International Mathematical Forum, 4, 2009, no. 3, 127 - 134
On Dark Matter J. A. de Wet Box 514, Plettenberg Bay, 6600, South Africa
[email protected] Abstract A recent paper [12] demonstrated that quarks are branes on a manifold with su3 holonomy. In fact they are conical singularities on an orbifold of E6 which in turn carries the tetrahedral symmetry enjoyed by the nucleon. Thus we are looking at singularities on a covering Kaehler manifold. In this scenario quarks and anti-quarks would appear as dark matter possibly generated continuously as a quantum foam.
Mathematics Subject Classification: 22E70, 81T30, 83C45 Keywords: E6 ,Standard Model,Orbifolds,Quantum Gravity
1
Introduction
In several papers [8,10,11,12] de Wet has investigated the symmetry of a representation of the nucleon provided by equation (1) which is Lorentz invariant and can be shown to be idempotent and therefore following Boole [5] a fundamental theorem of logic. Also in 1994 Barth and Nieto [2] related the operators E23 , E05 and E14 to 3 pairs of ’fix-lines’on the ±i eigenspaces defined 2 = −1. The 6 fix-lines are the edges of a fundamental tetrahedron that by Eμν may be inscribed in a cube so (1) implies tetrahedral and cubic symmetry governed by the quaternion groups Td , O(cf.Coxeter [7]). Furthermore Td , O are isomorphic to the Lie algebras E6 , E7 by the MacKay correspondence [16]. In this way we have graduated from (1),which is also an irreducible representation of the Lorentz group [8], to tetrahedral symmetry and a theory of elementary particles based on a study of of the exceptional Lie algebras E6 ⊂ E7 ⊂ E8 that constituted the core of [11].In particular Fig.1 shows the famous 27 lines on a cubic surface which correspond to the 27 fundamental weights of E6 (Hunt [14],Section 4.1.1). These weights were associated with the members of the Standard Model by Slansky [17] in 1982 and a possible allocation of the 18 quarks (u; d; s; u; d; s) appears in Fig.1 taken from a paper by Coxeter [6]. The 18 quarks lie on 6 tritangent planes which are tangent to
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the cubic surface at 3 points. Fig.1 is a section of a 6 dimensional space so all the tritangents do not lie on the same plane and Fig.2 shows the tritangents u,u,d ,constituting a proton, and the anti-quarks u, u, d as the faces of two tetahedra with a common vertex at O lying on a plane between them. This is an asyzygy which inverts the vectors meeting at O that will be seen to cause the isospin, or charge operators, E05 to change sign. Thus converting quarks into anti-quarks. There is another tritangent in Fig.2 labeled by the leptons e, e+ , νe which do not enjoy tetrahedral symmetry and this could account for the mass differences between electrons and u,d quarks. For example the dimensions of E6 and su2 are respectively 27 and 2. So if following de Wet [9] we associate representations of e with su2 then a simple dimensional argument would yield a ratio of 13.5 for the e mass m of the d quark and the electron mass of 0.511 MeV i.e. m ≈ 6.9 MeV which is near a current estimate by the Particle Data Group. Now if we rotate Fig.2 through 80 degrees then uud → sss and there will be another inner tritangent that we will label by the mesons μ− , μ+ , νμ that are associated with strangeness production in heavy ion collisions. In fact both could be just maximum entropy states washing out all memory of what happened before according to Biro [3]. Even though pion production is the dominant dissipative reaction these quickly decay into muons which can therefore occur together with strangeness. So if we are indeed looking at a maximum entropy state then the meson is just a heavy electron with almost the same mass as the strange particles s which have the same electric charge of -1/3 as the down quark. Entropy arguments were also used by de Wet [11] to explain the mass ratios of the top and down quarks and the charm and down quarks. However in view of the reduced current estimate of the mass of the up quark; a comparison of the number of elements in the subalgebras of E8 and E6 with the masses of top and up quarks predicts a pair of top quarks and an up quark mass of approximately 3.2 MeV contrary to [11] where only a single up quark was found. In this way we can account for the mass differences in the Coxeter multiplets shown in Figs.1,2 and also quark mass differences in the generations. In particular Figs.1,2 exhibit 18 conical singularities and the tritangents are orbifolds that have appeared in Type II String Theory (cf.[13] Section 9.5). They will be considered in more detail when we will look at a covering Kaehler manifold with SU(3) holonomy. However for the moment these singularities are the origin of the quantum foam of Fig.3 which is a surface invariant under the tetrahedral group Td namely the Clebsch cubic (6).Two tritangents at the center of the figure are blown up for clarity. In fact they cannot be described by the Sing Surf program used to plot equation (6) and can be orders of magnitude smaller than the quark branes that determine the sizes of protons u,u,d and antiprotons u, u, d.
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Thus we are analyzing behavior in the small and in this connection there are two references that have recently considered a type of tetrahedral symmetry structure in tiny space. In[15] Iqbal et. al.find a tetrahedral shape by exploiting an analogy with crystal melting; while in [1] Ambjorn et.al.consder space as composed of causal dynamical triangulations.
2
The Fundamental Idempotent Relation
The equation 1 Ψ = (iE4 ψ1 + E23 ψ2 + E14 ψ3 + E05 ψ4 )e (1) 4 is a minimal left ideal of the center of the Dirac ring describing spin about x3 and is an irreducible representation of the Lorentz group [8]. It has been used to model a nucleon because E23 , E05 are rotational operators of spin and isospin through half-angles ψ2 , ψ4 while E14 is a parity operator completing the triality. iE14 ψ1 is the identity operator for rotations mod 2π. Eddington’s E-numbers are mapped into the 4 × 4 Dirac matrices by 2 Eμν
γν = iE0ν , Eμν = Eρμ Eρν = −Eνμ , = −1, Eμν Eστ = Eστ Eμν = iEλρ , μ < ν = 1, . . . , 5
(2)
where E4 is the unit matrix. Two more equivalent representations may be obtained by a cyclic interchange of the indices 1,2,3 to express rotations about x2 , x3 . Returning to (1)there are four primitive idempotents e, namely e1 = − 4i (E03 + E12 + E45 + E16 ) e2 = − 4i (−E03 − E12 + E45 + E16 ) e3 = − 4i (−E03 + E12 − E45 + E16 ) e4 = − 4i (E03 − E12 − E45 + E16 )
(3)
which satisfy ei ej = 0 and therefore have the properties e1 E23 e1 = E23 e2 e1 = e1 E14 e1 = E14 e4 e1 = e1 E05 e1 = E05 e3 e1 = 0 so that
(Ψ/4)2 = (Ψ/4)iE4 ψ1
(4) (5)
which is idempotent if iE4 ψ1 = 1. The idempotents e3 , e4 are supposed to represent neutrons. In [10] de Wet associated the u,u;u,d lines of the principal triangle of Fig.2 with the operators E05 = −E50 with isospin ±1/2 and showed that this recipe would yield the quark charges 1/6 ± 1/2. The third edge was assigned to the
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parity operator E14 so that a change E14 → E41 from a Left hand to a Right hand quark is accompanied by charge conjugation. The 3 remaining skew-lines meeting at O were labeled by E41 and E23 , E32 with spin ±1/2. In particular the asyzygy that inverts vectors meeting at O will cause E05 → E50 = −E05 thus changing quarks into anti-quarks.
3
Quantum Foam
In [17] Slansky exploits the subalgebra su3 × su3 × su3 of E6 to describe the Standard Model, where a possible allocation of the 27 weights of E6 to the 27 vertices of the famous lines on a cubic surface is provided by Figs. 1,2. The 9 vertices of Fig.2 also carry Coxeter’s labels (λ, μ, ν) for the powers of ω = exp(2iπ/3) where λ, μ, ν can independently assume the values 0,1,2 [6].Thus the apex of each equilateral triangle is simply the previous one multiplied by ω and equivalent to an SO(6) rotation through 120 degrees. Referring to Fig.2 this twist will rotate separately u(012) → u(023) → u(012); d(013) → u(021) → d(013); e(011) → e+ (022) → e(011) which may be accomplished by su3 ×su3 ×su3 where su3 is composed of only the diagonal elements 1+ω+ω 2 = 0. Color symmetry implies, for example, that if say u(012) is red and u(023) is green, then red → green → red and so on. In this way E6 /(su3 × su3 × su3 ) has su3 holonomy on a 6 dimensional cubic surface with su3 × su3 × su3 gauge. A hyperplane section of the Segre cubic in 6 dimensions is the Clebsch Diagonal Cubic 81(x3 + y 3 + z 3 ) − 189(x2 y + x2 z + y 2x + y 2z + z 2 x + z 2 y) +54xyz + 126(xy + xz + yz) − 9(x2 + y 2 + z 2 ) −9(x + y + z) + 1 = 0
(6)
found by utilizing tetrahedral coordinates for the tritangents (cf.Hunt[14],Section 4.1.3). This is plotted in Fig.3 believed to be a picture of quantum foam on a Ricci-flat Kaehler manifold with SU(3) holonomy which is discussed in Section 16.3 of [13] where the orbifold of Figs.1,2 is ’covered’by a 3 dimensional complex manifold Y(r) glued over the 27 ’holes’ or singularities.Here Y(r)is conformally invariant and dependent only on the parameter r. If we extend Y(r) to a ’World-brane’ then quarks and anti-quarks would appear as Dark Matter generated continuously by singularities on Y possibly as a foam. Perhaps quarks were the first objects created by the ’Big Bang’ and later united to form the nucleons constituting Dark and Star Matter as an ongoing process fueled by thermal energy. By adding the bulk of such dark matter to Y(r) space Boehmer and Harko and Harko and Mak [4] modified the Schwarzchild metric to describe accurately the halos of exotic matter found around galaxies.
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References [1] J.Ambjorn, J.Jurkiewicz and R.Loll, arXiv:hep-th/0509010v3 14 Oct 2006.
The universe from scratch,
[2] W.Barth and I.Nieto, Abelian surfaces of type (1,3) and quartic surfaces with 16 skew lines, J.Algebraic Geometry, 3 (1994) 173-222. [3] Tamas S. Biro, Strange quark matter, arXiv:hep-ph/0304265v1 28 April 20023. [4] C.G.Boehmer and T.Harko, Galactic dark matter as a bulk effect on the brane, arXiv:0705.2496v3 [gr-qc] 10 Jun 2007; T.Harko and M.K.Mak, Conformally symmetric vacuum solutions of the gravitational field equations in brane-world models, Annals of Physics 319(2005)471-492. [5] George Boole, An Introduction to the Laws of Thought, Dover,Reprinted in God Created the Integers by Stephen Hawking, Running Press,Philadelphia,2005. [6] H.S.M.Coxeter, The polytope 221 where 27 vertices correspond to the lines on the general cubic surface, American J.of Mathematics 62(1940),457486. [7] H.S.M.Coxeter, Regular Complex Polytopes, 2nd ed.,Cambridge University Press,1991. [8] J.A.de Wet, A group theoretical approach to the many nucleon problem, Proc. Camb.Phil.Soc, 70(1971),485-496. [9] J.A.de Wet, The many electron problem, 72(1972),123-134.
Proc.Camb.Phil.Soc.,
[10] J.A.de Wet, Tetrahedral symmetry and the graviton, Int.Mathematical Forum, 2 (2007)no.31,1537-1551. [11] J.A.de Wet, A logical basis for the Standard Model and quark mass ratios, Int.Mathematical Forum, 3 (2008)no.16,777-782. [12] J.A.de Wet, Are quarks branes on E6 ? Int.Mathematical Forum,3(2008) On Google, Under quarks and branes [13] M.B.Green,J.H.Schwarz and E.Witten, Vol.2,Cambridge University Press,1993.
Superstring
Theory
[14] Bruce Hunt, The Geometry of Some Arithmetic Quotients, Lecture Notes in Mathematics,1637,Springer,Berlin,Heidelberg,1966.
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[15] A.Iqbal,N.Nekrasrov,A.Okrounkov and C.Vafa, Quantum foam and topolgical strings, arXiv:hep-th 0312022v2 22 Dec 2003. [16] John Mackay, Cartan, finite groups of quaternions and Kleinian singularities, Proc.AMS, (1981) 153-154. [17] R.Slansky Group theory for unified model building Reprinted inUnity of Forces in the Universe,Ed.A.Lee, World Scientific, 1982. Received: July 1, 2008
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Figure 1: The Coxeter Polytope
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