A topological model of composite preons from the minimal ideals of two Clifford algebras
aa r X i v : . [ phy s i c s . g e n - ph ] M a y A topological model of composite preons from theminimal ideals of two Clifford algebras
Niels G. GresnigtMay 8, 2020
Abstract
We demonstrate a direct correspondence between the basis states of theminimal ideals of the complex Clifford algebras C ℓ (6) and C ℓ (4), shown earlierto transform as a single generation of leptons and quarks under the StandardModel’s unbroken SU (3) c × U (1) em and SU (2) L gauge symmetries respectively,and a simple topologically-based toy model in which leptons, quarks, and gaugebosons are represented as elements of the braid group B .It was previously shown that mapping the basis states of the minimal leftideals of C ℓ (6) to specific braids replicates precisely the simple topologicalstructure describing electrocolor symmetries in an existing topological preonmodel. This paper extends these results to incorporate the chiral weak symme-try by including a C ℓ (4) algebra, and identifying the basis states of the minimalright ideals with simple braids. The braids corresponding to the charged vec-tor bosons are determined, and it is demonstrated that weak interactions canbe described via the composition of braids. Grand unified theories (GUT) and preon models represent two approaches tomotivating the Standard Model’s (SM) symmetry group, SU (3) c × SU (2) L × U (1) Y , and particle content from more fundamental principles. The formerapproach merges the gauge groups of the SM into a single semi-simple Liegroup. Such GUTs, including the famous SU (5) and Spin(10) theories [1],invariably predict additional gauge bosons, interactions, and proton decay,none of which have thus far been observed. Even if the unification of the SMgauge groups is successful, the choice of Lie group still requires justification, iven the infinite possibilities. Preon models were developed with the hope ofderiving the properties of the SM particles from a smaller set of constituentparticles. The most famous of these is the Harari-Shupe preon model, basedon just two fundamental particles [2, 3].Using the Harari-Shupe model as inspiration, it was shown in [4] that onegeneration of SM fermions can be represented in terms of simple braids com-posed of three twisted ribbons and two crossings, connected together at the topand bottom to a parallel disk. In this model,the twist structure of the ribbonsaccounts for the electrocolor symmetries, with charges of ± e/ ± π on the ribbons, and the permutations of the twisted rib-bons representing color. The braid structure of the ribbons on the other handencodes the weak symmetry and chirality. The weak interaction is then repre-sented topologically via braid composition. The twist and braid structure arenot individually topologically conserved, but rather are interchangeable [5, 6].There has recently also been some efforts to generate the SM symmetriesand particle content from Clifford algebras, particularly those that arise whentensor products of division algebras act on themselves from the left or fromthe right . The basis states of the minimal left ideals of the Clifford algebra C ℓ (6), generated via the left adjoint actions of C ⊗ O , transform precisely as asingle generation of leptons and quarks under the electrocolor group SU (3) C ⊗ U (1) EM [10]. Furthermore, the chiral weak symmetry can be described via theminimal right ideals of C ℓ (4) [16]. In these models, the finite particle contentin the model is derived from the basis states of the finite-dimensional minimalleft and right ideals of Clifford algebras. These ideals are constructed from aWitt decomposition of the algebra, and the gauge symmetries then correspondto the unitary symmetries that preserve this Witt decomposition.In [17] (see also [18]) it was shown that by identifying the basis states ofthe minimal left ideals of C ℓ (6) with particular braids in the circular braidgroup B c , and subsequently exchanging the resulting braiding for twisting,that the twist structure responsible for the electrocolor symmetry in the braidmodel [4] is replicated. This paper extends this result to include the chiralweak symmetry, by identifying the basis states of the minimal right ideals of C ℓ (4) with suitable braids in B . The minimal ideal basis states then generatethe appropriate braid structure, which differs from the original model [4], withit no longer being the case that all particles are represented by braids with twocrossings. The weak force is represented topologically via braid composition. There are many related approaches that look at the division algebra as a basis for SM physics,see for example [7, 8, 9, 10, 11, 12, 13, 14, 15]. In the circular Artin braid group, the strands composing the braid are attached at top andbottom to a disk. Standard Model particle states from theminimal ideals of Clifford algebras C ℓ (6) In [10] it was shown that a Witt decomposition of the complex Clifford algebra C ℓ (6) decomposes the algebra into two minimal left ideals whose basis statestransform as a single generation of leptons and quarks under the unbrokenelectrocolor SU (3) c × U (1) em . A Witt basis of C ℓ (6) can be defined as α ≡
12 ( − e + ie ) , α ≡
12 ( − e + ie ) , α ≡
12 ( − e + ie ) , (1) α † ≡
12 ( e + ie ) , α † ≡
12 ( e + ie ) , α † ≡
12 ( e + ie ) , (2)satisfying the anticommutator algebra of fermionic ladder operators n α † i , α † j o = { α i , α j } = 0 , n α † i , α j o = δ ij . (3)From these nilpotents one can then construct the minimal left ideal S u ≡ C ℓ (6) ωω † , where ωω † = α α α α † α † α † is a primitive idempotent. Explicitly: S u ≡ νωω † +¯ d r α † ωω † + ¯ d g α † ωω † + ¯ d b α † ωω † u r α † α † ωω † + u g α † α † ωω † + u b α † α † ωω † + e + α † α † α † ωω † , (4)where ν , ¯ d r etc. are suggestively labeled complex coefficients denoting theisospin-up elementary fermions. The conjugate system gives a second, lin-early independent, minimal left ideal of isospin-down fermions S d ≡ C ℓ (6) ω † ω spanned by the states { , α , α , α , α α , α α , α α , α α α } ω † ω. (5)The Witt decomposition is preserved by the electrocolor symmetry SU (3) c × U (1) em , with each basis state transforming as a specific lepton or quark as indi-cated by their suggestively labeled complex coefficients. The reader is directedto [10] for the explicit representation of the SU (3) c × U (1) em generators. following the convention of [10]. .2 Including weak symmetries via C ℓ (4) Including the SU (2) L weak symmetry requires an additional Clifford algebra,whose Witt decomposition is preserved by SU (2). This algebra is C ℓ (4) with anilpotent basis { β , β , β † , β † } of ladder operators. Following the constructionof minimal left ideals of C ℓ (6), two four complex-dimensional minimal rightideals are given by ΩΩ † C ℓ (4) and Ω † Ω C ℓ (4), where Ω = β β and Ω † = β † β † .Explicitly the ideals are spanned by the states { Ω † Ω , Ω † Ω β † , Ω † Ω β † , Ω † Ω β † β † } , { ΩΩ † , ΩΩ † β , ΩΩ † β , ΩΩ † β β } . (6) C ℓ (6) electrocolor and C ℓ (4) weak states To fully describe weak transformations requires that we combine the C ℓ (6)minimal left ideal states with the C ℓ (4) minimal right ideal states. Each statethen simultaneously belongs to a C ℓ (6) minimal left ideal and a C ℓ (4) minimalright ideal.The neutrino ν is represented by the C ℓ (6) minimal left ideal basis state ωω † . Via the C ℓ (4) right ideals, we can now include chirality, so that ν R = ωω † Ω † Ω , ν L = ωω † Ω † Ω β † . (7)Similarly, the neutrino’s weak doublet partner, the electron e − in its left- andright-handed form can now be written as e − L = α α α ω † ω Ω † Ω β † , e − R = α α α ω † ω Ω † Ω β † β † . (8)Notice that the neutrino and electron live in different C ℓ (6) minimal left ideals,but in the same C ℓ (4) minimal right ideal. One can write down the quark statesin a similar manner (see [19] for details). In summary, the eight weak-doubletsare identified as (cid:18) ν L e − L (cid:19) = ωω † Ω † Ω β † α α α ω † ω Ω † Ω β † ! , u (3) L d (3) L ! = α † j α † i ωω † Ω † Ω β † ǫ ijk α k ω † ω Ω † Ω β † ! , (9) (cid:18) e + R ¯ ν R (cid:19) = (cid:18) α † α † α † ωω † ΩΩ † β ω † ω ΩΩ † β (cid:19) , ¯ d (3) R ¯ u (3) R ! = (cid:18) α † i ωω † ΩΩ † β ǫ ijk α j α k ω † ω ΩΩ † β (cid:19) (10)All of the other physical states are weak singlets. The appropriate SU (2)generators that transform the states correctly via the weak symmetry SU (2) L are now given by T ≡ − β β † ω − β β † ω † ,T ≡ iβ β † ω − iβ β † ω † , (11) T ≡ β β † β β † ωω † − β β † β β † ω † ω. Mapping minimal ideal basis states tobraided matter states
With one generation of fermions identified algebraically in terms of the min-imal ideals C ℓ (6) and C ℓ (4), we now wish to map these algebraic states totopological braid states. In the braid model [4], a red quark state u r is writ-ten as [1 , , σ σ − , where the vector [1 , ,
1] denotes the twist structure; inthis case 2 π clockwise twists on the first and third ribbon, and σ σ − is thebraid structure. The antiparticle state ¯ u r is written σ σ − [ − , , − , , σ σ − = σ σ − [0 , , C ℓ (6) withspecific braids in B c (as was done in [17]). Second, we include braid structureby identifying the C ℓ (4) ladder operators with suitable braids in B , in sucha way that the weak force is represented topologically via braid composition.That is, the C ℓ (6) ideals are responsible for generating twist structure, whereasthe C ℓ (4) ideals are responsible for generating braid structure. This resultsin all states being written as [ a, b, c ] B , with the twist structure first and thebraid structure second. The final step is then to change the order of the twistand braid structure for half of the states. The key to replicating the twist structure from the C ℓ (6) minimal ideals is theobservation that the twist structure and braid structure in the braid model arenot individually topological invariants, but rather are interchangeable [6, 5].One finds that:[1 , ,
0] = σ σ [0 , , , [0 , ,
0] = σ σ [0 , , , [0 , ,
1] = σ σ [0 , , , and it follows that the twist structure of every state in S u can be written as abraid in B c . If one now maps each α † i to a product of two braid generators, α † ( σ σ ) , α † ( σ σ ) , α † ( σ σ ) , (12)then each basis state in S u (see (4)) maps uniquely to the twist structure of abraid model state, in a one-to one manner. As an example, consider a greenup quark u g : u g : α † α † ωω † ( σ σ )( σ σ )[0 , ,
0] = [1 , , . (13) e can do the same for the second ideal, S d , via the maps α ( σ − σ − ) , α ( σ − σ − ) , α ( σ − σ − ) . (14)The braids associated with α † i and α i are simply braid inverses of one another .It is readily checked that ωω † [0 , ,
0] = ω † ω [0 , , ωω † = [0 , , ω † [0 , ,
0] = [0 , , ω † = [1 , , ω [0 , ,
0] = [0 , , ω = [ − , − , − S u and S d . We next focus on the braid structure, and identify the C ℓ (4) ladder operatorswith suitable braids in B , in such a way that the weak force is topologicallyrepresented via braid composition . This means that we want to define β † i and β i in terms of σ , σ , σ , and σ − . This section contains the main results ofthis paper.There seems to be no a priori reason to choose one braid structure overanother. As with the C ℓ (6) ladder operators, we would like β i and β † i to mapto inverse braids. One choice would be to map β † σ σ , and β † σ σ ,however in that case both α † and β † map to the same braid product. Takinginspiration from the braid model we define the following maps β † σ σ − , β † σ − σ , (15) β σ σ − , β σ − σ . (16)With this choice, ΩΩ † = Ω † Ω = I where I represents the identity (unbraid).Subsequently, every physical states in C ℓ (6) ⊗ C ℓ (4) can now be mappedto a specific braid with both twist structure and braid structure. For example,consider a weak doublet consisting of u rL and d rL . We have (cid:18) u rL d rL (cid:19) = α † α † ωω † Ω † Ω β † α ω † ω Ω † Ω β † ! (cid:18) [1 , , σ σ − [0 , − , σ − σ (cid:19) = (cid:18) [1 , , σ σ − σ − σ [ − , , (cid:19) . (17)It is here that it is important to remember that the twist structure is permutedby the braid structure, so that [0 , − , σ − σ = σ − σ [ − , , This is a slight but important deviation from the original construction in [17] C ℓ (6) has a total of six ladder operators, and the circular braid group B c has six generators. C ℓ (4) has four ladder operators and we therefore need a braid group with four generators. This isthe braid group B . This is also consistent with the braid model [4] where the braid structure isstrictly in B and not B c . he associated weak singlets we find (cid:0) u rR (cid:1) = (cid:16) α † α † ωω † Ω † Ω (cid:17) (cid:0) [1 , , I (cid:1) , (18) (cid:0) d rR (cid:1) = (cid:16) α ω † ω Ω † Ω β † β † (cid:17) (cid:0) σ σ − σ − σ [0 , , − (cid:1) . (19)The full list of particle states are listed in the Appendix.A couple of important deviations from the original model [4] are apparent.First, it is no longer true that the braid structure of all particles is the samelength. However it remains true that all weakly interacting particle states havethe same length, and are equivalent to their representations in [4]. Secondly,the right-handed neutrino and left-handed anti-neutrino both correspond tothe unbraid, or equivalently, the vacuum ν R ωω † Ω † Ω [0 , , I , ¯ ν L ω † ω ΩΩ † I [0 , , . (20) Finally we want to identify the charged vector bosons W + and W − with spe-cific braids in such a way that the weak interaction may be topologically rep-resented via braid composition. We can find the relevant braid representationsof the W + and W − bosons by looking at the SU (2) L generators (11). The keypoint is that charged vector bosons must change both the braid and the chargestructure. This was the motivation behind the construction of [19] where the SU (2) L generators are defined such that the change in electric charge in weaktransformations is apparent. Algebraically,[ T , u rL ] = − u rL T = u rL ( β β † ω + β β † ω † ) , = ( α † α † ωω † Ω † Ω β † )( β β † ω )= ( α † α † ω ) ω † ω Ω † Ω β † = α ω † ω Ω † Ω β † = d rL , (21)where we have used the fact that ω (and similarly ω † ) commutes with the C ℓ (4) ladder operators .The W − and W + bosons may then be identified as W − = β β † ω σ σ − σ − σ [ − , − , − , (22) W + = β β † ω † σ − σ σ σ − [1 , , , (23) See [6, 5] nd it follows that W + W − = W − W + = I [0 , , d rL → u rL W − as d rL → u rL W − σ − σ [ − , , , = σ − σ [0 , , − , − , − , = ([1 , , σ σ − )( σ σ − σ − σ [ − , − , − , (24)and subsequently W − → ¯ ν R e − L as W − → ¯ ν R e − L σ σ − σ − σ [ − , − , − , , σ σ − )( σ − σ [ − , − , − . (25)Algebraically, the weak force is automatically chiral. In terms of braids wehave u rR W − [1 , , I σ σ − σ − σ [ − , − , − σ σ − σ − σ [0 , − , , (26) d rR W + ( σ σ − σ − σ [0 , , − σ − σ σ σ − [1 , , , , σ σ − σ σ − . (27)In neither case does the right-handed particle transform into a physical state,because at the algebra level, the transformed states do not live in a C ℓ (4)minimal ideal. We have demonstrated that there exists a direct correspondence between thebasis states of the minimal ideals of the complex Clifford algebras C ℓ (6) and C ℓ (4), and a simple topologically-based toy model in which leptons, quarks,and gauge bosons are represented as simple braids composed of three ribbons.The basis states of the minimal left ideals of C ℓ (6) and minimal right idealsof C ℓ (4) were previously shown to transform as a generation of leptons andquarks under the SU (3) c × U (1) em and SU (2) L gauge symmetries respectively[19, 16].The twist structure, which topologically encodes the electrocolor symme-tries of leptons and quarks in the braid model [4] is replicated from the minimalleft ideals of C ℓ (6). Each ladder operator obtained from a Witt decompositionof C ℓ (6), and subsequently each basis state of the minimal left ideals S u and S d , is identified with a simple braid in the circular braid group B c , after whichthe resulting braiding is exchanged for twisting. This exchange of braiding fortwisting is only possible in B c [5, 6]. he braid structure, which topologically encodes the weak symmetry andchirality of leptons and quarks was then replicated from the minimal right idealstructure of C ℓ (4). The ladder operators arising from a Witt decompositionof the algebra are again identified with simple braids, but this time in thebraid group B which has four generators, compared to six for B c . Finally,the braids corresponding to charged vector bosons were determined and it wasdemonstrated what weak interactions can be described via the composition ofbraids.The braid structure in our model differs slightly from that of the originalmodel [4]. In the original model, the braid structure of all leptons and quarksconsisted of two braid generators. This is no longer the case in the presentmodel, however it remains true that all weakly interacting particles retain thisfeature. Furthermore, the braid representation of the charged vector bosonsis no longer trivial. Together, these two new features of our model (which aredictated from the underlying algebraic structure of the minimal right ideals of C ℓ (4)) explain the chiral nature of the weak force in the braid model, somethingwhich the original model did not address.A final interesting observation is that both the right-handed neutrino andleft-handed anti-neutrino both correspond to the unbraid, or equivalently, thevacuum. Acknowledgments
This work is supported by the Natural Science Foundation of the JiangsuHigher Education Institutions of China Programme grant 19KJB140018 andXJTLU REF-18-02-03 grant. ppendix ν L [0 , , σ σ − ν R [0 , , I e − L σ − σ [ − , − , − e − R σ σ − σ − σ [ − , − , − u rL [1 , , σ σ − u rR [1 , , I u gL [1 , , σ σ − u gR [1 , , I u bL [0 , , σ σ − u bR [0 , , I d rL σ − σ [ − , , d rR σ σ − σ − σ [0 , , − d gL σ − σ [0 , − , d gR σ σ − σ − σ [ − , , d bL σ − σ [0 , , − d bR σ σ − σ − σ [0 , − , d rR [1 , , σ − σ ¯ d rL [1 , , σ − σ σ σ − ¯ d gR [0 , , σ − σ ¯ d gL [0 , , σ − σ σ σ − ¯ d bR [0 , , σ − σ ¯ d bL [0 , , σ − σ σ σ − ¯ u rR σ σ − [ − , , −
1] ¯ u rL I [ − , , − u gR σ σ − [ − , − ,
0] ¯ u gL I [ − , − , u bR σ σ − [0 , − , −
1] ¯ u bL I [0 , − , − e + R [1 , , σ − σ e + L [1 , , σ − σ σ σ − ¯ ν R σ σ − [0 , ,
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