

Adam B. Gillard, Niels G. Gresnigta Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou HET, Jiangsu 215123, China

Received: 24 April 2019 / Accepted: 17 May 2019 / Published online: 27 May 2019 © The Author(s) 2019

Abstract We show that three generations of leptons and constant. Octonions also describe geometry in 10 dimen- quarks with unbroken Standard Model gauge symmetry sions [5], which have made them useful in supergravity and SU(3)c × U(1)em can be described using the algebra of superstring theories [6–9]. Recently there has been a revived complexiﬁed sedenions C ⊗ S. A primitive idempotent is interest in using the division algebras to attempt to construct constructed by selecting a special direction, and the action of a theoretical basis for the observed Standard Model (SM) this projector on the basis of C ⊗ S can be used to uniquely gauge groups and observed particle spectrum [10–17]. split the algebra into three complex octonion subalgebras A Witt decomposition of the adjoint algebra of left actions ∼ C ⊗ O. These subalgebras all share a common quaternionic (C ⊗ O)L = C(6) of C ⊗ O, generated from all composed subalgebra. The left adjoint actions of the 8 C-dimensional left actions of the complex octonions on themselves decom- C ⊗ O subalgebras on themselves generates three copies of poses (C⊗O)L into minimal left ideals [18]. The basis states the Clifford algebra C(6). It was previously shown that the of these ideals were recently shown to transform as a single minimal left ideals of C(6) describe a single generation of generation of leptons and quarks under the unbroken uni- fermions with unbroken SU(3)c ×U(1)em gauge symmetry. tary symmetries SU(3)c and U(1)em [11]. In a complemen- Extending this construction from C ⊗ O to C ⊗ S naturally tary work by [19] it was shown that by considering the right leads to a description of exactly three generations. adjoint action of C(6) on the eight minimal left ideals of the algebra, one can also include the spinorial degrees of freedom.1 One copy of C(6) = C(8) therefore provides the full 1 Introduction degrees of freedom for one generation of Dirac spinors. The main result presented here is that if we instead consider a The role of division algebras, particularly the octonion alge- decomposition of (C ⊗ S)L , where the sedenions S are the bra, in particle physics has a long history. Theorems by Hur- next Cayley-Dickson algebra after the octonions, then these witz [1] and Zorn [2] guarantee that there are only four earlier results are naturally extended from one single gener- normed division algebras, which are also the only four alter- ation to exactly three. native division algebras; the reals R, complex numbers C, Given that a considerable amount of the SM structure for quaternions H, and the octonions O. The ﬁrst results relating a single generation of fermions can be realised in terms of the the octonions to the symmetries of (one generation of) quarks complex octonions C ⊗ O, and its associated adjoint algebra ∼ goes back to the 1970s [3,4]. The hypothesis that octonions of left actions (C⊗O)L = C(6), a natural question is how to might play a role in the description of quark symmetries fol- extend these encouraging results from a single generation to lows from the observation that Aut(O) = G2 contains the exactly three generations. Although it is possible to represent physically important subgroup SU(3), corresponding to the three generations of fermion states inside a single copy of subgroup that holds one of the octonionic imaginary units C(6) [20,21], in that case the physical states are no longer

1 [11] describes one generation of fermions in terms of two out of eight minimal left ideals of C(6), without any mention of spinorial degrees of freedom. These can be acocunted for by including a copy of the complex quaternions C ⊗ H.[19] on the other hand writes down all eight of the minimal left ideals, which explicitly includes the spinorial a e-mail: [email protected] degrees of freedom. 123 446 Page 2 of 11 Eur. Phys. J. C (2019) 79 :446 the basis states of minimal left ideals as for the case of a 2 One generation of fermions as the basis states of the ∼ single generation. A three generation representation in terms minimal left ideals of (C ⊗ O)L = C(6) of C(6) therefore comes at the cost of giving up the elegant construction of minimal left ideals giving rise to a single One of the main observations in [11]2 and [19] is that the generation. Instead, one might therefore look for a larger basis states of the minimal left ideals of C(6) transform mathematical structure to describe three generations. as a single generation of leptons and quarks with unbro- Since describing a single generation requires one copy of ken SU(3)c and U(1)em gauge symmetries. In [11] the alge- the octonions, it seems reasonable to expect a three gener- bra C(6) is identified as the adjoint algebra of left actions ation model to require three copies of the octonions. This (C ⊗ O)L whereas in [19] C(6) is taken without any ref- makes the exceptional Jordan algebra J3(O) a natural can- erence to division algebras. In this section the minimal left ∼ didate. This intriguing mathematical structure and its role ideals of (C ⊗ O)L = C(6) are constructed using the Witt in particle physics has recently been explored by various decomposition for C(6). The basis states of these ideals are authors [22–24]. then shown to transform as a single generation of leptons and A different approach is taken here. C and O are both divi- quarks with unbroken SU(3)c and U(1)em. More details of sion algebras, and starting from R, all the other division alge- can be found in sections 4.5 and 6.6 of [11]. bras can be generated via the Cayley-Dickson process. Given Leptons and quarks are Dirac spinors. One way of includ- that this process continues beyond the octonions, one may ing the spinorial degrees of freedom is to enlarge the algebra wonder if the next algebra in the sequence, the sedenions S, from C ⊗ O to C ⊗ H ⊗ O and to consider both right and may be a suitable mathematical structure to represent three left adjoint actions. The approach taken in [11] it to con- generations of fermions. This paper confirms that the answer sider the combined left and right adjoint action of C ⊗ H ∼ is yes. which is C(2) ⊗C C(2) = C(4), providing a representa- Starting with C⊗S we repeat the initial stages of the con- tion of the Dirac algebra. On the other hand, in [19] all the struction of [11]. We define a projection operator by choos- spinor degrees of freedom are contained in a single copy of ing a special direction, and use it to obtain a split basis of C(6). The left actions of C(6) on its eight minimal left 14 nilpotent ladder operators, defining two maximal totally ideals generates the unbroken gauge symmetries, whereas isotropic subspaces (MTIS). But instead of using these nilpo- the right actions contain the Dirac and isospin symmetries ∼ tents to construct two minimal left ideals of (C ⊗ S)L ,we C(6) = C(4)Dirac ⊗C C(2)isospin. Therefore, C(6) is instead observe that these 14 nilpotents uniquely divide into large enough to contain the full degrees of freedom for a three sets of ladder operators, in such a way that each set pro- single generation of leptons and quarks. vides a basis for a C ⊗ O subalgebra of C ⊗ S. Our approach The octonions O are spanned by the identity 1 = e0 and therefore gives rise to three copies of the octonions from the seven anti-commuting square roots of minus one ei satisfying sedenions. In this construction one raising operator and one =−δ + , lowering operator is common to all three sets. The shared ei e j ije0 ijkek (1) ladder operators generate a common quaternionic subalge- where bra C ⊗ H =∼ C(2). From the three sets of ladder operators = = , 2 = , we construct six ideals. These ideals transform as three gen- ei e0 e0ei ei e0 e0 (2) erations of fermions and antifermions under the unbroken and is a completely antisymmetric tensor with value +1 gauge symmetry SU(3)c × U(1)em. ijk = , , , The next section reviews the construction of one gener- when ijk 124 156, 137, 235, 267 346 457. The multiplication of octonions is shown in Fig. 1. The multiplication ation of fermions with unbroken SU(3)c × U(1)em gauge symmetry starting from the complex octonions C ⊗ O. Then structure here follows that of [11], but is not unique. In par- in Sect. 3 it is shown that the same construction can be ticular, the Fano plane in the earlier work [3] is based on a used to describe three generations of leptons in terms of different multiplication structure. three C(2) subalgebras of C(6), generated from three Each pair of distinct points lies on a unique line of three C ⊗ H subalgebras of C ⊗ O. The main results of the points that are cyclically ordered. If ei , e j and ek are cyclically present paper are presented in Sect. 4, where three genera- ordered in this way then tions of fermions with unbroken SU(3) × U(1) gauge c em e e = e , e e =−e . (3) symmetries are constructed from the complex sedenions i j k j i k C⊗S. The generators for these symmetries are constructed in Sect. 5. 2 This work is also consistent with the works of Dixon [10,25–27]. Those works include a copy of the quaternion algebra, recovering the full SM gauge group from the algebra T = R ⊗ C ⊗ H ⊗ O.The quaternionic factor in T is responsible for the broken SU(2) chiral symmetry. 123 Eur. Phys. J. C (2019) 79 :446 Page 3 of 11 446

1 1 α ≡ (−e + ie ), α† ≡ (e + ie ), 1 2 5 4 1 2 5 4 1 1 α ≡ (−e + ie ), α† ≡ (e + ie ), 2 2 3 1 2 2 3 1 1 1 α ≡ (−e + ie )α† ≡ (e + ie ). (5) 3 2 6 2 3 2 6 2

The hermitian conjugate simultaneously maps i →−i and ei →−ei . The new basis vectors span the two MTIS satis- fying4

α ,α = , α†,α† = , α ,α† = δ . i j 0 i j 0 i j ij (6) Fig. 1 Octonion multiplication represented using a Fano plane

Care should be taken here (and later) not to confuse the complex i satisfying i2 =−1, and the index i = 1, 2, 3. If one now considers these basis vector operators as elements of ∼ (C ⊗ O)L = C(6), then one can construct the primitive ωω† = α α α α†α†α† idempotent 1 2 3 3 2 1. The ﬁrst minimal left ideal is then given by Su ≡ C(6)ωω†. Explicitly:

Su ≡ νωω† + ¯r α†ωω† + ¯gα†ωω† + ¯bα†ωω† r α†α†ωω† d 1 d 2 d 3 u 3 2 + gα†α†ωω† + bα†α†ωω† u 1 3 u 2 1 + +α†α†α†ωω†, e 3 2 1 (7)

where ν, d¯r etc. are suggestively labeled complex coefﬁcients denoting the isospin-up elementary fermions. The conjugate Fig. 2 Fano planes of three octonionic subalgebras in the sedenions system analogously gives a second linearly independent minimal left ideal of isospin-down elementary fermions:

Every straight line in the Fano plane of the octonions (taken Sd ≡¯νω†ω together with the identity) generates a copy of the quater- r † g † b † r † +d α1ω ω + d α2ω ω + d α3ω ωu¯ α3α2ω ω nions, for example {1, e4, e6, e3}. Furthermore {1, e1, e2, e4} g † b † also gives a copy of the quaternions, making for a total of +¯u α1α3ω ω +¯u α2α1ω ω − † seven copies of the quaternions embedded within the octo- +e α3α2α1ω ω. (8) nions. One starts by choosing a privileged imaginary unit from One can show that these representations of the minimal the standard octonion basis e , e , ..., e and uses this to 0 1 7 left ideals are invariant under the color and electromagnetic resolve the identity into two primitive idempotents. The com- symmetries SU(3)c and U(1)em, and each of the basis states mon choice is e , so that 7 in the ideals transforms as a specific lepton or quark under these symmetries as indicated by their suggestively labeled 1 1 1 = ρ+ + ρ− = (1 + ie7) + (1 − ie7). (4) complex coefficients. 2 2 In terms of the Witt basis ladder operators, the SU(3) generators take the form Multiplying the standard octonionic basis elements on the left by ρ+ defines a split basis of nilpotents, and allows one to define the new basis vectors,3 along with their conjugates as 4 The split basis in [11] differs slightly from the one given in [3]. The split basis of the latter uses the complex conjugate ∗ instead of the hermitian conjugate †, and satisfies the additional multiplication rules α α = α∗ α∗α∗ = α i j ijk k , i j ijk k . These additional multiplication rules 3 Following the convention of [11]. are not satisfied by the split basis of [11]. 123 446 Page 4 of 11 Eur. Phys. J. C (2019) 79 :446

Λ =−α†α − α†α ,Λ= α†α − α†α , ( ) ( ) 1 2 1 1 2 2 i 2 1 i 1 2 erations of leptons with U 1 em symmetry. The U 1 em gen- 5 Λ = α†α − α†α ,Λ=−α†α − α†α , erator is given by 3 2 2 1 1 4 1 3 3 1 Λ =−iα†α + iα†α ,Λ=−α†α − α†α , = α α† + α α† + α α†. 5 1 3 3 1 6 3 2 2 3 Q 1 1 2 2 3 3 (15) Λ = α†α − α†α , 7 i 3 2 i 2 3 ∼ Starting from (C⊗O)L = C(6), one can represent either −1 Λ = √ (α†α + α†α − α†α ). a single generation of leptons and quarks with unbroken 8 1 1 2 2 2 3 3 (9) 3 SU(3)c × U(1)em symmetry, or alternatively three generations of leptons with U(1) symmetry. The former results ( ) em The U 1 generator can be expressed in terms of the Witt from constructing the minimal left ideals of C(6), whereas basis ladder operators as the latter corresponds to constructing the minimal left ideals of three (in this case independent) C(2) subalgebras of = 1(α†α + α†α + α†α ). C( ) Q 1 1 2 2 3 3 (10) 6 . Choosing a preferred octonionic unit e7 singles out 3 three natural quaternionic subalgebras of O which contain g e7. These may be identified as describing three generations As an illustrative example we consider [Λ1, u ]: of leptons. g † † † † This same description of three generations of leptons was Λ1, u = −α α1 − α α2 α α 2 1 1 3 considered in [28,29], although there these three genera- † † † † tions emerged from a dimensional reduction of an octonionic −α α −α α1 − α α2 , 1 3 2 1 massless Dirac equation in ten dimensions to four dimen- =−α†α†α α† + α†α†α†α , sions. This reduction likewise leads to three quaternionic sub- 2 3 1 1 3 2 1 1 = α†α† α α† + α†α , algebras, singled out because they contain e7, each identified 3 2 1 1 1 1 as describing a generation of (massive and massless) leptons. = α†α† = r . In the next section we demonstrate that the same approach 3 2 u (11) for the sedenions singles out three octonionic algebras, each of which describes a generation of leptons and quarks with electrocolour symmetry. 3 Three generations of leptons from three minimal left ∼ ideals of (C ⊗ H)L = C(2) ⊂ C(6) 4 From octonions to sedenions: one generation to three Instead of defining the MTIS of Eq. (6), one can instead generations define the six totally isotropic (but not maximal) subspaces The previous section reviewed earlier results that a single α† , {α } , i = 1, 2, 3. (12) i i generation of fermions with unbroken SU(3)c × U(1)em gauge symmetry can be described starting from C ⊗ O. Each of these totally isotropic subspaces is a MTIS not of the Whereas C and O are each individually division algebras, full C(6), but rather of a C(2) subalgebra of C(6).Note the tensor product C ⊗ O is not. Nor is C(6), the adjoint ∼ that C(2) = (C ⊗ H)L , and so each C(2) subalgebra is algebra of left actions of C⊗O on itself. Indeed, the construc- generated from a C⊗H subalgebra of C⊗O. Associated with tion of minimal left ideals is very dependent on defining a these subspaces one can define six primitive idempotents nilpotent basis of ladder operators. The zero divisors in C⊗O therefore play an important role. ω ω† = α α†,ωω† = α α†,ωω† = α α†, C, H, O 1 1 1 1 2 2 2 2 3 3 3 3 (13) The division algebras can each be generated via the Cayley-Dickson process. This process however does not together with their conjugates, and subsequently three ideals terminate with the octonions, but can be used to generate a2n-dimensional algebra for any positive integer n.For ν ω ω† + +α†ω ω†, n > 3 however, the Cayley-Dickson algebras are no longer e 1 1 e 1 1 1 division algebras. However, given the observations above, ν ω ω† + μ+α†ω ω†, μ 2 2 2 2 2 this should not dissuade us from considering larger Cayley- ν ω ω† + τ +α†ω ω†, τ 3 3 3 3 3 (14) Dickson algebras. with their conjugates. Here as before, the suggestively 5 In the present case we can define a single U(1) generator Q which labeled complex coefficients denote the isospin-up leptons. acts on all three generations. This will not work in general, where one The twelve states of these six ideals transform as three gen- should instead define one U(1)em generator for each generation. 123 Eur. Phys. J. C (2019) 79 :446 Page 5 of 11 446

4.1 The sedenion algebra Unlike the four division algebras, the sedenions have not been the focus of much study in relation to particle physics, The 16-dimensional sedenion algebra is the ﬁfth Cayley- presumably due to the fact that they are not a division algebra Dickson algebra A4 = S. The algebra is not a division and are neither associative nor alternative. Tensor products algebra, and is non-commutative, non-associative, and non- of division algebras however also contain zero divisors. The alternative. However it is power-associative and distributive. constructions of particle symmetries from C⊗O [11], C(6) One can choose a canonical basis for S to be E16 ={ei ∈ [19], and T = R ⊗ C ⊗ H ⊗ O [10], all contain zero divi- S|i = 0, 1, ..., 15} where e0 is the real unit and e1, ..., e15 are sors, and these play an important role. It seems reasonable mutually anticommuting imaginary units. In this canonical therefore to continue the Cayley-Dickson algebra into the basis, a general element a ∈ S is written as non-division algebras. The symmetries of these algebras can be extracted from 15 15 their automorphism groups. Interestingly, for the sedenions A = a e = a + a e , a ∈ R. (16) i i 0 i i i one ﬁnds that i=0 i=1

The basis elements satisfy the following multiplication rules Aut(S) = Aut(O) × S3. (20)

e0 = 1, e0ei = ei e0 = ei , The only difference between the octonions and sedenion 2 = 2 = ... = 2 =− , automorphism groups is a factor of the permutation group e1 e2 e15 1 S3. This permutation group can be constructed from the tri- e e =−δ e + γ k e , (17) i j ij 0 ij k ality automorphism of Spin(8), the spin group of C(8) [32]. This automorphism is of order three. Interestingly then, what where the real structure constants γ k are completely anti- ij Eq. (20) suggests is that the fundamental symmetries of S are symmetric. The multiplication table of the sedenion base ele- the same as those of O, although the factor of S suggests ments is given in Appendix A. For two sedenions A, B, one 3 we get three copies. This is exactly what we want in order to has describe the observed three generations of fermions. 15 15 15 For higher Cayley-Dickson (n > 3) algebras one has AB = ai ei b j ei = ai b j (ei e j ), i=0 i=0 i, j=0 Aut(An) = Aut(O) × (n − 3)S3. (21) 15 = γ k , This tells us that the underlying symmetries are G2,the fij ijek (18) i, j,k=0 automorphism group of the octonions. The higher Cayley- Dickson algebras only add additional trialities, and perhaps where fij ≡ ai b j . Addition and subtraction is component- no new fundamental physics should be expected beyond wise. C ⊗ S. Because the sedenion algebra is not a division algebra, it S contains zero divisors. For these are elements of the form 4.3 From one generation to three generations

(ea + eb) ◦ (ec + eb) = 0, ea, eb, ec, ed ∈ S. (19) A natural way to extend the intriguing results from Section There are 84 such zero divisors, and the subspace of zero (2) beyond a single generation of elementary fermions is to divisors of unit norm is homeomorphic to G2 [30]. consider the complex sedenions C ⊗ S instead of complex octonions C ⊗ O. This larger algebra also generates a larger (C⊗S) 4.2 Why sedenions? adjoint algebra of left actions L . The key point of this paper is to demonstrate: One appealing reason for considering division algebras in C ⊗ O → C ⊗ S relation to particle symmetries (both spacetime and internal) (C ⊗ O) → (C ⊗ S) is that, unlike Lie algebras and Clifford algebras, there are L L only a finite number of division algebras.6 At the same time, 1 generation → 3 generations. each division algebra generates very specific Lie and Clif- ford algebras. Starting with a division algebra, the physical As in section (2), we start by choosing e15 as a special symmetries are dictated. imaginary unit from the sedenion basis, mimicking the common choice of e7 for O. One can then define the primitive idempotents 6 There are many convincing arguments that the four division algebras should play a special role in particle physics. Many of these arguments 1 1 ρ+ = (1 + ie ), ρ− = (1 − ie ), (22) can be found in the extensive works of Dixon [10,31]. 2 15 2 15 123 446 Page 6 of 11 Eur. Phys. J. C (2019) 79 :446 satisfying singles out three quaternionic subalgebras. In [28,29] these 2 are identified with three generations of leptons. Instead here ρ+ + ρ− = 1,ρ± = ρ±,ρ+ρ− = 0. (23) we have found that selecting e15 as the special unit imag- We now act with ρ+ on the sedenion basis elements ei , inary singles our three octonionic subalgebras. This allows and subsequently define a split basis for C ⊗ S. This gives us to describe three generations of leptons and quarks with a set of seven nilpotents ηk, k = 1, 2, ..., 7 of the form unbroken SU(3)c × U(1)em symmetry. These three C ⊗ O 1 (− + ) ∈ , , , , , , 2 ea ie15−a where a 5 7 13 1 4 6 12, together subalgebras are however not independent. All three subal- † 1 with seven nilpotent conjugates η of the form (ea+ie15−a). gebras share a common quaternionic subalgebra spanned by k 2 { , , , } Considered as elements of (C ⊗ S)L , these define two MTIS 1 e1 e14 e15 . The three intersecting octonion subalgebras satisfying together with the multiplication rules of their base elements may be represented by the three Fano planes in Fig. 2. η†,η† = η ,η = , η†,η = δ . i j i j 0 i j ij (24) In addition to the standard fermionic ladder operator algebras satisfied by the α’s, β’s, and γ ’s separately, we also have η† η These i and i uniquely divide into three sets of ladder {α†,β†}={β†,γ†}={γ †,α†}= , operators that do i j i j i j 0 1 1 {αi ,βj }={βi ,γj }={γi ,αj }=0, α† = (e + ie ), β† = (e + ie ), 1 2 5 10 1 2 7 8 {α†,β }={β†,γ }={γ †,α }=δ . i j i j i j ij (27) 1 γ † = (e + ie ), 1 2 13 2 From each of the three C ⊗ O subalgebras we can now 1 construct an ideal following the procedure or Section 2. These α† = β† = γ † = (e + ie ), 2 2 2 2 1 14 ideals will be minimal left C(6) ideals embedded within the (C ⊗ S) α† = 1( + ), β† = 1( + ), full L algebra. Using the first set of ladder operators 3 e4 ie11 3 e6 ie9 {α†,α†,α†,α ,α ,α } 2 2 1 2 3 1 2 3 , we can now construct the first ideal u † † † † † † 1 as S = C(6)ω1ω , where ω1ω = α1α2α3α α α is a γ = (e12 + ie3), (25) 1 1 1 3 2 1 3 2 C(6) primitive idempotent. together with u † S ≡ νeω1ω 1 1 1 1 ¯r † † ¯g † † ¯b † † r † † † α1 = (−e5 + ie10), β1 = (−e7 + ie8), +d α ω ω + d α ω ω + d α ω ω u α α ω ω 2 2 1 1 1 2 1 1 3 1 1 3 2 1 1 + gα†α†ω ω† + bα†α†ω ω† 1 u 1 3 1 1 u 2 1 1 1 γ1 = (−e13 + ie2), + 2 +e α†α†α†ω ω†, (28) 1 3 2 1 1 1 α2 = β2 = γ2 = (−e1 + ie14), 2 The complex conjugate system gives a second linearly inde- 1 1 C( ) α = (−e + ie ), β = (−e + ie ), pendent 6 minimal left ideal 3 2 4 11 3 2 6 9 d † 1 S ≡¯νeω ω1 γ = (−e + ie ). (26) 1 1 3 2 12 3 + r α ω†ω + gα ω†ω + bα ω†ω ¯r α α ω†ω d 1 1 1 d 2 1 1 d 3 1 1u 3 2 1 1 Here, {α†,α }, {β†,β }, and {γ †,γ } each individually form +¯gα α ω†ω +¯bα α ω†ω i i i i i i u 1 3 1 1 u 2 1 1 1 a split basis for C ⊗ O and satisfy the anticommutator alge- − +e α α α ω†ω . (29) bra of fermionic ladder operators exactly as in section (2).7 3 2 1 1 1 η† By uniquely we mean that any other division of the seven i One can likewise construct the ideals of the other two and seven ηi into equally sized sets does not result in three C( ) {β†,β } 6 subalgebras from the sets of ladder operators i i C ⊗ O subalgebras. Thus, what we find is that the action {γ †,γ } and i i . In total we find six minimal left ideals which of the projector ρ+ on the standard sedenion basis uniquely { u, d , u, d , u, d } u d we write as S1 S1 S2 S2 S3 S3 . Whereas Si and Si are divides the algebra into three sets of split basis elements for linearly independent, Su and Su are not. C ⊗ O i j . This reflects what happened earlier when consider- So far we have not commented on the spinorial degrees C ⊗ O ing . There, choosing e7 as the special unit imaginary of freedom. In our construction we have followed [11], making use of only two of the eight ideals of C(6).Thefull 7 There is a slight deviation here in the definition of ladder operators C(6) =∼ C(8) algebra however contains eight minimal left as in [11]. Here we have multiplied the standard basis elements e on i i = , ··· , the left by ρ+ and from the result have immediately identified ηi and ideals. Compactly, for 1 8 these eight ideals can be η† α† β† γ † written as: i . This means that the real parts i (and similarly for i and i )form a quaternion subalgebra. In [11] it is the imaginary parts that form a = u { ,α ,α ,α ,α ,α ,α ,α } quaternion subalgebra. Pi S1 1 23 31 12 123 1 2 3 (30) 123 Eur. Phys. J. C (2019) 79 :446 Page 7 of 11 446 ( ) for the first generation, and similarly for the second and third As two illustrative examples, consider Λ 2 , sr and 1 generations with the α’s replaced by β’s and γ ’s respectively. ( ) Q 3 , t¯b : As demonstrated by [19], incorporating all eight of the minimal left ideals allows one to include the spinorial degrees of Λ(2), r =−(β†β + β†β )β 1 s 2 1 1 2 1 freedom. In particular, the chiral assignments are as follows: +β (β†β + β†β ), 1 2 1 1 2 = , = , = , = , P1 P1R1 P2 P2R2 P3 P3L1 P4 P4L2 = β†β β + β β†β , 1 1 2 1 1 2 = , = , = , = , P5 P5L1 P6 P6L2 P7 P7R1 P8 P8R2 † † g = (β β1 + β1β )β2 = β2 = s , (34) 1 1 u d ( ) 1 where P1 = P1R and P5 = P5L are the two ideals S and S 3 , ¯b = (γ †γ + γ †γ + γ †γ )γ γ 1 1 1 1 Q t 1 1 2 2 3 3 2 1 explicitly considered above and in [11]. All the results from 3 [19] for a single generation generalize to three generations −1γ γ (γ †γ + γ †γ + γ †γ ), 2 1 1 1 2 2 3 3 in the current construction. 3 =−1γ γ γ †γ + 1γ γ γ †γ , 3 2 1 1 1 3 1 2 2 2 1 † 1 † ( ) × ( ) =− γ2γ1(−γ1γ + 1) + γ1(−γ γ2 + 1)γ2, 5 Unbroken SU 3 c U 1 em gauge symmetries 3 1 3 2 2 2 ¯b C⊗O =− γ2γ1 =− t . (35) As for the case of a single generation constructed from , 3 3 the physical states of the three generation extension presented here transform as three generations of SM fermions 5.1 Mixing between generations under SU(3)c × U(1)em. Each generation has its own copy of SU(3) and U(1), which we label SU(i)(3) and U (i)(1), The ladder operators for each generation in our three genera- i = 1, 2, 3. tion model belong to a Fano plane. Each Fano plane gives the The SU(1)(3) generators can be represented in the first projective geometry of the octonionic projective plane OP2 C(6) ⊂ (C ⊗ S)L as [5], which is the quantum state space upon which the quantum exceptional Jordan algebra J3(O) acts [33]. One might Λ(1) =−i ( − ), Λ(1) = i ( − ), therefore expect a relationship between the approach pre- 1 e1e10 e14e5 2 e14e10 e5e1 2 2 sented here, based on (C ⊗ O)L and three generation models ( ) i ( ) i Λ 1 =− ( − ), Λ 1 =− ( − ), based on J3(O) [22–24]. 3 e5e10 e1e14 4 e5e11 e10e4 2 2 For the case of three non-intersecting Fano planes, the total (1) i (1) i state space is given by the tensor product of the three individ- Λ =− (e10e11 − e4e5), Λ =− (e4e14 − e11e1), 5 2 6 2 Λ(1) ual state spaces. This would mean that what we called i Λ(1) = i ( − ), is really Λ ⊗ 1 ⊗ 1, and what we called a blue top quark tb is 7 e11e14 e1e4 2 really 1⊗1⊗tb. It follows that any inter-generational commu- ( ) −i (1) Λ 1 = √ ( + − ). tator such as [Λ , tb] vanishes trivially. In our construction 8 e5e10 e1e14 2e4e11 (31) i 3 however, the three Fano planes are not truly independent of one another, but actually share a common complex quater- This can be rewritten in terms of the ladder operators, taking nionic line of intersection. This means that the tensor product the form structure is not perfectly commutative, and this can lead to Λ(1) =−(α†α + α†α ), Λ(1) =−(α†α − α†α ), mixing between generations. For example 1 2 1 1 2 2 i 2 1 1 2 Λ(1) = α†α − α†α ,Λ(1) =−(α†α + α†α ), Λ(2),α† = β†. 3 2 2 1 1 4 1 3 3 1 1 2 1 (36) Λ(1) = (α†α − α†α ), Λ(1) =−(α†α + α†α ), 5 i 1 3 3 1 6 3 2 2 3 What this curious behaviour corresponds to physically is cur- Λ(1) =−(α†α − α†α ), rently being investigated. One interesting proposal is that the 7 i 3 2 2 3 − mixing between generations might form a basis for the weak (1) 1 † † † Λ = √ (α α1 + α α2 − 2α α3). (32) PMNS and CKM mixing matrices.8 8 3 1 2 3

The explicit SU(3) generators for the other two generations 8 It may also be that the tensor product gets braided. The relation are given in Appendix B. Under the action of SU(1)(3), one between braids and the minimal left ideals of C(6) were reported on in u d [12]. Embedding braided matter into Loop Quantum Gravity is possi- ﬁnds that the ideals S1 and S1 transform as [3,11,26] ble when the underlying group representations that label the edges of a spin-network are q-deformed. Such a q-deformation has an associated u ∼ ⊕ ¯ ⊕ ⊕ ¯, d ∼ ¯ ⊕ ⊕ ¯ ⊕ . S1 1 3 3 1 S1 1 3 3 1 (33) R-matrix that braids any tensor products. This idea is still speculative. 123 446 Page 8 of 11 Eur. Phys. J. C (2019) 79 :446

6 Discussion The sedenions should not be ruled out as playing a role in particle physics on the basis that they do not constitute a The adjoint algebra of left actions of the complex octonions division algebra. Whereas R, C, H and O are by themselves was previously shown to have the right mathematical struc- division algebras, tensor products such as C ⊗ H and C ⊗ O ture to describe one generation of SM fermions with the are not, and in fact the zero divisors of these algebra play unbroken gauge symmetries SU(3)c × U(1)em [11]. In this a crucial role in the constructions of left ideals [10,11,25]. paper we have extended these results to exactly three gen- Because Aut(S) = Aut(O)×S3, the fundamental symmetries erations by instead starting from the complex sedenions. A of S are the same as those of O. The factor of S3 is generated split basis for this algebra naturally divides fourteen nilpotent by the triality automorphism of Spin(8). In a future paper ladder operators into three sets, each of which gives a split the relation between S, triality, and Spin(8) will be presented basis for a complex octonion subalgebra. Each of these sub- in more detail. The combination of the octonion algebra and algebras generates, via the minimal left ideals of their adjoint triality leads naturally to the exceptional Jordan algebra, a 27- algebra of left actions, a single generation of fermions. dimensional quantum algebra. This should make it possible The minimal left ideals in our case are minimal left ideals to relate our work to the exceptional Jordan algebra [23,24]. ∼ of (C × O)L = C(6), but not of the full (C ⊗ S)L .Our Beyond the sedenions, one might not expect new physics. construction therefore generalizes the construction outlined This is because Aut(An) = Aut(O) × (n − 3)S3, where An in Section 3, which follows [11], making use of only two of is the n-th Cayley-Dickson algebra. The fundamental sym- the eight ideals of C(6). As demonstrated by [19], when the metries remain those of the octonions, with only additional remaining six minimal left ideals are considered, one finds factors of the permutation group S3 being added. ∼ that there are exactly enough degrees of freedom to describe The three copies of (C ⊗ O)L = C(6) live inside the three generations of fermions, including the spinorial degrees larger algebra (C ⊗ S)L . This larger algebra may make it of freedom. possible to include additional particle degrees of freedom Instead of defining the minimal left ideals, via the MTIS, (e.g spin). This idea is currently being investigated. Finally, ∼ of C(6) one can instead divide the split basis for C ⊗ O into the three copies of (C ⊗ O)L = C(6) are not independent, three split bases for three C ⊗ H subalgebras. These three but share a common C(2) subalgebra. One interesting idea quaternionic subalgebra all contain the special imaginary unit is that this C(2) may form a theoretical basis for the PMNS e7 used to construct the projectors. In earlier works these three and CKM matrices that describe neutrino oscillations and quaternionic subalgebras have been identified with three gen- quark mixing. erations of leptons [28]. For the case of C⊗S, instead of con- (C ⊗ S) Data Availability Statement This manuscript has no associated data or structing the minimal left ideals of L ,wehaveused the data will not be deposited [Authors’ comment: There is no additional the special imaginary unit e15 to define three C ⊗ O sub- data to be deposited.] algebras. These three subalgebras share a common C ⊗ H algebras. Open Access This article is distributed under the terms of the Creative The sedenions are not a division algebra. Unlike R, C, H Commons Attribution 4.0 International License (http://creativecomm O ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and they are not alternative. However the lack of alterna- and reproduction in any medium, provided you give appropriate credit tivity, like the lack of associativity, does not affect the con- to the original author(s) and the source, provide a link to the Creative struction of minimal left ideals. The fermion algebra satis- Commons license, and indicate if changes were made. 3 fied by the split basis requires only two basis elements to Funded by SCOAP . be multiplied together at any time. In any such calculation, associativity or alternativity play no role. The ideals are constructed from the adjoint algebra of left actions of the algebra Appendix A: Sedenion multiplication table on itself and is always associative, alternative, and isomor- phic to a Clifford algebra.

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Appendix B: C(6) minimal left ideals in (C ⊗ S)L

Generation 1 1 1 α† = (e + ie ), α = (−e + ie ), 1 2 5 10 1 2 5 10 1 1 α† = (e + ie ), α = (−e + ie ), 2 2 1 14 2 2 1 14 1 1 α† = (e + ie ), α = (−e + ie ), ω ω† 3 2 4 11 3 2 4 11 1 1 = α α α α†α†α†. 1 2 3 3 2 1 (37) u ≡ C( ) ω ω† The ﬁrst minimal left ideal is given by S1 6 1 1 1. Explicitly, u ≡ ν ω ω† S1 e 1 1 + ¯r α†ω ω† + ¯gα†ω ω† + ¯bα†ω ω† r α†α†ω ω† d 1 1 1 d 2 1 1 d 3 1 1u 3 2 1 1 + gα†α†ω ω† + bα†α†ω ω† u 1 3 1 1 u 2 1 1 1 + +α†α†α†ω ω†, e 3 2 1 1 1 (38) The complex conjugate system gives a second linearly independent minimal left ideal d ≡¯ν ω†ω S1 e 1 1 + r α ω†ω + gα ω†ω + bα ω†ω ¯r α α ω†ω d 1 1 1 d 2 1 1 d 3 1 1u 3 2 1 1 +¯gα α ω†ω +¯bα α ω†ω u 1 3 1 1 u 2 1 1 1 + −α α α ω†ω . e 3 2 1 1 1 (39) The SU(3), written in terms of the ladder operators, are given by Λ(1) =−(α†α + α†α ), Λ(1) =−(α†α − α†α ), 1 2 1 1 2 2 i 2 1 1 2 Λ(1) = α†α − α†α ,Λ(1) =−(α†α + α†α ), 3 2 2 1 1 4 1 3 3 1 Λ(1) = (α†α − α†α ), Λ(1) =−(α†α + α†α ), 5 i 1 3 3 1 6 3 2 2 3 Λ(1) =−(α†α − α†α ), 7 i 3 2 2 3 − (1) 1 † † † Λ = √ (α α1 + α α2 − 2α α3). (40) 8 3 1 2 3 The U(1) generator, written in terms of ladder operators is

( ) 1 Q 1 = (α†α + α†α + α†α ). (41) 3 1 1 2 2 3 3

Generation 2 1 1 β† = (e + ie ), β = (−e + ie ), 1 2 7 8 1 2 7 8 1 1 β† = (e + ie ), β = (−e + ie ), 2 2 1 14 2 2 1 14 1 1 β† = (e + ie ), β = (−e + ie ), 3 2 6 9 3 2 6 9 ω ω† = β β β β†β†β†. 2 2 1 2 3 3 2 1 (42)

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u ≡ C( ) ω The complex conjugate system gives a second linearly The second minimal left ideal is given by S2 6 2 2 ω† independent minimal left ideal 2. Explicitly, d † r † g † b † u ≡ ν ω ω† S ≡¯ντ ω ω3 + b γ1ω ω3 + b γ2ω ω3 + b γ3ω ω3 S2 μ 2 2 3 3 3 3 3 ¯r † ¯g † ¯b † +¯r β†ω ω† +¯gβ†ω ω† +¯bβ†ω ω† r β†β†ω ω† t γ3γ2ω ω3 + t γ1γ3ω ω3 + t γ2γ1ω ω3 s 1 2 2 s 2 2 2 s 3 2 2c 3 2 2 2 3 3 3 − † + gβ†β†ω ω† + bβ†β†ω ω† +τ γ3γ2γ1ω ω3. (49) c 1 3 2 2 c 2 1 2 2 3 +μ+β†β†β†ω ω†, ( ) 3 2 1 2 2 (43) The SU 3 , written in terms of the ladder operators, are given by The complex conjugate system gives a second linearly independent minimal left ideal Λ(3) =−(γ †γ + γ †γ ), Λ(3) =−(γ †γ − γ †γ ), 1 2 1 1 2 2 i 2 1 1 2 d † r † g † b † ( ) ( ) S ≡¯νμω ω + s β ω ω + s β ω ω + s β ω ω Λ 3 = γ †γ − γ †γ ,Λ3 =−(γ †γ + γ †γ ), 2 2 2 1 2 2 2 2 2 3 2 2 3 2 2 1 1 4 1 3 3 1 r † g † b † (3) † † (3) † † c¯ β3β2ω ω2 +¯c β1β3ω ω2 +¯c β2β1ω ω2 Λ = (γ γ − γ γ ), Λ =−(γ γ + γ γ ), 2 2 2 5 i 1 3 3 1 6 3 2 2 3 − † ( ) +μ β3β2β1ω ω2. (44) Λ 3 =−(γ †γ − γ †γ ), 2 7 i 3 2 2 3 − The SU(3), written in terms of the ladder operators, are (3) 1 † † † Λ = √ (γ γ1 + γ γ2 − 2γ γ3). (50) given by 8 3 1 2 3 Λ(2) =−(β†β + β†β ), Λ(2) =−(β†β − β†β ), ( ) 1 2 1 1 2 2 i 2 1 1 2 The U 1 generator, written in terms of ladder operators Λ(2) = β†β − β†β ,Λ(2) =−(β†β + β†β ), is 3 2 2 1 1 4 1 3 3 1 (2) † † (2) † † (3) 1 † † † Λ = i(β β3 − β β1), Λ =−(β β2 + β β3), Q = (γ γ + γ γ + γ γ ). (51) 5 1 3 6 3 2 3 1 1 2 2 3 3 Λ(2) =−(β†β − β†β ), 7 i 3 2 2 3 − (2) 1 † † † Λ = √ (β β1 + β β2 − 2β β3). (45) 8 3 1 2 3 References ( ) The U 1 generator, written in terms of ladder operators 1. Adolf Hurwitz, Über die komposition der quadratischen formen. is Mathematische Annalen 88(1–2), 1–25 (1922) 2. Max Zorn, Theorie der alternativen ringe. in Abhandlungen aus (2) = 1(β†β + β†β + β†β ). Q 1 1 2 2 3 3 (46) dem Mathematischen Seminar der Universität Hamburg, volume 8, 3 pages 123–147. Springer (1931) 3. Murat Günaydin, Feza Gürsey, Quark structure and octonions. J. Generation 3 Math. Phys. 14(11), 1651–1667 (1973) 4. M. Günaydin, F. Gürsey, Quark statistics and octonions. Phys. Rev. γ † = 1( + ), γ = 1(− + ), 1 e13 ie2 1 e13 ie2 D 9(12), 3387 (1974) 2 2 5. John Baez, The octonions. Bull. Am. Math. Soc. 39(2), 145–205 1 1 γ † = (e + ie ), γ = (−e + ie ), (2002) 2 2 1 14 2 2 1 14 6. M.J. Duff, Jonathan M Evans, Ramzi R Khuri, J.X. Lu, Ruben Minasian, The octonionic membrane. Nuclear Phys. B- Proc. γ † = 1( + ), γ = 1(− + ), 3 e12 ie3 3 e12 ie3 Suppl. 68(1–3), 295–302 (1998) 2 2 7. John C Baez, John Huerta, Division algebras and supersymmetry ω ω† = γ γ γ γ †γ †γ †. 3 3 1 2 3 3 2 1 (47) i. Superstrings, geometry, topology, and C*-algebras 81, 65–80 (2009) u ≡ C( ) ω ω† 8. John C Baez, John Huerta et al., Division algebras and supersym- The third minimal left ideal is given by S3 6 3 3 3. Explicitly, metry II. Adv. Theor. Math. Phys. 15(5), 1373–1410 (2011) 9. Alexandros Anastasiou, Leron Borsten, M.J. Duff, L.J. Hughes, u † Silvia Nagy, Super yang-mills, division algebras and triality. J. High S ≡ ντ ω3ω 3 3 Energy Phys. 2014(8), 80 (2014) + ¯r γ †ω ω† + ¯gγ †ω ω† + ¯bγ †ω ω† r γ †γ †ω ω† b 1 3 3 b 2 3 3 b 3 3 3t 3 2 3 3 10. Geoffrey M. Dixon, Division algebras: octonions quaternions complex numbers and the algebraic design of physics, vol. 290 + gγ †γ †ω ω† + bγ †γ †ω ω† t 1 3 3 3 t 2 1 3 3 (Springer, Berlin, 2013) + +τ γ †γ †γ †ω ω†, (48) 11. Cohl Furey. Standard model physics from an algebra? arXiv 3 2 1 3 3 preprint arXiv:1611.09182, (2016) 12. Niels G Gresnigt, Braids, normed division algebras, and standard model symmetries. Phys. Lett. B 783, 212-221 (arXiv preprint arXiv:1803.02202) (2018) 13. Niels G Gresnigt, Braided fermions from hurwitz algebras. arXiv preprint arXiv:1901.01312 (2018) 14. Tevian Dray, Corinne A Manogue, Octonionic cayley spinors and e6. Comment. Math. Univ. Carolin 51(2), 193–207 (2010) 123 Eur. Phys. J. C (2019) 79 :446 Page 11 of 11 446

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