1. Identification of elementary particles
1.1 Original picture
The original view about family replication phenomenon assumed that fermions correspond to single boundary component of the space-time surface (liquid bubble is a good analogy) and thus characterized by genus g telling the number of handles attached to the sphere to obtain the bubble topology.
- Ordinary bosons would correspond to g=0 (spherical) topology and the absorption/emission of boson would correspond to 2-D topological sum in either time direction. This interpretation conforms with the universality of ordinary ew and color interactions.
- The genera of particle and antiparticle would have formally opposite sign and the total genus would be conserved in the reaction vertices. This makes sense if the annihilation of fermion and anti-fermion to g=0 boson means that fermion turns backwards in time emitting boson. The vertex is essentially 2-D topological sum at criticality between two manifold topologies. In the vertex 2-surface would be therefore singular manifold. The analogy to closed string emission in string model is obvious.
Later the original picture was replaced with a more complex identification.
- Fundamental particles - partons - serving as building bricks of elementary particles are partonic 2-surfaces identified as throats of wormhole contacts at which the Euclidian signature of the induced metric of the wormhole contact changes to Minkowskian one. The orbit of partonic 2-surface corresponds to a light-like 3-surface at which the Minkowskian signature of the induced metric changes to Euclidian, and carries fermion lines defining of boundaries of string world sheets. Strings connect different wormhole throats and mean generalization of the notion of point like particle leading to the notion of tensor network (see this).
Elementary particles are pairs of two wormhole contacts. Both fermions and bosons are pairs of string like flux tubes at parallel space-time sheets and connected at their ends by CP_{2} sized wormhole contacts having Euclidian signature of induced metric. A non-vanishing monopole flux loop runs around the extrenely flattened rectangle loop connecting wormhole throats at both space-time sheets and traverses through the contacts.
- The throats of wormhole contacts are characterized by genus given by the number g of handles attached to sphere to get the topology. If the genera g_{a},g_{b} of the opposite throats of given wormhole contact are same, one can assign genus to it : g=g_{a}=g_{b}. This can be defended by the fact, that the distance between the throats is given by CP_{2} length scale and thus extremely short so that g_{a}≠ g_{b} implies strong gradients and by Uncertainty Principle mass of order CP_{2} mass.
If the genera of the two wormhole contacts are same: g_{1}=g_{2}, one one can assign genus g to the particle. This assumption is more questionable if the distance between contacts is of order of Compton length of the particle. The most general assumption is that all genera can be different.
- There is an argument for why only 3 lowest fermion generations are observed (see this). Assume that the genus g for all 4 throats is same. For g=0,1,2 the partonic 2-surfaces are always hyper-elliptic allowing thus a global conformal Z_{2} symmetry. Only these 3 2-topologies would be realized as elementary particles whereas higher generations would be either very heavy or analogous to many-particle states with a continuum mass spectrum. For the latter option g=0 and g=1 state could be seen as vacuum and single particle state whereas g=2 state could be regarded as 2-particle bound state. The absence of bound n-particle state with n>2 implies continuous mass spectrum.
- Fundamental particles would wave function in the conformal moduli space associated with its genus (Teichmueller space). For fundametal fermions the wave function would be strongly localized to single genus. For ordinary bosons one would have maximal mixing with the same amplitude for the appearance of wormhole throat topology for all genera g=0,1,2. For the two other u(3)_{g} neutral bosons in octet one would have different mixing amplitudes and charge matrices would be orthogonal and universality for the couplings to ordinary fermions would be broken for them. The evidence for the breaking of the universality (see this) is indeed accumulating and exotic u(3)_{g} neutral gauge bosons giving effectively rise to two additional boson families could explain this.
What about gauge bosons and Higgs, whose quantum numbers are carried by fermion and anti-fermion (or actually a superposition of fermion-anti-fermion pairs). There are two options.
- Option I: The fermion and anti-fermion for elementary boson are located at opposite throats of wormhole contact as indeed assumed hitherto. This would explain the point-likeness of elementary bosons. u(3) charged bosons having different genera at opposite throats would have vanishing couplings to ordinary fermions and bosons. Together with large mass of g_{a}≠ g_{b} wormhole contacts this could explain why g_{a}≠ g_{b} bosons and fermions are not observed and would put the Cartan algebra of u(3)_{g} in physically preferred position. Ordinary fermions would effectively behave as u(3)_{g} triplet.
- Option II: The fermion and anti-fermion for elementary boson are located at throats of different wormhole contacts making them non-point like string like objects. For hadron like stringy objects, in particular graviton, the quantum numbers would necessarily reside at both ends of the wormhole contact if one assumes that single wormhole throats carries at most one fermion or anti-fermion. For this option also ordinary fermions could couple to (probably very massive) exotic bosons different genera at the second end of the flux tube.
Option I: Since only the wormhole throat carrying fermionic quantum numbers is active and since fundamental fermions naturally correspond to u(3)_{g} triplets, one can argue that the wormhole throat carrying fermion quantum number determines the fermionic u(3)_{g} representation and should be therefore 3 for fermion and 3bar anti-fermion.
At fundamental level also bosons would in the tensor products of these representations and many-sheeted description would use these representations. Also the description of graviton-like states involving fermions at all 4 wormhole throats would be natural in this framework. At gauge theory limit sheets would be identified and in the most general case one would need U(3)_{g}× U(3)_{g}× U(3)_{g}× U(3)_{g} with factors assignable to the 4 throats.
- The description of weak massivation as weak confinement based on the neutralization of weak isospin requires a pair of left and right handed neutrino located with ν_{L} and νbar_{R} or their CP conjugates located at opposite throats of the passive wormhole contact associated with fermion. Already this in principle requires 4 throats at fundamental level. Right-handed neutrino however carries vanishing electro-weak quantum numbers so that it is effectively absent at QFT limit.
- Why should fermions be localized and su(3)_{g} neutral bosons delocalized with respect to genus? If g labels for states of color triplet 3 the localization of fermions looks natural, and the mixing for bosons occurs only in the Cartan algebra in u(3)_{g} framework: only u(3)_{g} neutral states an mix.
- The model for CKM mixing (see this) would be modified in trivial manner. The mixing of ordinary fermions would correspond to different topological mixings of the three states su(3)-neutral fermionic states for U and D type quarks and charged leptons and neutrinos. One could reduce the model to the original one by assuming that fermions do not correspond to generators Id, Y, and I_{3} for su(3)_{g} but their linear combinations giving localization to single valued of g in good approximation: they would correspond to diagonal elements e_{aa}, a=1,2,3 corresponding to g=0,1,2.
- p-Adic mass calculations (see this) assuming fixed genus for fermion predict an exponential sensitivity on the genus of fermion. In the general case this prediction would be lost since one would have weighted average over the masses of different genera with g=2 dominating exponentially. The above recipe would cure also this problem. Therefore it seems that one cannot distinguish between the two options allowing g_{1}≠ g_{2}. The differences emerge only when all 4 wormhole throats are dynamical and this is the case for graviton-like states (spin 2 requires all 4 throats to be active).
3. Reaction vertices
Consider next the reaction vertices for the option in which particles correspond to string like objects identifiable as pairs of flux tubes at opposite space-time sheets and carrying monopole magnetic fluxes and with ends connected by wormhole contacts.
- Reaction vertex looks like a simultaneous fusing of two open strings along their ends at given space-time sheets. The string ends correspond to wormhole contacts which fuse together completely. The vertex is a generalization of a Y-shaped 3-vertex of Feynman diagram. Also 3-surfaces assignable to particles meet in the same manner in the vertex. The partonic 2-surface at the vertex would be non-singular manifold whereas the partonic orbit would be singular manifold in analogy with Y shaped portion of Feynman diagram.
- In the most general case the genera of all four throats involved can be different. Since the reaction vertex corresponds to a fusion of wormhole contacts characterized in the general case by (g_{1},g_{2}), one must have (g_{1},g_{2})=(g_{3},g_{4}). The rule would correspond in gauge theory description to the condition that the quark and antiquark su(3)_{g} charges are opposite at both throats in order to guarantee charge conservation as the wormhole contact disappears.
- One has effectively pairs of open string fusing along their and and the situation is analogous to that in open string theory and described in terms of Chan-Paton factors. This suggests that gauge theory description makes sense at QFT limit.
- If g is same for all 4 throats, one can characterize the particle by its genus. The intuitive idea is that fermions form a triplet representation of u(3)_{g} assignable to the family replication. In the bosonic sector one would have only u(3)_{g} neutral bosons. This approximation is expected to be excellent.
- One could allow g_{1}≠ g_{2} for the wormhole contacts but assume same g for opposite throats. In this case one would have U(3)_{g}× U(3)_{g} as dynamical gauge group with U(3)_{g} associated with different wormhole contacts. String like bosonic objects (hadron like states) could be therefore seen as a nonet for u(3)_{g}. Fermions could be seen as a triplet.
Apart from topological mixing inducing CKM mixing fermions correspond in good approximation to single genus so that the neutral members of u(3)_{g} nonet, which are superpositions over several genera must mix to produce states for which mixing of genera is small. One might perhaps say that the topological mixing of genera and mixing of u_{3}(g) neutral bosons are anti-dual.
- If all throats can have different genus one would have U(3)_{g}× U(3)_{g}× U(3)_{g}× U(3)_{g} as dynamical gauge group U(3)_{g} associated with different wormhole throats. This option is probably rather academic.
Also fermions could be seen as nonets.
- If g is same for all 4 throats, one can characterize the particle by its genus. The intuitive idea is that fermions form a triplet representation of u(3)_{g} assignable to the family replication. In the bosonic sector one would have only u(3)_{g} neutral bosons. This approximation is expected to be excellent.
4. What would the gauge theory description of family replication phenomenon look like?
For the most plausible option bosonic states would involve a pair of fermion and anti-fermion at opposite throats of wormhole contact. Bosons would be characterized by adjoint representation of u(3)_{g}=su(3)_{g}× u(1)_{g} obtained as the tensor product of fermionic triplet representations 3 and 3bar.
- u(1)_{g} would correspond to the ordinary gauge bosons bosons coupling to ordinary fermion generations in the same universal manner giving rise to the universality of electroweak and color interactions.
- The remaining gauge bosons would belong to the adjoint representation of su(3)_{g}. One indeed expects symmetry breaking: the two neutral gauge bosons would be light whereas charged bosons would be extremely heavy so that it is not clear whether QFT limit makes sense for them.
Their charge matrices Q_{g}^{i} would be orthogonal with each other (Tr(Q_{g}^{i}Q_{g}^{j})=0, i≠ j) and with the unit charge matrix u(1)_{g} charge matrix Q^{0}∝ Id (Tr(Q_{g}^{i})=0) assignable to the ordinary gauge bosons.These charge matrices act on fermions and correspond to the fundamental representation of su(3)_{g}. They are expressible in terms of the Gell-Mann matrices λ_{i} (see
this).
- One could start from an algebra formed as a tensor product of standard model gauge algebra g= su(3)_{c}× u(2)_{ew} and algebraic structure formed somehow from the generators of u(3)_{g}. The generators would be
J_{i,a}= T_{i} ⊗ T_{a} ,
where i labels the standard model Lie-algebra generators and a labels the generators of u(3)_{g}.
This algebra should be Lie-algebra and reduce to the same as associated with standard model gauge group with generators T^{b} replacing effectively complex numbers as coefficients. Mathematician would probably say, that standard model Lie algebra is extended to a module with coefficients given by u(3)_{g} Lie algebra generators in fermionic representation but with Lie algebra product for u(3)_{g} replaced with a product consistent with the standard model Lie-algebra structure, in particular with the Jacobi-identities.
- By writing explicitly commutators and Jacobi identifies one obtains that the product must be symmetric: T_{a}• T_{b}= T_{b}• T_{a} and must satisfy the conditions T_{a}• (T_{b}• T_{c})= T_{b}• (T_{c}• T_{a})= T_{c}• (T_{a}• T_{b}) since these terms appear as coefficients of the double commutators appearing in Jacobi-identities
[J_{i,a},[J_{j,b}],J_{k,c}]]+[J_{j,b},[J_{k,c}],J_{i,a}]] + [J_{k,c},[J_{i,a}],J_{j,b}]]=0 .
Commutativity reduces the conditions to associativity condition for the product •. For the sub-algebra u(1)^{3}_{g} these conditions are trivially satisfied.
- In order to understand the conditions in the fundamental representation of su(3), one can consider the product the su(3)_{g} product defined by the anti-commutator in the matrix representation provided by Gell-Mann matrices λ_{a} (see this and this):
{λ_{a},λ_{b}}= 43δ_{a,b} Id + 4d_{abc}λ^{c} , & Tr(λ_{a}λ_{b}) =2δ_{ab} , & d_{abc}= Tr(λ_{a}λ_{b},λ_{c})
d_{abc} is totally symmetric under exchange of any pair of indices so that the product defined by the anti-commutator is both commutative and associative. The product extends to u(3)_{g} by defining the anti-commutator of Id with λ_{a} in terms of matrix product. The product is consistent with su(3)_{g} symmetries so that these dynamical charges are conserved. For complexified generators this means that generator and its conjugate have non-vanishing coefficient of Id.
Remark: The direct sum u(n)⊕ u(n)_{s} formed by Lie-algebra u(n) and its copy u(n)_{s} endowed with the anti-commutator product • defines super-algebra when one interprets anti-commutator of u(n)_{s} elements as an element of u(n).
- Could su(3) associated with 3 fermion families be somehow special? This is not the case. The conditions can be satisfied for all groups SU(n), n≥ 3 in the fundamental representation since they all allow completely symmetric structure constants d_{abc} as also higher completely symmetric higher structure constants d_{abc...} up to n indices. This follows from the associativity of the symmetrized tensor product: ((Adj⊗ Adj)_{S}⊗ Adj)_{S} =(Adj⊗ (Adj⊗ Adj)_{S})_{S} for the adjoint representation.
See the article Topological description of family replication and evidence for higher gauge boson generations.
For a summary of earlier postings see Latest progress in TGD.
1 comment:
Sometimes questions leave a good trace. Thanks.
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