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PoS(EPS-HEP2019)615 ⊗ c ) 3 ( SU , this one S https://pos.sissa.it/ common to H . By going beyond the ⊗ O C , and three ⊗ ) C 8 ( ` C . It is speculated that the shared ) O ( 3 J , however these three are not independent of one another. S ⊗ C ∗† can be described in terms of complex algebra of em O ) [email protected] 1 ⊗ ( Speaker. This work is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China Recently there has been renewedwith interest their in associated using Clifford , togenerations products identify of of the leptons structures and algebras, of together the transforming Standard correctly Model. under One the full electrocolor division algebras, and consideringgeneration the model larger is Cayley-Dickson extended algebraC to of exactly three generations.This three Each generation generation model is canthe be contained exceptional related in Jordan to an algebra an alternativeall model three of generations three might generations form based a on for CKM mixing. U † ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Programme grant 19KJB140018 and XJTLU REF-18-02-03Geoffrey grant. Dixon, Cohl The Furey, author and wishes Adam to Gillard thank for Carlos many Castro insightful Perelman, discussions. Niels G. Gresnigt Xi’an Jiaotong-Liverpool University, Department of MathematicalSuzhou Sciences, HET, 111 Jiangsu, Ren’ai China Road, 215123. E-mail: Sedenions, the European Physical Society Conference on High10-17 Energy July, 2019 - EPS-HEP2019 - Ghent, Belgium fermion generations PoS(EPS-HEP2019)615 ], ∼ = 11 L ) chiral (2.2) (2.3) (2.1) ) O 2 ( ⊗ SU 137, 235, C ] to exactly ( , 2 , . , C 1 ) Niels G. Gresnigt 156 2 em , ) ie 1 , ( + ) , 124 2 6 0 U e e ie = ⊗ − = c ( + ) 2 0 6 1 2 i jk 3 e e ( ( , ≡ i ]. 1 2 e 3 9 SU ]. α ≡ = 8 , complex is responsible for the broken i † , 3 e R 7 T α 0 , e 6 , ) , is nonassociative, its left adjoint algebra = 1 and seven anti-commuting roots of 5 0 , , 0 ie O e ) 4 e i as a starting , was concurrently developed in [ 1 e + , ) ⊗ = 3 3 6 ie ( e , C ` + 2 1 − C , 3 ( is found starting from the complex e 1 1 2 ( where 1 2 em ]. Some ongoing research and speculations relating to ] it was shown that a Witt decomposition of ) ≡ 2 1 2 ≡ 10 ( , α k † 2 U e α . In [ . The quaternionic factor in ) ⊗ i jk O 6 c ε ( ) , ⊗ as a special direction, a split basis of can be defined ` 3 + ) ( , 4 , it is possible to extend the one generation results of [ H C 7 0 into minimal left ideals whose basis states transform as a single ) ] one full generations of leptons and quarks transforming correctly e e S ie 4 ]. ⊗ 2 L SU i j 2 ie ) C + δ 5 ]. Those works include a copy of the algebra, recovering the full SM gauge O ⊗ + − 1 e 5 R . In [ ⊗ − e = ( ( are spanned by the identity 1 O = j C 2 2 1 1 e ( T i O e ≡ ≡ , and three generations of fermions † 1 1 ) α 8 α satisfying ( i ` e 457. By selecting C generated via the left adjoint actions of the algebra on itself is associative, and isomorphic , is a completely with value +1 when L ) decomposes (itself not a ). That work builds on initial results from the 1970s relating the i jk 346 O ) The It is worth mentioning that a similar approach, using following the convention of [ Although the algebra of complex octonions The establishment of the (SM) as a description of the structure of particles There are only four normed division algebras; the reals One prominent open question in the division algebra program is how to extend the many ε O 6 , 1 2 ⊗ ( ⊗ 2 , and the octonions ` C ( to the complex Clifford algebra C minus one group from the algebra symmetry. this approach are then discussed. 2. Three generations from complex sedenions generation of leptons and quarks under the unbroken electrocolor H C octonions to the symmetries of (one generation of) quarks 1970s [ three generations in a very natural way [ and which is also consistent with [ under the electrocolor group and their interactions hasdespite been the one model’s of overwhelming the success,from great more its fundamental achievements underlying principles. in mathematical Finding theoreticalunderlies structure a physics. the is theoretical basis SM not However, for remains derived theof a mathematical embedding structure prominent the that challenge SM gauge inusing group the physics. into fundamental some and Instead larger generativemathematical group, of structures framework recently of the for there the particle common physics has four approach [ been normed an division algebras interest as in a simple results from a single generation toare exactly three reviewed generations. which In show this short thatDickson paper by algebra some of recent going results sedenions beyond the division algebras, and including the Cayley- Sedenions, 1. Introduction 267 as PoS(EPS-HEP2019)615 0 ∼ = = e i L ) , (2.5) (2.4) (2.8) (2.6) (2.7) (2.9) = i S η ⊗ C , allows for ( ) 6 , , ( ) ) ` Niels G. Gresnigt 2 3 C ie ie ∼ = + + L ) 13 12 is selected as a special e e O each individually form a 7 − − . ( ( e ⊗ } . i † i j 1 2 1 2 i j γ C as the special unit imaginary which defines a minimal left δ , δ ( † i = = † 1 15 ωω = γ = 1 3 e † α 1 { † 12, together with seven γ γ † o 2 o , α j j † α 6 2 are spanned by the identity 1 α , † η 3 ωω α , , 4 † S 3 b α † i † , and , i , 3 , , u α † } 1 α η ) ) i b α , ¯ 8 9 + n 2 β n satisfying the table given in d , † ie ie i 13 α ωω † i + , e 1 † 1 + + β 7 † α α , { 7 6 ωω + † , with each basis state in the ideals transforming , 2 5 † , e e 3 , † = 0 0 α } 2 ωω α em − − i ∈ † † 3 † † 2 ) ( ( 1 = = α a 1 α α 1 2 1 2 α , ). That is, selecting ( g

, g νωω +

† i ¯ j ωω j ) u e d U = = α . The sedenions α 2.4 η 14 1 3 { , , , satisfying ⊗ S i + + + i β β ) ie c where α a η ) † † ≡ + −   ) 3 naturally extends the above construction of a single generation ( a 1 u 15 L e = − = S ωω ωω ) , , ie † † 2 1 SU − 15 ) ) S o o ( α α j † j † + ie 10 11 r † 2 1 3 ⊗ ¯ a α η d ie ie e α , , + C r ( † † = i i ( a + + u 2 1 e 2 η α 5 4 γ n e e n − ( = − − 1 2 . Explicitly: and satisfy algebra ( ( ( 2 † ]. This larger algebra also generates a larger left adjoint algebra 2 1 2 1 β as a special (in the same way that O 10 uniquely divide into three intersecting sets ωω = = = ⊗ 15 , and three generations of fermions ) † i ) 3 1 2 e of the form 6 C 8 ( η α α α † ( i ` etc. are suggestively labeled complex coefficients denoting the isospin-up elementary ` η r C C ¯ d and ≡ , i 7 of the form ν u η . A decomposition of S ) One way of extending this model from a single generation to exactly three generations is to Selecting 8 ,..., ]. This is the algebra of sedenions ( 2 ` , 10 go beyond the division algebras, and[ consider the next Cayley-Dickson algebra after the octonions as a specific lepton or as indicated by their suggestively labeled complex coefficients. where fermions. The conjugate system analogously gives aof second isospin-down linearly elementary independent fermions. minimal These left representations ofunder the the minimal electrocolor left symmetry ideals are invariant the construction of a primitiveideal idempotent C singles out three intersecting octonionic subalgebras. This allows us to describe three generations to exactly three generations. Considering the basis vectors as elements of the left adjoint algebra imaginary unit in the octonion1 case), this time one can define a basis of seven nilpotents split basis for These together with their conjugates. The sets Sedenions, satisfying the anticommutator algebra of fermionic ladder operators and seven anti-commuting square roots ofAppecndix minus A one of [ conjugates PoS(EPS-HEP2019)615 , . . ) S ) (as 8 16 ⊗ ( (3.1) ( . The O ` ` C } C ⊗ C subalge- 15 e H ∼ = , O subalgebras L ⊗ 14 ⊗ ) nor by e O C S , 1 C 1 Niels G. Gresnigt O β ⊗ e ⊗ i , ⊗ C − 1 ]. O ] they are added by { C 2 = 12 ⊗ i H † 3 γ ]. An alternative way of ⊗ , ) 1 10 , the same as the left adjoint C ( 1 ) ( , Λ 8 2 ( h ` , 3 C β i . From each of these algebras one = ) − 6 L ( ) ` = O . Each of the three octonionic subalgebras C i symmetry. These three 1 † 1 ⊗ γ em , H , for an overall model based on ) ) 1 1 ( 1 H ⊗ ( 3 Λ generator constructed from the first ⊗ C U h ) ( , C 3 ⊗ 1 ( c γ ) i 3 SU ( = share a common complex quaternionic line of intersection. , instead of just two as in [ i ) † SU 1 3 6 β ( , ` ) 1 C 1 ( generators as above. That is, each generation has its own copy of is the first Λ 10. Associated with this algebra is the group (10), which forms ) h ` 2 em , ) C 3 α algebra of the anomoly free 10D heterotic string [ 1 γ † Fano planes of three octonionic subalgebras in the sedenions. 1 i 8 ( ∼ = grand unified theory. It is however not clear why a copy of the octonions α e ] on the other hand, all the degrees of freedom are included by considering U L = ) + ) ⊕ 11 i 1 S 8 associated with it. 10 † 1 e α ( and β ⊗ † 2 , c em ) . In [ α Figure 1: ) ) SO H , and three generations of fermions 1 ( S 3 1 ) 1 ( ( ( 8 ⊗ − Λ ( ⊗ h ` U C = SU C ( C ) 1 ( 1 and Λ c ) ]). The left adjoint algebra in this case is Including a copy of the quaternions into the model would enlarge the left adjoint The spinorial degrees of freedom and the weak force are not captured by The three Fano planes in Figure 3 1 ( algebra of which contains the the basis of the is missing in this case. Including a copy of the octonions, one gets algebra to in [ describing the full degrees of freedom is to therefore consider all 16 minimal left ideals of where bra associated with the first generationone of may fermions. be To eleminate able these to unphysical consider transformations linear combinations of fermion states, symmetry generators, or both. SU 2.1 Spinorial degrees of freedom This reflects the fact thatanother. the This three leads to generations unphysical in transformations theas between sedenion generations not model observed are in not nature independent such of one all eight minimal left ideals of constructs There are several ways toincluding include a these copy of additional the degrees complex of quaternions freedom. In [ 3. The intersectionality of three generations as a basis for CKM mixing above generates via its adjoint left actions a copy of Sedenions, of leptons and quarks withare unbroken not independent but share a common quaternionic subalgebra spanned by three intersecting octonion subalgebras together with themay multiplication be rules represented of by their base the elements three Fano planes in Figure PoS(EPS-HEP2019)615 . . ) S are O acts ⊗ ( S ) 3 C J O ( ]. 3 J 14 , 13 Niels G. Gresnigt subalgebra . For the sedenions 2 O G ]. There one likewise ⊗ ]. One expects a close 15 C 14 [ , models. has been (2016) . ) . Unlike the smaller algebra S 14 one can generalize the results ) O , 8 ( ⊗ S ( , indicating the higher Cayley- 3 13 3 ` J . Since describing a single gener- C ⊗ S ) C . This permutation group can be ) 6 C 3 3 ( S ∼ = ` and to − L C ) ]. The fundamental symmetries of n O O S ( ∼ = 18 ⊗ ⊗ L [ ⊗ × ) C and three generation models based on ) ) arXiv:1611.09182 ) H C O , 8 8 L ( suggests one obtains three copies. For higher O ( ) ( ⊗ ( subalgebras of ⊗ ` 3 S ∼ = C 4 ) S C C L ⊗ ( Aut O ) ( C O 2 ( J also contains three copies of the octonions, making this ) = ⊗ ) n H O A ( ( . The only difference between the octonion and sedenion 3 ⊗ 3 J S C gives the projective of the octonionic projective ( × 1 ) O , this larger algebra admits a triality associated with ( O Aut ⊗ subalgebra. This intersectionality of three generations may provide a basis , although the factor of . C 3) algebras Aut S O , vol. 290. Springer Science & Business Media, 2013. H ) = and its left adjoint algebra > S ⊗ ⊗ ( n O Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic C C , and three generations of fermions ⊗ ) Standard model physics from an algebra? 8 C ( ` C ], which is the quantum state upon which the exceptional associated with 16 [ ) Design of Physics The exceptional Jordan algebra The of the octonions is the exceptional A considerable amount of the SM structure for a single generation of fermions can be realised It was mentioned earlier that 6 2 ]. ( P ` [1] G. M. Dixon, [2] C. Furey, 17 One finds that the threeshare generations a common are not independentfor of including one CKM quark another mixing but into all the three model. generations algebra another natural candidate to describe three generations [ Spin(8). Triality is a non-linearesting outer to automorphism interpret of this Spin(8) triality of physically in three. the It would be inter- Cayley-Dickson ( References C ation requires one copy of therequire octonions, three it copies seems of reasonable the to octonions. expectfrom Extending a from a three single generation generation model to to exactly three, with each generation living in a Indeed each in Figure the same as those of Dickson algebras only add additional trialities,expected and beyond perhaps no new fundamental physics should be associated with three generations in the context of the exceptional Jordan algebra [ relationship between the approach based on [ O starting from 5. Discussion finds that the three generationstion are corresponds not to truly one of independent three of canonical one another, but rather each genera- one finds that Aut automorphism groups is a factor of the permutation group Sedenions, This at the same time mayinvestigated. provide a basis for the CKM mixing of quarks. This4. is currently Three being generations from the triality of constructed from the triality automorphism of Spin PoS(EPS-HEP2019)615 , 51 14 912 Nuclear , Niels G. Gresnigt Phys. Lett. B, 783, (2018) 81. , Advances in Applied , (2019) 58. 28 Advances in Applied , 29 (2002) 145. Nuclear Physics B , 39 Journal of Physics: International Journal of , Journal of Physics: Conference , in (2019) 446. , in Comment. Math. Univ. Carolin 79 , Journal of Mathematical Physics Journal of Physics: Conference Series , , in 5 Quantum symmetries: From clifford and hurwitz . Cambridge University Press, 2001. 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