PoS(CORFU2019)163 , 8 3 J , in- , the 8 8 2 ⊂ 2 J J 8 2 J of ) C https://pos.sissa.it/ ( 16 H ⊗ ) C ( 16 H that respects the lepton-quark 8 3 J 3 hermitian octonionic matrices, × of 4 F , consisting of 3 8 3 J . Its restriction to the special Jordan subalgebra 3 Z / ) to 8-vectors corresponds to the Yukawa coupling of the ) ew ) 8 ( 3 ( originated in the paper arXiv:1604.01247v2 by Michel Dubois-Violette. SU 8 3 J × c ) 3 of the . The Euclidean extension ( )) 2 SU ( ( ∗† U × ) 3 ( [email protected] U ( splitting is associated with a single generation of fundamental fermions, is precisely the symmetry group appears to be tailor madeThe maximal for rank the subgroup internal of space the authomorphism of group the three generations of quarks and leptons. subalgebra of hermitian matrices of the complexification of the associative envelope of volves 32 primitive idempotents givingrelating the left states and of right Higgs the boson first to generation quarks and fermions. leptons. TheThe triality present study of It further develops1805.06739v2; ongoing 1808.08110. work with him and with Svetla Drenska: arXiv:1806.09450; S Speaker. Extended version of a talk at the: 13-th International Workshop "Lie Theory and Its Applications in Physics" The exceptional euclidean † ∗ 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). (LT13), Varna, Bulgaria, June 2019; Workshop "GeometrySofia, and mathematical July physics" 2019; (to the Simplicity memory"Frontiers of in III Physics, Vasil Tsanov), from Workshop Electroweak at to Planck PerimeterGeometry scales", Corfu, and Institute, September the 15-19, Standard Canada, 2019; Model", Conference September "Noncommutative Jagiellonian University, 9-12, Krakow, 8-9 2019; November 2019. Humboldt Kolleg Institute for Nuclear Research and NuclearBulgarian Energy Academy of Sciences Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria E-mail: Ivan Todorov Jordan algebra approach to finite quantum geometry Corfu Summer Institute 2019 "School and(CORFU2019) Workshops on Elementary Particle Physics and31 Gravity" August - 25 September 2019 Corfù, Greece PoS(CORFU2019)163 (1.5) (1.1) (1.2) (1.3) (1.4) Ivan Todorov 1 3 2 3 ) − 2 . ) , 1 )) 3 . , , ; ( 2 5 ¯ 3 . 5 ¯ 1 , ( 3 5 Λ )) 1 R ↔ = , − 2 doublets, respectively (and d ⊕ ) ( 4 L 3 2 ) = ¯ 3 2 R ? d U , ) ) ( Λ , ¯ 3 2 = ( L 3 4 u d ⊕ × ( ( ¯ ν ⊕ ) L ) 1 ), look rather baroque for a funda- ¯ 1 1 8 ( ⊕ 3 − , = SU ( , Y 5 3 Λ 3 ) 5 ) ) yields a proton decay rate that con- ( U 2 ( = Λ − ν , S e and , L 2, 1) , 1.5 3 ↔ , 1 ( L = ¯ ( R 10 4 16 − ) u ) ⊕ , ) 2 = = 2 ( R 0 1 = ( , ) ( ) carry, besides the expected eight gluons and 1 + 2 , 1 ) 1 16 R Λ 2 e ( SU U , ¯ 8 1 − , 1 10 ⊕ ⊕ × ) J.C. Pati - A. Salam (1973). ]. It is noteworthy that the Pati-Salam GUT is the ( ( ν ↔ )( L 1 ) 3 4 Λ 2 , ⊕ L 2 ) combines the three colours with the lepton number. − , 0 16 1 ( 2 0 6 u ( 1 ( ) 5 = ) Spin , Z ⊕ PS ν = 3 ¯ − ⊕ SU 4 SU H. Georgi - S.L. Glashow (1974); ν , ↔ e G 3 1 × 1 = (of ) PDG114 × 32 ( ) − R 2 5 ( = ( 2 ) ⊂ ) : ( 5 e Z ⊕ 3 ( C ) R ) u 45 d ( SU 0 8 4 Λ ) × ( 10 , ( ) SU 1 ( ) SU 4 = , ( 1 , = 8 SU , ) ) and = 3 ? SU 2 32 ) 0 , Spin : 5 Λ ) = ( 4 ( SM ) 1 5 , ) = G 24 ( 1 4 SU = ( ( ( L will appear in Sect. 4.1. SU 8 (of ↔ PS Spin R g ν 24 × ( of ) 6 ) ( 2 H. Georgi (1975), H. Fritzsch - P. Minkowski (1975); ( ) su Spin ⊕ 10 If the fermions fit nicely in all GUTs, the gauge bosons start posing problems. The adjoint The first two GUTs, based on simple groups, gained popularity in the beginning, since they The gauge group of the Standard Model (SM), ( = ) 4 ( PS four electroweak gauge bosons, unwanted leptoquarks; for instance, Moreover, the presence of twelve gauge leptoquarks in ( mental symmetry. Unsatisfied, the founding fathers proposed(semi)simple Grand symmetry Unified groups: Theories (GUTs) with conversely for the antifermions): representations The quark-lepton symmetry playssu a pivotal role in our approach, too, and the Lie subalgebra Spin (The question marks on the sterilerectly (anti)neutrino - indicate from that the their neutrino existencelepton oscillations.) is symmetry: only The inferred Pati-Salam the indi- GUT group The is left the only and one right to fermion exploit octets the are quark formed by G tradicts current experimental bounds [ (the subscript standing for the value of the weak hypercharge naturally accommodated the fundamental fermions: and its (highly reducible) representation forantifermions), the first generation of 16 basic fermions (and as many 1. Motivation. Alternative approaches Jordan algebra approach to finite quantum geometry PoS(CORFU2019)163 ] ]) (1.6) (1.7) (2.1) Woi CS19 (called so does not ]). F S ). The only Ivan Todorov A A (i.e., a Jordan , the prime ex- ]. Concerning the ], the lack of exper- W Jordan algebras CCS ) i ]. X ◦ B12 i , X suited for a quantum theory - M )) one can of course work with , 2 ]. There are but a few papers on the = : ], one arrives at the finite algebra observable algebra 1.6 DJ . 2 i DM ) , X CC A ( ◦ equipped with a commutative product 0 special Jordan algebra GPR ] B , J 3 = [ n C X ], was preceded by the analysis (by Dirac, Jordan and ) (= ⊕ T85/16 = ] (for a pedagogical exposition see [ , H BA GUT. Intriguingly, a version of this symmetry is ... , however, are the real numbers, so 2 JvNW ⊕ + ) advocated authoritatively in [ L18 B08 H = , , C 10 1 AB ( ( X = BF B93 2 1 F ], "at the height of the string revolution" (to cite [ ⇒ matrices with entries in the coordinate ring Spin A = ]). 0 n CL H B , ], that did not get a Nobel Prize. The work on = × ◦ J 2 n n A formal reality condition X ]): + DKM ], who axiomatized the concept of ], was posted where an alternative approach, closer to Connes’ real spectral triple, ... PJ CS19 ,[ + BF19 1 supersymmetric GUTs 2 1 X ]) and by followers [ ]), see [ C , C17 , CCS DKM1 , ] (Proposition 3 of [ standing for the algebra of with a unit 1 satisfying the CCM The only one of the "Boys’ Club", [ An euclidean Jordan algebra is a real vector space Recently, a new paper, [ The noncommutative geometry approach, was started in 1988 (according to the dates of sub- The algebraic approach to quantum theory has, in fact, been initiated back in the 1930’s by Pas- ] , 1 2 Y n [ ◦ A CCM CC involving a different Jordan algebra, is been developed. subalgebra of an associative algebra equipped with the product ( [ hermitian elements of the quaternionappear as algebra the associative envelope ofour an interesting treatment observable on algebra. an We shall, appropriatepermitting, by finite in contrast, dimensional particular, base a Jordan spectral algebra decomposition of observables. von Neumann) of quantumapplications transformation of Jordan theory algebras reviewed to insightfully quantum theory, in [ [ 2. Euclidean Jordan algebras ( most popular nowadays imental evidence for any superpartner makes us share the misgivings expressed forcefully in [ and pursued vigorously by[ Alain Connes and collaborators (work that can be traced back from Such a (finite dimensional) Jordan algebra shouldof appear smooth as an functions "internal" counterpart of of the classical algebra fields. In the case of a X by A.A. Albert, 1946), of 1932-1934, that culminated in [ mission of [ only one which does not predictinteractions a with scalar gauge fields triggered that proton would decayappears permit (albeit in such it a a allows preferred decay). model reduction dependent Accordingly, of the Pati-Salam the group cual Jordan (1902-1980) Jordan algebra approach to finite quantum geometry ample of which is the algebraspace) of equipped complex with hermitian the matrices symmetrized (or product self-adjoint operators in a Hilbert its associative envelope, -geometry i.e., approach to with the the SM, based corresponding on a real algebra. spectral triple [ In the noncommutative also encountered in the noncommutative geometry approach to the SM, [ (see also the recent popular account [ PoS(CORFU2019)163 , . ) ) r f X Y Y X are ( ( + f (2.3) (2.2) (2.4) (2.5) (2.6) U U e − spectral < ) and regular e , it can be e X ( Ivan Todorov . Y has a U ◦ )) J − Y X is smaller than ) ◦ ∈ hermitian matrices = Y X X X . ( n + Y J L of hermitian elements ) is another idempotent. X × ∈ X ( f − Y ( n , ) if it is a Jordan subalgebra Y is defined unambiguously. U L + . Each , is an associative involutive X we conclude that X stands for the (associative) : J . ( e , = X X 2 ) : ) . L . Two idempotents , f X 0 r A ) saying that ) ) X XY λ ( Y Y J X = > r ( , . special a ◦ f ≤ ) L in X r i α ). If rank of ( ) e Y ... ( 1 )( i j U ) + where ◦ 1.6 δ − X ≤ Y 2 ( ) ( idempotent 2 k = X L λ if it cannot be decomposed into a sum of + ( a commute and j ) k YX is a set of orthogonal primitive idempotents e f also maps a pair X ... ≤ α ◦ ( ) = + 1 i 2 + L e λ A ( partial order 1 ( X and ) = , 2 3 XY − ◦ ∈ 1 r e ( X R t of multiplication by 2 1 primitive Y ) Y α ( = ∈ ) = ) , ( X ) is not the only one which preserves hermiticity. The i r ◦ k = Y ( or X X e λ , a ( 1 , X coincides with the degree of the characteristic polynomial , Y 1.6 , a L X + i Jordan frame ( J ◦ R e − i ⇔ ... U r X λ ∈ XYX , 0 is called a (non zero) t 1 ) ]) and its polarized form + ) r ) ), 2 = = 1 ∑ minimal i 0 X e X ( ) allows to define ) = ( Y )] = : k 1.6 = ̸= ) ) are different the spectral decomposition is unique. Such 2 ) X McC L a ( ) are necessary and sufficient to have spectral decomposition of any , 2.1 X e X X X t ( − ( ( ( is associative. In particular, any power of 2.5 r ) = L L 2.2 U J . The rank of F , 2 X ) J in ( e ( of 2 ; then multiplication by i X ): ( 0 L λ J X L 2 [ ) and ( = ∈ f 2.1 X ) = can be written as a sum of squares. Noting that ◦ ) operator of the form X e satisfying X X ( and thus treat it as an observable. J if − U J .) A prototype example of a Jordan algebra is the space of ) ∈ Y satisfying X e for which all ( r , if e L X To begin with, power associativity means that the subalgebra (including the unit) generated by Y < ,..., 1 e (nontrivial) orthogonal idempotents. A can be defined in terms of X A non-zero idempotent is called (defined for any In order to introduce spectralelement decomposition we need the algebraicorthogonal counterpart of a projector. An The formal reality condition ( in a Jordan frame is independent of its choice and is called the For a form a dense open set in Each such frame gives rise to a complete set of commuting observables. The number of elements an arbitrary element written in the form: . More generally, a Jordanof algebra is an called associative algebra with(star) Jordan algebra product the defined symmetrized product by ( ( The conditions ( element of 2.1 Spectral decomposition, characteristic polynomial decomposition (In general, non-associativity ofmaps the Jordan product is encoded in the noncommutativity of the quadratic (in and power associativity. Jordan hasassociativity. found Introducing a simple the necessary operator and sufficient condition for power Jordan algebra approach to finite quantum geometry with anticommutator product ( into a hermitian element.(whose role For is a emphasized in general [ (not necessarily special) Jordan algebra the map PoS(CORFU2019)163 , of r (2.8) (2.7) (2.9) a (2.12) (2.13) (2.10) (2.11) in a Jordan , is a linear Ivan Todorov ) X ( the idempotents tr X , . ) 1 , ) ) ], can be found in Chapters X K X ( ( 1 . L − ) one obtains the basic third 0 r )) a ̸= Y 1 2.2 . The ( , , − J , L r 0 0 r Peirce decomposition ) ) 1 X , defined as the trace of the Jordan , f or X , , ( . − 1 ) = ) = ], that follows [ 1 ]) = r L X − 1 X λ r . Y ...... ( i ( = − , + ( = 0 3 X ... λ tr ) r Z r 1 L JvNW e ... , , determined from the system of equations e , Y > = 2 λ r 1 X ◦ ) + + r )) λ e ([ 1 X − X X 1 ) + ... L , + ) = − ( ( r r − ]): 1 X L X r X r + inner product ( 4 ... ( a ( X ( λ 1 ) L 2 )] + FK e ) + X + ⇒ Z 2 (= det 1 ( i ( ) e 1 X ) = λ ... L 1 ( a Y , Y λ i L ) = ( + ◦ of degree )). The )] ∑ − 3 1 L X 1 Y X ) X e ( ( − ( 2.6 − 2 r 1 r L ) ) = a − tr X , 3 r X for a formally real Jordan algebra. The last coefficient, 1 ( ) X is non zero for distinct and its inverse is given by ( of ( ones (which cannot be further decomposed into nontrivial direct X ( λ = (some of which may coincide). Given a regular L . The argument is based on the X ) : ( r 3 ) r ( tr ) L ) − X λ L X 1 ( ) Y ( [[ : , 1 − Y ( a X X det simple ,..., ◦ ( 1 2 euclidean λ invertible = X : ( is 1 =) L − ) (and t X ( X ]. 2.5 taking value 1 on primitive idempotents: are FK then J r F 0 determinant of ̸= ) Vandermonde determinant X given by ( ( To begin with, by repeated manipulation of the Jordan identity ( A streamlined pedagogical version of the original argument, [ The finite dimensional simple euclidean Jordan algebras were classified at the dawn of the We are now ready to define a trace and an inner product in ) is the 3 X can be expressed as polynomials in det i 2.6 II-V of the book [ algebra which we are going to sketch. degree formula (see Proposition II.1.1 (iii) of [ that is equivalent to theory, in 1934, by Jordan et al. The theory of euclideanwritten Jordan as algebras a direct is sum simplifiedsums). of by the fact that any such2.2 algebra Simple can Jordan algebras. be Euclidean extensions If functional on product, is positive definite: This justifies the name for whose The roots of Jordan algebra approach to finite quantum geometry ( e PoS(CORFU2019)163 ) r is e ( ) e L ( (2.14) (2.18) (2.16) (2.17) (2.15) 1 of J J , the first ) have the ) r 2 ⊂ of the form J Ivan Todorov e can take any ) annihilated by ) the subspace e , we can intro- Y d r ( 1 (= ) J 1 e e J ( . > 0 . 0 = r j J , X 0 ̸= ) = is minimal then which involve matrices i e ). The first three algebras , e ( ) r ) = 1 2 the degree 27 1 H J . ( ) 2 ) r ; r − ,..., 1 1 2 ) 2; ) , 1 ) = = Jordan algebra H 1 + e 8 2 3 − r ( r ≥ J / = d of real numbers - no room for off ) + ( L ′ 1 r j ) e simple (of rank d , . , , and )( R r ( 2 ( i 0 ) are the dimensions 1, 2, 4, 8 of the i j 1 J ( L ) dim , ( . The eigensubspace C E ) d , ( (for a given simple algebra 1 ) C − ji r 2 ( ) j jk r E e < in a rank ) = JSpin , the subspace of elements δ + ) = ( H ( ] . For d . For an idempotent, r ) ) e H ̸= e J d L J r 2 = ( ). If the idempotent + . Similarly, the subspace = i ( 2 e 1 2 e of commuting operators: )[ e ](= d , − 2 ik 2 r r e J )( J } δ i j 2.3 ( J J ) ) = 1 e r ( 5 d (= dim 2 E 1 2 ( )[ ,..., U j 1 2 e J = 1 L ( e e ′ − , 1; ,..., , ( 2; < e = i ) associator ) 1 e e 0 ) = i j . It turns out that the two positive numbers, the rank ≥ dim ≥ ( ( O e 0 = , ( E ) (including r L r ( for. U 2 ) 3 ( , , k to three possibilities L r k ] > ) ) , e = ) is the J H ) ( R ⇒ e d H k 2.18 )[ ) ( i j ( ( L ) + the allowed values of ] e ′ only the dimensions 1, 2, 4 of the associative division rings are = r r ( e E L e J )) = Y ( 8 4 3 3 ( L , ◦ )[ H 2 C ] H J ′ { r ( U L e ≥ Z = r J = , 3 = ( r r )[ e 1 r 4 i H is the quadratic map ( ◦ r ( − J ) = ( e J ) e ( X U U ( e ( 0 U − dim 3 J , ) = L = Y ) e are one-dimensional while 2 ii 1 ◦ ( orthogonal subspaces, a counterpart of Weyl’s matrix units: 2 ii where 1 2 + ) E r ) J E Z J , determine all finite dimensional simple euclidean Jordan algebras, to be, hence, ( 1 2 d ◦ + ∈ be a nontrivial idempotent: ): ) takes the form: r ( X J X )) = , . (The single rank one Jordan algebra is the field ∈ 2.13 R d r X = ( ( : e J ) r e ] ( has to be nontrivial as well: Y H can be written as , ( Given a frame of primitive idempotents Let U ) ) Z e , ( e = ( 1 2 dim X positive integer value.(normed) For division rings. For permitted. The resulting simple Jordanalgebra algebras (proven split to into have four noAlbert infinite algebra series associative and envelope one by exceptional A. Albert also in 1934 and often called the same dimension, called the degree, and the degree diagonal elements and no need for a degree in this case.) For denoted corresponding to eigenvalue 1 coincides with duce a set of ( in the above list consist of familiar hermitianstress matrices (with once entries in more associative division that rings).We all items in ( They are eigenspaces of the set The subalgebras L J thus restricting the eigenvalues of [ Jordan algebra approach to finite quantum geometry one-dimensional: it us spanned by real mutiples of equation ( Y PoS(CORFU2019)163 ⊕ ) R - n ( 9 = ℓ of ]). 2 16 C b d B 2 ˜ , J H (2.24) (2.22) (2.23) (2.19) (2.20) (2.21) a , ] b Γ ,..., spin factor , ) Ivan Todorov a H Γ ( 2 . , ] ] = [ H 4 , [ 16 ab ) [ as the corresponding )). In fact they admit H Γ (see the review [ C ) R ( are special requires an n = 2 ( ⊗ ) only belong to the corre- 6 2.18 n 1 H ℓ ℓ , , − µ 2 ab C ) n 2 C x , Γ , . J ] i ] 0 JSpin R ] 2 = ( . m d = m [ ∑ 2 ) detX 8 2 µ 2 [ [ H + C )(= H − n ( C C n ⊕ ℓ . ( ) = X ] 16 1 - as a Jordan subalgebra of the ( ∼ = C ⊕ x 2 ξ ] ; ( i j [ ) ; H ] 2 δ m ) N H C 9 2 ⊕ , ( 16 = [ ℓ JSpin = [ ) 1 = m 2 C C ) 2 R + C 5 x ℓ ] X special ( ℓ j ( ⊂ ∼ = , ⊕ C C 16 ] Γ 0 ) ].) We will identify the optimal extension N , )( , , ) i ] C 6 matrices of the form ) is H O 16 = = ( 2 Γ C 8 [ 3 ( [ [ 1 ˆ 2 ( 2 x B12 J 2 = R ˆ 1 2 1 + x H tr × + , m H 2 16 , = d 2 ) 2 = = J ℓ ℓ x 9 R j = 4 ( ℓ C C ⊕ ℓ 8 ]: 2 Γ ∈ N C J (unlike ∈ C , ◦ 2 16 ] ξ , i . − 1 J ] 8 2 9 , 2 Γ + 2 ˆ J ℓ x 16 ) we observe that it involves three obvious repetitions: the spin [ d = [ ξ : ℓ C D-V T C + R . 8 2 C = ˜ 1 J = 2.18 = ∈ ξ 3 8 d ℓ 2 ℓ is isomorphic to the Jordan subalgebra of the real Clifford algeb ra = J C detX ) C ]). The fact that the algebras coincide with the first items in the three families of matrix algebras in the , , X n ] ] ( 4 2 8 [ , [ TD 2 R C , stands for the nonassociative division ring of 1 = = JSpin ), however, such a definition would exclude the most important obsevables: the 9 2 7 = O ℓ ℓ ℓ are antihermitian matrices; the hermitian observables ) d C C 8 C 3 n can be thought of as the set of J ℓ euclidean extensions 1 for C and + d 2 . In the case of real symmetric matrices (including the euclidean Jordan subalgebras of d J 8 of ) ℓ ]): ℓ R C n L Coming back to the list ( The algebra C ( , with the corresponding subalgebra of hermitian matrices. We shall exploit, in particular, ⊂ 2 16 ℓ ,..., d d 2 2 hermitian counterparts of the symmetry generators. Indeed the derivations H 1 sponding matrix Jordan algebra ifmetric) matrices. we More are generally, dealing we with shallthe complexify complex spin from hermitian factors the as (rather outset postulated the than in assoiciative real [ envelope of sym- above list. We could also write The (10-dimensional) spin factor dimensional) associative algebra factors here (as in (See the insightful discussion in (Sect. 3 of) [ argument (while it is obvious for the first three series of matrix algebras ( It seems natural to define the euclidean extensions of the spin factors To define an appropriate euclidean extension wee.g. use [ the classification of real Clifford algebras (see (cf. Remark 3.1 of [ with complex and quaternionic entries) are regarded as algebras over the reals. The Jordan algebra approach to finite quantum geometry J interesting spanned by the unit element and an orthonormal of gamma matrices with Jordan product subalgebras of hermitian (in the real case - symmetric) matrices: J C PoS(CORFU2019)163 = of d + state , (2.28) (2.29) (2.26) (2.25) (2.27) (2.30) R d r J whose Lie transforms . There is a Ivan Todorov ) ) d : r C ) J ( d ( r state of maximal J ( str Aut 0 . , ) is the semi-direct product of ) str . r ) 26 1 2 u r 2 . − , J ( ( 4 consisting of all positive ( ) ∗ 6 f r of trace one defines a e ) = 2 su C Aut density matrices ( J > ) = 1 ) = 8 3 , ⊂ (in fact, to the boundary) of the 8 ) = 3 J . usp 4 ¯ 4 J r ( < ¯ } , C ( J ¯ C 2 . 0 ( C , ∈ (= 0 ) = der ) are not invertible. )) 1 4 r 1 ∈ str , ρ 1 ρ J ( str , open cone = ( )) tr X > , . For the simple Jordan algebras tr 1 ) )) d , ∀ , the product of a central subgroup J r , the group that preserves the determinant ¯ ) are (mixed) ) 1 der , C ) = 0 + C , J ∗ , , J C r ( 1 r ) d ( ( ∈ > r ( 0 ( ]. Xu + ) )( sl : ρ Str d J X , X su : + Str 7 ( , ) ( spin B12 ) = ρ of X . An element [ tr ; they are extreme points in the convex set of nor- 2 uX ( r ) = ∗ ) = r 4 ◦ 1 ; J spin 2 r ( J C )(= C ρ J → 0 ( 1 = ( ( ∈ 0 = X tr + str )(= ) ρ > contains a convex, Aut der , 1 d C { expectation value ∋ ) , , ( J X , ) 1 ) = d r R r = pure states so , J an : it has a transitively acting symmetry group that defines the ( X < r + ( ∗ , ( J : so d ρ , the normalized unit matrix, called by Baez the C sl ( ) = u ∈ d r d 2 J so J X = ( ) = self-dual ( ) = 1 r 1 > r J J ( ) = der ( d X 2 0 J der ( < homogeneous itself, as primitive idempotents (for 0 . Here is a list of the corresponding simple Lie algebras str generated by complex conjugation. J is 2 C str Z of the Jordan algebra, are (normalized) positive linear functionals on the space of observables, so they C (i.e., all invertible elements that can be written as sums of squares, so that all their state observable correspondence J with a : r states Z / of hermitian matrices over an associative division ring an element ) r ( The corresponding automorphism group need not be connected. For instance, The structure acts by automorhisms on The cone The Remarkably, an euclidean Jordan algebra gives room not only to the observables of a quantum Each euclidean Jordan algebra 4 4 , SU 2 , assigning to any observable algebra coincides with the derivation algebra of each element of distinguished mixed state in of The stabilizer of the point 1 of the cone is the maximal compact subgroup of eigenvalues are positive). Jordan frames belong to the closure Any other state candisplaying be a obtained by multiplying it by a (suitably normalized) observable - thus elements of open cone, not to The primitive idempotents define malized states. All positive states (in the open cone ignorance structure group uniform dilation with a (semi)simple Lie group 1 belong to the closure of the dual cone In fact, the positive cone is 2.3 , states, structure group theory, it also contains its states:to these more are, roughly precise speaking, definitions. the positive observables. We proceed Jordan algebra approach to finite quantum geometry hermitian matrices into hermitian by the formula: PoS(CORFU2019)163 , ) ) ⊂ 3 B ( ) 3.1 d (3.1) (3.2) r J SU ( ] and in der D14 Ivan Todorov ], 1974, Sect. into complex , G , O j F D10 i j ], to apply it to the , , ) 4 δ 7 G ∗ 4; e 2 , x , 4 − ∗ 2 ), can be identified with ,... Z , y 1 1 ( 1 GG + 2 3.1 = = 2 + = i j e k j F and in the exceptional Jordan 2 F in ( , i , 1 2 Z 8 ◦ 2 7 , C i J e + ik ) E 7 δ 1 ) = , e i 2 y 1 ( E mod − 1 Z ( i j j ] that, conversely, the unimodularity of + should belong to the observable algebra δ i 3 = = 8 F x 3 i J = e DV ◦ Z whose euclidean extensions match each one k + j ) , j e 8 7 2 x E x ( J e i + ◦ 7 ) "could be related to leptons". Interpreting ( i F k belongs to the derivation subalgebra = x e ], Sect. 1), were originally introduced as pairs of , 8 E j i ) u 3.1 + 8 e , 3 2 Z 0 J , TD + )) x i ) i in ( 7 x E . If ( = C i    u + z mod F 3 ∗ 1 2 ( 1 , x x ξ 3 + + Z graded commutator (regarding the hermitian matrices as odd i + i 2 i ∗ 3 1 2 E e x x ξ E + ], and in [ i Z z )( 1 ∗ 2 3 ξ L = y ( ) becomes a commutator (thus annihilating the Jordan unit). In x x ξ , , 1 1 = of the octonions, that fixes the first x 3 = + ∑ x    i i 2 , e P95 2.30 i G 3 = e C ) = ) = ( y x ⊕ ( , and ( i , the non-associatyive 8-dimensional composition algebra (reviewed in [ ξ C F u ( O ◦ = − X ) x O = ( i ∗ F ) can be viewed as a u whose elements obey the following Jordan product rules: is the hermitian conjugate of 2.30 octonions 8 3 ∗ then J u ) ] in Dubna. The subject has been later pursued by G.M. Dixon, - see e.g. [ d r ] among others. M. Dubois-Violette pointed out [ The We shall argue that the exceptional Jordan algebra J ], in Chapters 19, 23 of [ ( Gov F16 spaces, [ (the "Cayley-Dickson construction"). But it was the decomposition of that led Feza Gürsey (and his student Murat Günaydin) back in 1973, [ quarks (then the newly proposed constituents ofof hadrons). the They automorphism figured group out that thethe subgoup quark colour group.confinement Gürsey - tried the unobservability to of relate free quarks. theVII) Only non-associativity "as hesitantly of another did speculation" he the that propose octonions the (in [ first toas a the manifestation quark of the quark-lepton[ symmetry was only taken seriously in 1987 by A. Govorkov the quark’s colour symmetry yields -unique through an product associated with invariant a volumenot form multiplicative represent - norm. an an The observable essentially octonions algebra. (just They like take the part, quaternions) however, in do 3.1 Why octonions? CS 3. Octonions, quark-lepton symmetry, where Jordan algebra approach to finite quantum geometry general, ( elements). of the SM. It has threefamily (special) of Jordan basic subalgebras fermions. algebra str PoS(CORFU2019)163 4 F (4.1) (3.5) (3.6) (3.7) (3.3) (3.4) of the 4 F into itself 8 3 ⊂ Ivan Todorov J has the form: ω 4 . ) and its spinorial . F ∗ x ) ) , of rank 4 and di- 3 7 9 ξ ( 4 ω . ( : , F ∗ i , 8 UZV 3 so X 2 spinors. The group 3 x = J i x ξ ) + x 1 ∗ 2 ; i 8 ∗ x , so it corresponds to the R + ξ ( ) , = 8 ) and on the flavour index V 1 2 . 1 − 4 , ) 3 ] so 3 ξ = , ) ∑ z 3 i E ⊕ 3 2 , [ ω ξ 3.5 C + , L ( 1 − ξ ( C 1 8 1 ( ξ 3 ) 7 ξ 3 e ∈ ⊕ + x H 3 (= ) , 2 VX ∗ 1 V j ) x 3 x 8 ) = √ = 1 ). , 1 2 1 → 2 x X 2 E 3 x ⊕ + ( ( , 26 J ) 8 2 ) ˜ and its three (inequivalent) 8-dimensional 1 1 − x − J tr 8 ∈ , Re ) 3 1 ( − = 2 ) 8 ( ξ that will be related - developing an idea of ξ 0; z ( ( 8 3 2 so r , = + J ξ , X so ) ξ 3 7 ∼ = 4 1 : ( ) = ξ + , ω 9 2 E ) 2 ∗ 3 X X , 16 ξ , x ( , V triality 1 − 3 1 , )) Z ξ i x 1 ) + x U det = ( 7 for the left and right chiral − 9 j acts on the first term in ( ) + ( − 2 invariant and transforms the traceless part of ω ) = , R z ∋ j ( ξ , r ) = ( (of order three): 3 8 i so X X V 1 1 Z ( ξ ) F E 4 is the compact exceptional Lie group ew ξ ( ( ∼ = ) F X 8 2 and observables in 8 + + 3 , orthogonal, say, to the projector 3 X ) yields the following decomposition of ( 4 and J det i = ( J and its symmetry 8 ( ) f 2 3 S ⊂ ) = L E J 9 incorporates 8 E Z i 3 ) 3.1 ( 8 X + J ξ ) = 8 SU 8 3 3 ( ⊂ + ( 2 ) = ( x J S 1 8 , Z 1 8 3 × 2 X 3 = J J ∑ ξ c ) E i Spin ( which respects this decomposition is the commutant ( ) acts on each quark’s colour index X Spin 3 = X ) ( factors (with their common centre acting trivially): ( 8 der 3 ) = U 1 ⊂ ) tr J x 3 ( 2 , SU ( − ξ G 3 ( = Aut SU X ∈ X , and can be expressed in terms of ω 4 ω ω F 16 colour symmetry while subgroup of c and is identified with (an extension of) the broken electroweak symmetry as suggested ) SM 3 ) ( G 3 stands for the 8-vector, , ] - to the three generations of basic fermions. 2 SU V , 8 The Jordan subalgebra The automorphism group of The exceptional algebra The lepton-quark splitting ( 1 (= D-V T r by its restriction to the first generation of fermions (Sect. 4). automorphism 4. The first generation algebra 4.1 The We see that the factor, It consists of two The subgroup of here leaves the unit element (under its lowest dimensional fundamental representation exact 3.2 Quark-lepton splitting of (indices being counted mod 3). The (order three) characteristic equation for Jordan algebra approach to finite quantum geometry [ mension 52 whose Lie algebra is spanned by the (maximal rank) subalgebra representation representations: PoS(CORFU2019)163 . j 9 e ℓ the C - (4.4) (4.3) (4.6) (4.7) (4.2) (4.8) (4.5) )) are , be the the her- algebraic , whose 7 4 8 2 2.19 F J Ivan Todorov ,..., ⊂ . explains 1 (cf.( ) of ) 1 , 9 0 8 2 − ( J ) 4.6 0 whose generators ( = 2 P . = . , ; 8 2 1 and their product: 0 ⇔ a ) 3 3 R ν 2 J . ˆ 7 P ⊗ e c Spin 6 ) σ , R . P ⊗ 3 16 16 ( − ) ν . , ⊗ C 3 γ − = ; its Dirac repre- is 1 ℓ e ( 0 3 9 16 ba σ ⊕ )) c C 5 ℓ = ⊗ σ 16 Γ 2 L = H − ( = C 8 ) = c is obtained by replacing the − = H 1 16 3 Γ + 8 U 2 = = ] γ = , 8 j J = − ˆ 2 = ∗ that generate P = × 7 1 ( P . 1 γ 16 c , 7 ) has another appearance in the , − 1 J a 2 16 , 8 c 7 3 γ ˆ ) , = 32 e Γ ) of J 2 1 1 ( 0 P 9 ) ,..., , [ ( γ σ σ . The generators of the the symmetry 9 , generating 1 i i ∗ ab U 8 9 ℓ = ) ⇒ ( j Γ ℓ = ⇒ 0 2.23 ) S C = C = P = ( Spin : 9 C ( P − 7 6 j 5 ,..., c a ℓ ˆ 9 , e ⊂ 8 = P 2 ˆ , γ isomorphic to ˜ ℓ 1 , P ≃ ( 0 j J C , , 7 ... ) ω P ... ) C P 1 3 16 ](= 0 0 ) 1 1 1 ∼ = , ˆ J b e , 9 P ⊗ , ⊗ ⊂ 2 ( = Γ 9 )( 9 1 ,..., , 2 ⊕ , ( = ( 1 matrices = 2 γ 1 a 1 16 σ a ℓ : 0 , i 8 1 J 16 , 10 8 ℓ Γ ˆ 9 7 C e J a = = = matrices Spin ( ˆ ]). The group = P C ⊕ . Its automorphism group P × 0 j j , 0 j 9 ˆ ℓ , , → P 8 2 ℓ jk 16 16 ⊗ j j Γ ] = [ C ] , J ) = , ⇒  δ 1 b C σ TD-V 7 9 × 0 ce 2 σ Γ ⊂ ( 16 = P ab 1 [ ⊗ − ) 8 = δ 16 2 − = 1 ˜ R ⊗ 1 ,..., J j 2 . We make correspond to the generators Γ , a σ = 1 ) ˆ 1 e Spin , generate an irreducible component of the ⊕ 9 γ j Γ 1 , = a are given by the antihermitian matrices ( ] → , = ) P ∩ 8 = a Γ 8 k = 9 j 2 1 8 2 ˆ j 1 e 16 ( ω J P γ J 4 0 [ b , ). The anticommutation relations of the imaginary octonion units ℓ − , j ˆ = F e Spin Γ ,..., 2 R + Γ C P , 0 1 : 1 k 3.1 0 + = e , = P ⊗ ( P b 0 j = [ ⊗ matrices ) = ˆ 9 : 0 e c 8 P ∗ 2 SM ℓ ⊗ e a = J c of the even subalgebra ˆ 1 e 0 ab 32 = G C ( = a a γ σ 0 : j Γ , that respects the quark-lepton splitting, coincides with - and = , ˆ ] back in 1949 (see also [ × Γ ]. ⊂ P a ω 0 + 1 = ˆ = 4 R ] P 8 Str 2 γ − F 1 k 32 0 J ( BdS ˆ P Γ e − , j Γ of the spin factor P [ ) 9 ( so of P. Ramond [ Another instance of a (closed connected) maximal rank subgroup of a compact Lie group that have been studied Here is an explicit realization of the above embedding/extension. Let 5 gauge group (1.1) of the SM octonionic units satisfying ( Then the ten real by mathematicians [ represented by similar real symmetric generate the Clifford algebra dreams The nine matrices The nine two-by-two hermitian traceless octonionic matrices intersection with sentation splits into two chiral (Majorana-)Weyl spinors: can be realized by the real skew-symmetric mitian elements algebra The resulting (reducible) Jordan algebraternal of quantum rank numbers) 32 of gives room fundamental"sterile" precisely fermions neutrino. to of the one In state generation fact, space - (of it including in- the is right acted handed upon by the simple structure group of As argued in Sect. 2.2 the optimal euclidean extension is special, its associative envelope being Jordan algebra approach to finite quantum geometry belong to the even part of the Clifford algebra real symmetric by complex hermitianassociative matrices envelope: in the maximal euclidean Jordan subalgebra of its PoS(CORFU2019)163 ) L 9 ( − , the (4.9) which B ) (4.14) (4.10) (4.12) (4.13) (4.11) ) 3 1 ( Spin ( u so ⊕ Ivan Todorov ⊕ L and the com- ) ) 2 6 ( , )) ( 3 ) 1 su we identify them , I 7 , . so ) R 3 ) 9 I 2 ( 1 A 2 mod ( ) of pairs of complex − j so = + ⊂ ) ( 3 and their expressions in , 2 j L 8 L ) U 8 γ . , 2.23 Γ 9 A ( − − , ( A 4 )) )) , ∗ 8 2 8 2 B . It is easily verified that the B 1 ˜ , / γ so ∈ ( ( J , R 1 which also involves the spinor ∑ γ ) 8 9 ∩ = = , 8 so , both viewed as Lie subalgebras 3 j ( 3 2 1 ( ) J ( Y PS i − )(= . )( 3 9 so , U preserves the quark lepton splitting g ( 8 1 ( 1 Γ su ) , ) = R 3 A ⊂ so U 9 − ⊕ : 9 I ( ] ) ( L ) ) = L 0 ⊂ ⊂ 1 − ) 2 L 45 ℓ , ] − 4 ( 2 3 Γ 9 C with I ( g − i 45 B ( i ( involve Spin su , R ℓ ∈ Γ B 4 , su ) i ∈ . ( 26 Y ⊕ 1 8 C , of 2 4 γ 3 = Γ ) 1 ( I F i 3 . The electric charge is a linear function of 9 26 = 4 , 4 into ( ⊗ su 07 Γ 2 11 i g of 13 5 ) ))( Γ 6 , ) + su ⊕ i γ 3 Γ L I ⊂ L i L 13 26 , 2 ) − ∼ = ) = − Γ ( − 2 i 8 } ) 07 , 8 , ( , B 6 1 3 B ) Γ Γ ( ( ( i − 07 R su 3 γ [ I 1 2 ... Γ ,... Γ so i R 1 2 ⊕ only belong to the subalgebra + 1 , − ) Γ = ⊕ 8 : 1 + = , 0 4 = ab ) 1 ( = L )) 3 Q Γ 8 6 Γ k span a 16-dimensional vector subspace of the 32-dimensional real I − 1 9 i ( L = ( 3 , su and the hypercharge 2 A ( i 8 is a convenient global label for fundamental fermions, it is I Γ − [ , so 4 L ) 07 Γ = A so . In accord with our statement in the beginning of this section only / ik 2 R (= Γ ) ( R 3 = i Γ = I : 9 PS ⊂ = 07 su { ( ( g 1 4 − , 4 Γ ∈ 9 g i are more complicated: 8 g , A ) does respect the required symmetry for both nontrivial IRs of is embedded diagonally 3 ω 1 : I γ − ) Spin Span − in 2 4.2 = A given by the Coxeter element of = Γ c : ( ( matrices. ) of 3 and γ I in 3 = su )(= ( 2 .) To reveal the physical meaning of the generators of L SM 8 16 6 R 16 3 acting on the vector representation of hermitian elements of the universal enveloping algebra ( I Γ su G − A 4 × 4 so g (the difference between the baryon and the lepton number) is given by the commutant B A of , 16 L 3 8 in . Here I ) c A − ) : 10 3 B ( 3 ( . It does not preserve this splitting in the full Albert algebra We recall that while All elementary fermions can be labeled by the eigenvalues of two commuting operators in the In order to identify a complete commuting set of observables we introduce a maximal abelian I 8 6 2 so su J Lie algebra in representation its subgroup with the Cartan elements ofintersection the the maximal rank semisimple Lie subalgebra of and The right and left isospins terms of carries the local gauge symmetry of weakly interacting bosons. Here of and of the centralizer of of the Pati-Salam algebra contained in the fundamental representation (The multilinear functions of vector space The corresponding observables Jordan algebra approach to finite quantum geometry hermitian subalgebra mutant PoS(CORFU2019)163 , of ) . The 1 (4.16) (4.15) (4.17) , three L , j .) The S 9 8 spanned contains 3 a ( J ∗ i 8 ) 0 3 a ℓ J 1 and , Ivan Todorov of C 9 , by contrast, R ( ) 4 ) S ; ℓ . 1 5 F . ) (nine ) , ) C Γ 1 1 j generate the split 9 i representations as 1 ( − , a − c 5 ℓ , + , Γ 3 1 ) . ∗ i Γ 4 C 1 , i 3 8 a , + , ( 1 Γ 1 6 . ( 8 of ( − , 1 Γ 2 1 Γ ( SU i ,..., 8 µ ( , , 1 ↔ ) in his attempt to make the Γ 2 1 3 ↔ = , 4 ↔ (which allow to treat spinors 7 ) , Γ singlet. These are encoded in 0 4 1 i ) , ) 2 = + R a c , L , = 1 n  e , ) 7 8 d , 1 ( ) 3 n a Γ ( − , 6 n , ( = i , , ( 0 ) , and the basic observables in terms Γ ) ) ℓ i = L ) 1 1 1 SU = 8 µ ). C , + , α − = − Γ ] (for ν  1 3 4 2 , B , γ , α Γ = , , − 2 = µ ( ¯ 3 Γ 1 , , 2 , i K18 2 1 γ for an 1 1 − , , , ↔ µν 1 − 1 4 0 = ( , δ = 1 3  2 2 = L can be identified with the subspaces of , a = − 12 , = 1 L | ↔ ) µ − + S 3 j 3 L ( ] = I = ¯ B u ∗ ν µ Γ 2 , j L i a subgroup of the automorphism group Γ . The split forms 1 , , and , ) − + ) ) µ 1 , in particular the generators R 7 = >< 1 8 C ) a B − S ( [ ( j , 3 1 Γ L Γ I , ¯ , ( 3 u 10 2 0 | jmod 9 1 2 ℓ ( ( 4 3 ), respectively. (As we shall recall in Sect. 5.1 below, they , . The set of 15 quadratic combinations of Spin ↔ 1 C = 2 Γ 8 , ℓ = i , 1 1 3 3.2 + ∑ 1 of C 4 | ↔ ] = obey the canonical anticommuttation relations, + for an antiquark, and j a , L ν j = , 2 µ ∗ and two signs ( )) ν ]. To begin with, we note that the complexification of ¯ a 3 matrices ) , Γ , = a 1 L 7 ) : ( 1 , µ 3 Γ , ) 1 2 9 − i a 32 F18 / 0 (anticommuting creation and annihilation operators - updating Sect. 5 of [ ( >< L , 1 ¯ ℓ B u of Eq. ( + × L = = ( C  0 j ν ) | , Γ µ 3 a 32 A95 1 ∼ = ( . x , , thus commute with chirality and preserve individually the spaces = : ( e 1 2 :  µ )( . 3 L ]) i a 5 ) F ( , L µ = ∗ − 32 (= 5 ν a [ 0 ( . The L ( B a ℓ R and − C for a quark triplet, ∼ = ) B 2 and their conjugate 3 )( x 1 µ ( , 2 We now proceed to translate the identification of basic observables of Sect.4.1 in terms of the We shall now present an explicit realization of both The 16-dimensional vector spaces Remark As the quark colour is not observable we only have to distinguish a ]), inspired by [ 9 F fermionic oscillators ( ℓ TD anticommute with chirality and interchange theinternal left space and part right of spinors. the They Dirac can operator. serve to define the labels: the value of five products by of [ a five dimensional isotropic subspace spanned by the anticommuting operators equivalent to the defining Clifford algebra relations for SM natural. The We have eight primitive idempotents corresponding to theprimitive) left chiral and (anti)quark right idempotents (anti)leptons and (colour eight singlets (non of trace three); for instance, 4.2 Observables. Odd chirality operators transform as expected under the as differential forms) have been used by K. Krasnov [ but we won’t neednumber them. Remarkably, all quarks and leptons are labeled by a single quantum Jordan algebra approach to finite quantum geometry (extended) observables belong by definition to the complexification of the even part, elements of the odd subspace, real form C PoS(CORFU2019)163 . 8 µ ). ) , π 4 0 ( 4.9 ( = su (4.25) (4.23) (4.19) (4.21) (4.22) (4.18) (4.24) (4.20) µ ): A ) 1 a products of ; ; Ivan Todorov 2 , ) 1 . 5 ∗ 8 a 0 . a 0 2 4 a . 2 8 . a 4 ] a , 0 a ) ; ; Ω . 8 , a 4 4 ] = ] 0 0 + R 8 ∗ 4 2 a , a ∗ 0 = ν a a 8; e a , a a 2 a ∗ 0 ∗ 0 , ( π ∗ 8 ∗ 2 1 Ω , , ∗ 0 , a a 4 Ω a ∗ 4 a = a 0 1 , a ν µ ∗ 1 a ][ , a 2 − − 0 − a ∗ 2 a π 4 0 , [ = ( 8 ∗ 8 , a . a 1 , ) = ∗ 1 a a , 0 = , ¯ U 0 ∗ 8 8 ∗ 4 1 Ω a ] = ∗ 0 ] = [ 0 a a a 4 . Any of the = by consecutive action of the µ a − − ) = a ][ = I , = factors: 8 2 ¯ Ω 2 , ¯ ) = Ω ∗ 8 ¯ µ Ω = 0 π a a ] = ] = ν ¯ + L ν ∗ µ 16 a π 1 a , ¯ I ¯ γ L R π − − ∗ 0 ν a , ( ∗ 2 µ a + , or I I ] ( a (an odd number for the right chiral , : = , 8 a , , π 1 8 ] , π ¯ = [ a , L ][ L R ∗ 8 ν [ Ω )) + + 8 µ π Ω − I I 1 0 µ ¯ 3 a ( 1 a π π Ω R I ¯ − a , 4 a , ) π ∗ 8 , 2 8 = 8 tr 1 π + L a B = [ = [ ∗ 1 = , a , a [ 2 e = L a 8 L 1 R , = ( 3 π 3 j 2 I ][ I µ Γ ∗ ∗ 8 1 ) generate the Pati-Salam Lie algebra A − = [ − ¯ µ 0 ∗ 0 ) = 2 j 2 a a π ∗ π ) = ( a L a π , 8 , B 0 , a ∈ 1 3 are again idempotents. No product of less than , )): + L 8 , 0 ν ∗ i = Γ π µ ∗ 0 tr e ( a 1 a = 1 ↔ a µ X , only involves ( π a ∗ 0 ∗ 0 , 8 ) 2.3 − , π 8 = a R 0 ] = 13 , − a a = ⇒ R ¯ j I µ ∗ Γ π ∗ = now assumes the form: 0 a ¯ 4 Ω = [ ν 3 4 a a = ) = I ( 2 π 0 ¯ µ Ω + , 8 π 32 γ 2 4 Xa 2 a ∗ 0 π 2 R a L + L ( ( − − , π µ , ¯ a I I π = 8 1 L Ω a − µ , 1 − su I a ∗ 8 π ¯ 1 ∗ 8 = a ↔ ) = π = ∗ 8 0 , a B + : a 0 µ ∗ [ ) + R 0 a tr in ∗ 0 π ∗ 0 ¯ Ω µ a π e R , a − a c ) a ( I ∗ 8 − ν 1 ) 0 ⇒ , and as many , = , = a i 0 3 Xa a ] : = 0 0 ( a j , a = µ ∗ µ ) = j a a = ( ∗ 0 ∗ 0 = R ¯ − a but also using the "colourless" operator a ) Ω π 8 µ − R su a I a a , 8 ¯ L j + L e + ( π Γ ∗ π I − − correspond to the left and right isospins: Ω = ( e is a basis of idempotents of the abelian multilinear algebra − a 4 ( U [ j + 1 ) = = 8 ) of ∗ 0 π } X = j L , a µ 2 8 ) a R , the antiparticle of j ¯ j + ∑ ∗ = 0 ν I π ) µ π π ( ) = a a , 1 3 1 L i a = µ 4.11 − L 4 ¯ ¯ , + j π ν a ( π ∗ 8 e ( π 0 ∑ µ ∗ = L , ( a + , L π a I µ 3 2 , ∗ 2 ¯ L ( Ω a π − 8 (or only on Ω = = , U µ : − = : a 1 B ¯ a Ω B π + Y Ω I , ) = = 0 ) = R of different indices is primitive. We shall take as "vacuum" vector the product of all and their complements: π µ R ν and ). The indices ¯ { ( ν π 4 ν µ ( , ¯ Ω π π = 2 , commute among themselves, products of , µ π 1 µ Each primitive idempotent can be obtained from either All fermion states and the commuting observables are expressed in terms of five basic idem- π π = j states). In particular, here More economically, we obtain all eight lepton states by acting with the above operators for The left chiral states involve products of an even number of As which carry the quantum numbers of the right chiral ("sterile") neutrino: potents five factors is primitive; for instance on both The hypercharge and the chirality also involve ( independent products Jordan algebra approach to finite quantum geometry The centralizer five involutive chirality changing bilinear maps (cf. ( PoS(CORFU2019)163 − B (5.1) , ] that 3 (4.28) (4.26) (4.27) (4.29) I 2 displays DT ) 3 ; ( Ivan Todorov ] where they ) as the gamma 3 2 SO 1 , Sch . 1 3 ⊕ × ) , ) . 9 1 ; ) 0 6 ) ( L − ( R , . (The corresponding ¯ d u : 0 γ . Their splitting into the ; ↔ SO , 9 ) O 1 ℓ ¯ × ¯ D D 4 ) = ( C , , ) = ( g ) R with five simple roots corre- = ( − L D ¯ of ν . 3 2 e ) ( ) ( Q a q 1 1 − q , GUT (see, e.g. [ Γ , , U 0 U 9 ) = ( 6 ≡ , 1 3 ( 4 , ) E 0 = , ) L − e so 2 R , u ¯ L , d ) 0 )( j 1 ( − x a . ( , = B of the ) = ( j ↔ ∗ j ) = ( Re + R a = L , ( D 27 e 2 j appears as a subrepresentation of the defining ∗ ν ( Y ( U a q 8 2 = q , ⇒ 4 , j J U 0 j 2 U ↔ ∗ 14 , , which splits into three irreducible components of a ; x a 1 ; 16 + 9 +1 ) 8 ) 3 ∑ 2 ) = 2 3 J R − j L = = ( d ¯ = u is a real valued linear form on W of 6 = ∗ , : ] = 26 µ x 0 4 ; j ∗ e q ) F - the lowest weight vector of ) = ( a + µ 3 W ) = ( , U L , , the state that minimizes our choice of observables, , x ) x is spanned by the generators ¯ − R ] L ν 1 + − R e µ ( one can find the mixtures that define the (neutral) Z-boson and µ ) ( ( e q − also appear in the 9 a ∑ q ( W 1 ( , ) = ⊕ U ( B ¯ U x µ ∗ D [ , ) : = ( , , ) a , 1 SO is the lowest weight vector of [ ) j x R , tr D L a ¯ u 6 ) ) = of d 0 R 3 2 9 a ν , = ( ( − 8 ) = ( 9 ) = ( Γ + L , R ] = e ∗ 0 j ν ( a ( a q q , with the singlet U L U 3 = ( − 3 invariant (trivalent) quark states are obtained by acting with the operator . Each of the rank 16 extensions of B [ of the automorphism group c ) 3 26 : ( ) 9 The trace of an octonion As pointed out in the beginning of this subsection we shall view instead the We note that while Remark ( SU , is the right electron singlet γ , Spin sponding to the commutators highest weight vectors are given by the respective antiparticle states.) module the photon. The leptoquarks may enter the . 5. Triality and Yukawa coupling 5.1 Associative trilinear form. The principle of triality Combining the The vector representation defining representations of the twothe factors quantum of numbers its of maximal a pair rank of subgroup conjugate leptoquark and a triplet of weak interaction bosons: are treated as superheavy). matrices which anticommute with chirality and naturally appear as a Higgs connection [ on the corresponding lepton states: The Jordan algebra approach to finite quantum geometry L PoS(CORFU2019)163 : t 3 × of )). ) ) S 8 (5.2) (5.3) (5.8) (5.4) (5.7) (5.5) (5.6) form 8 3.4 ( ( ⊂ 3 SO in ( Z Spin R × there exists 8 ] (Chapter 23), ) Ivan Todorov ) L 8 8 ( ( . and 1 SO . L SO ) 8 = ∈ ∈ normed triality ) ∗ ) only takes place for yzx g ( − uu g tr 5.8 ) satisfies the invariance , ,, . ] (Chapter 24), [ ) + 5.2 vzv of quaternions. By contrast, ) = g ( R -invariant, the trilinear form P95 , ) ) = t g ∈ 7 8 4 zxy (denoted by ( ( z ( ( ( ∗ v belongs to the subgroup  . tr . B ∗ S SO xx SO . u , 1 ) O B z . yu , , = ) = ∈ u y ) = yx 3 , y x = R R , x ( ν , xyz y ( x ( u u t N ) is provided by left-, right- and bi-multiplication = , L tr ⇒ R , y ) , : ) that permutes the nodes of the Dynkin diagram ∗ y R 5.5 z u ) = x 3 − ) then the form ( ux z B g R 1 N S , )) = 15 ) = − ], Theorem 1.14.2). For any )( 5.4 → y g x xy x yz Y ( , u ( u L + y x N R L g ( − ) = , x g ) tr y ( satisfying ( , → z obeys ( - see [ L x , ) triality see, in order of appearance, [ u N x ) = ( y 1 + ) L , L − ) = − 8 g ≤ , such that x xy : z ( g ( g ) belong to the group of outer automorphisms of the Lie algebra ( ( ) , t 2 , t ν 8 | g + ( ) − − ) then the only other pair which obeys the principle of triality is xy g z Spin g g , , (( , , ) = y SO g 5.4 z + + , ] (Sects. 1.14-1.16). tr ] Theorem 1.16.2). The set of triples ∗ x g g Y u ( , ( Y t B g | subgoups of the full isometry group , ) = y z ) u , 3 y and the corresponding scalar product are ( R (see [ , , satisfies ( ) x Principle of triality x satisfying: x ( ( SO ) u t ( , the 8-vector and the two chiral spinors, say ) L − ) 2 N ] (Sect. 8.3), [ ( g 8 t , ( . If the triple xyz The associativity law expressed in terms of left (or right) multiplication reads ) of elements of CS ( + . − g ) ( g Re Spin , − . In particular, the map that permutes cyclicly g + ) g satisfying the principle of triality form a group isomorphic to the double cover ) = − 8 . which coincides with the symmetric group z , ( ) ) , For systematic expositions of the Remark. Theorem 5.1 Proposition 5.2 Corollary An example of a triple ) + 8 8 y 7 so 8 ( ( g , ( ] (Sect. 2.4), [ x − ( B for a pair ( so It is valid for complexreal numbers multiples and of for quaternions; powers for ofgenerate octonions different a Eq. single element. ( Left and right multiplications by unit quaternions by a unit octonion: The permutations among We proceed to formulating the more subtle trilinear invariance law. If the pair ( condition [ The normalization factor 2 is chosen to have: While the norm It allows to define an associative and symmetric under cyclic permutations Jordan algebra approach to finite quantum geometry corresponds to the invariant product of thetations three of inequivalent 8-dimensional fundamental represen- SO SO t PoS(CORFU2019)163 ) 3.3 anti- ( (5.9) (5.11) (5.12) (5.10) (5.13) 9 ) admits ℓ spinors, X 8 3 for some C ( J ) generate − u S should be a E of Ivan Todorov B det ) a ⊗ E or and Γ 8 ( 3 u J + R . S leaves the diagonal dim ∗ . as the internal sym- ) ) 4 ) 3 y F 3 ( , , x ] x 8 P − 8 , ; 1 ∗ [ of g 2 ) ) ( x 2 R x ) = 3 gives rise to a Jordan frame , corresponding to the Jordan ( E 8 1 F 0 i 8 2 ( ∼ = P x P J + . ( has the form 6 , t ) which appears in 1 3 ) + − α , 8 3 2 E ℓ ) = x Spin 2 J x ∗ 5.2 ∗ , C α )( y + 3 1 P ∗ g ∈ x x ( should transform as , ( = 2 α ∑ 2 3 i P F P x )) x invariant and transforms the off diagonal , , , = α ) 1 8 i x 3 ∗ )) x 4.5 ) + J 0 ) E ∗ 1 ( 1 and t x 7 = x ∑ ( ∗ 2 2 − α 1 2 gx x x P 1 ( ) of ( , ( 1 16 0 P 8 3 F ) = ) = ( P J 3.2 x 3 3 ∗ ) ( ( i x F → ( ) P E xx 1 2 ) x , respects all binary relations. We have, for instance (in F 3 , − 3 = x → ◦ ξ , ( 1 ∗ ( x 2 3 ) . The above consideration suggests that )) vector, then ]). , )) = F X vector, one for each generation x 2 2 ) ∋ E 1 ( x ) x 8 ( CS P ( ) = ( ( 8 O = i ) + ) ( F P ( 2 i x ( 8 ◦ x , 2 ( 2 i ( J ) SO P F F Spin 2 ] Theorem 2.7.1). The subgroup 1 ] any finite (unital) module over F x ◦ Y ( ) = )) J68 F as a ∗ 1 ) + ( 1 x 1 x xx ( x (see [ ( ( P 1 P ( )), F 1 F (see Sect. 8.4 of [ 3.2 as the finite geometry image of the Higgs boson, the associated Yukawa coupling ) ) fails, however, to preserve the trilinear form ( 8 1 in the generic element ( x i E 5.11 SO As demonstrated in Sect. 4 the natural euclidean extension of It would be attractive to interpret the invariant trilinear form Thus if we regard Proposition 5.3 commute with chirality and thus(the do internal not part belong of) to theno the Dirac associative operator. observable envelope algebra Unfortunately, and butmay according henceble serve to no substitute to Albert’s such it theorem, define an would euclidean beconstruction extension instructive of to either. Sect. see 4.1. To what The search exactly first for would step, a the go map possi- wrong (cf. if ( we try to imitate the finite dimensional real vector space multiple of three so thateach there generation. would be room for an octonion counterpart of afitting vector nicely current one for generation of fermions. We also observed that the generators The map ( for each of the arguments of the notation of ( the entire metry counterpart of the Yukawa couplingthe between a 8-vector vector and two (conjugate) spinors. Viewing According to Jacobson [ and in triple products like 5.2 Speculations about Yukawa couplings would be responsible for the appearance ofthree (the possible first choices generation) fermion for masses. the There are, in fact, products of upto 7 left multiplications of unit octonions (and similarly of upto 7 Jordan algebra approach to finite quantum geometry projectors elements as follows: subalgebra respectively. PoS(CORFU2019)163 4 is F J (5.14) ) = 8 ]. I thank 3 J ( Ivan Todorov D-V T Aut , of TD , ω 4 F TD-V with the automorphism ] should be reflected in , . ) ω 4 2 and the Yukawa couplings DV F module (following our dis- ) = E ] (one of the originators of 3 ) ) i 8 3 ∗ 2 X ( J PDG14 ( 8 2 X DV , 3 J ∗ 1 F X ◦ ∗ 3 ]). Here we survey the progress in X )) 2 + PDG12 X matrices: ( 3 2 CS19 X 8 F 2 , . ◦ X × 8 3 1 ) 8 BF J 1 , the corresponding (symmetric, traceless, mu- X , 1 X ( − 1 17 + ( F 1 that describes the three generations of fundamental C17 ( E 8 3 ) = ) J 4 2 have both square 1 and can be interchanged by an inner e X 4 of a single generation is just the gauge group of the SM. , 1 P 2 8 3 X 2 e more general J 3 , P 1 t 1 X e ⊂ P ( , + 8 2 t ) algebra 3 , J ∗ 3 P 0 2 X ∗ P 1 1 X P ) = Albert ∗ 2 3 X e , (( 2 , hence, a (commutative) euclidean Jordan algebra. As the deep ideas of e 1 4 , 1 e we find instead of ( t i X ], 1934) and the quark-lepton symmetry (Sect. 3.1) we argue that it is a multiple of of the subalgebra ], on one side, and connect with current phenomenological understanding of the SM, ) 9 ) = ( i x JvNW ( DV18 P Spin , The next big problem we should face is to fix an appropriate This paper can be viewed as a progress report on an ongoing project pursued in [ The idea of a finite quantum geometry appeared in the late 1980’s in an attempt to make the In order to apply properly the full exceptional Jordan algebra to particle dynamics we need Recalling (Sect. 2.2) the classification of finite dimensional simple euclidean observable alge- CDD that respects the quark-lepton splitting. Remarkably,group the intersection of cussion in Sect.module. 5.2) and to write down the Lagrangian in terms of fields taking values in this an alternative attempt, put forward by Michel Dubois-Violette, [ my coauthors in these papers and Cohlboth Furey her for and discussions, Svetla Michail Drenska Stoilov forGeorge for a Zoupanos critical their reading and help of Andrzej the in Sitarz manuscript, preparingKraków, and for respectively. these invitation notes. and hospitality I in am Varna, grateful at to the Vladimir Perimeter Dobrev, Institute, Latham in Boyle, Corfu and in the exceptional Jordan (or fermions. We postulate that the symmetry group of the SM is the subgroup Standard Model natural, avoiding at theaccompanying same time GUTs the and excessive number higher ofduring dimensional new the unobserved theories. states past thirty It yearstriple was in in developed the work and framework of of attained Alain noncommutative maturity Connes geometry and and others the real ([ spectral tually orthogonal) matrices a corresponding mixing of theirus finite fit geometry together counterpart. the euclidean Its extensions concrete of realization the should three help subalgebras 6. Outlook Moreover, while on the other. In particular,Maki-Nakagawa-Sakata the (PMNS) lepton-mixing Cabibbo-Kobayashi-Maskawa (CKM) matrices [ quark- and the Pontecorvo- setting further study of differential[ calculus and connection forms on Jordan modules, as pursued in Jordan algebra approach to finite quantum geometry automorphism. expressed in terms of the invariant trilinear form in the noncommutattive geometry approach,algebra too) of based observables on aPascual Jordan finite seem dimensional to counterpart have foundphysicists, of a we the more used receptive the audience opportunity among to mathematicians emphasize than (in Sect. among 2) how well suited a Jordan algebra bras (of [ for describing both the observables and the states of a quantum system. PoS(CORFU2019)163 B , (2018) 071, Ivan Todorov (2005) 213; 06 Oxford Univ. :3 (2010) 42 47 24 . JHEP ibid. Nucl. Phys. Proc. Suppl. . (2012) 819-855; The interface of and 42 (2010) 553-600; arXiv:1004.0464 58 Bull. Amer. Math. Soc. Found. 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