International Mathematical Forum, Vol. 7, 2012, no. 51, 2519 - 2524
A Standard Model Algebra J. A. de Wet Box 514, Plettenberg Bay, 6600, South Africa
[email protected] Abstract In this note we will present an algebraic derivation of the Standard Model without any appeal to a Langrangian or a strong force to bind the 3 quarks. Instead we will develop a formulation based on the exceptional Lie algebra E6 ,introduced by Slansky [13],which is an orbifold or Toric variety.Without appealing to a Higgs boson, mass is accommodated by relating the orders of the subalgebras of E6 to an entropy or information content in accord with the Holographic Principle [7]. If we agree that quarks were in a 6d Planck space, just after the Big Bang, before combining to form nucleons in 3-space after a huge expansion (with accompanying cooling) then the central contribution is the demonstration that the 6-space blows up to 3d simply by eliminating the complex dimensions.In this way it is not necessary to to postulate a 10d space with 6 ’curled-up’ dimensions as in String Theory.The Introduction has already been covered by de Wet [8],but Section 2 is believed to be new.
Mathematics Subject Classification: 14J32, 22E70, 81V25, 81E15, 83F05 Keywords: E6 ,Standard Model,Toric Variety,Entropy, Blow-up from Quark to Neutron Space, Higgs Boson, Dark Matter
1
Introduction
Fig.1 is the lattice of the su3 × su3 × su3 subalgebra of E6 . The 27 vertices are labeled by (u,u,d),(d,d,u)and (s,s,s) with their anti-particles together with 9 leptons (e± , νe ), (μ± , νμ ), (τ ± , ντ ).It is an orbifold or T 6 /Z toric variety [10] with most of the 27 cones meeting at the Origin. Some time ago Slansky [13] related the 27 vertices to the Standard Model but the labeling is by the Author (cf [8] and references therein). The lattice also appears in work by Coxeter and Hunt [5,6,11] as a Hessian polyhedron. The order of the Weyl group W of the subalgebra (su3)3 is 216 which is also the number of possible permutations of the up-quark pair uu in the subalgebra
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and believed to be a measure of the entropy S directly proportional to energy and mass [8].Experimental evidence for the Holographic Principle appears in [7,8]. This asserts that the origin of a universal relation between geometry and information must lie in the number of fundamental degrees of freedom (or entropy) involved in a unified description of space and matter [3]. In [8] the masses of the stable particles e, u, W ± are derived without any appeal to QCD or a Higgs Boson. Turning to Fig .1 the ’radius’ η is a Benkenstein Bound [3,4] with a value of 27.7 keV that relates the entropy, or degrees of freedom,to the mass [7,8]. In this way we find masses of about 6 MeV for the up-quark pair uu, 160.2 GeV for the W ± pair and 500 ke V for the electron if we use the orders |W (A2)|3 = 63 , 3|W (A2)| = 18 of subalgebras of E6 and |W (E7 + A1 )| = 5806080, a subalgebra of E8 , following Adams and Manivel [1,12].
2
Blow-up of CP 3
Fig.2 is a Clebsch cubic surface due to Xahlee [14].It is a complex projective 6space CP 3 with homogeneous coordinates that also carries the famous 27 lines and 27 vertices (cf [6] Section 12.3).Here the Weyl group W (E6 ) of permutations in E6 is invariant under the tetrahedral group that has a representation which is the Clebsch cubic (cf [5],[11] Ch.4).Algebraically Fig.2 is the dual of Fig.1 by Fulton [9] Ch.2.The torus is generated by 3 rotations of 120 degrees of the 3 tori labeled by p,n and s which are lined up by twists of 40 degrees. This 6d Planck symplectic space CP 3, home of the quarks,is believed to be blown-up just after the Big Bang into P 3 where 3 quarks unite to create the stable deuteron pair after an expansion of the order of 1020 accompanied by massive cooling. After the blow-up a picture of the 6 singularities corresponding to the 6 faces of the toric variety X(Λ) over an invariant cube appears in [9] p 50.The fan Λ of the tetrahedron defines the toric variety X(Λ) = P 3 . However there is a second possible counter-tetrahedron shown in Fig.3.Faces of these tetrahedra are thought to map the deuteron pair after the blow-up and still exhibit tetrahedral symmetry under rotations and reflections eg. d → d, d ← d. Such a symmetry has been considered by Ambjorn et.al.[2] in a model for quantum gravity that triangulates space with tetrahedra including time inversions at O of Fig.3 where there are 2 diagonally opposite vertices at the corners O which Ambjorn et.al. label by ±t corresponding to time-slices in a 3-space. Physically this just means that the nucleons are not created simultaneously. Finally there are far fewer possible permutations of the vertices in the 3d cube than in the Weyl group W (E6 ). But in 3-space there are also μ and τ leptons which decay rapidly into stable electrons and neutrinos so their mass
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A standard model algebra
is transferred into stable deuterium according to the relation mτ + mμ → mp + mn + me
(1)
which is accurate to within the current measurements of mτ = 1777M eV, mμ = 106M eV .
3
Conclusion
Neutrinos, being effectively massless, do not appear in equation (1) but are thought to be the particles belonging to the Dark Matter field possibly consisting of extremely dense deuterium atoms left over from the Big Bang. In this scenario the matter is dark simply because of the extremely high frequencies of neutrinos.
References [1] J.F.Adams, Lectures on Exceptional Lie Groups,The University of Chicago Press,1996. [2] J.Ambjorn, J.Juckiewicz and R. Loll, arXiv:hep-th/0002050 v3. [3] Raphael Bousso, The Holographic Principle, Rev.Mod.Phys.2002,74,No.3. [4] R.Casini, arxiv:hep-th/0804.20802 v3. [5] H.S.M. Coxeter, The polytope 221 where 27 lines correspond to the lines on a general cubic surface,American J. of Maths.,1940,62,457-486. [6] H.S.M. Coxeter, Edn.,1991.
Regular Complex Polytopes,Camb.Univ.Press,2nd
[7] J.A.de Wet, Experimental evidence for the Holographic Principle,To be published by Int.Mathematical Forum, Available from Hikari.Ltd. [8] J.A.de Wet, On the strong force without QCD and the origin of mass without the Higgs,Int.Mathematical Forum,2012,29,1419-1425. Available from Hikari.Ltd. [9] William Fulton, Introduction to Toric Varieties, Annals of Mathematical Studies,No.131 Princeton Univ.Press,1993. [10] M.B.Green,J.B.Schwarz and Camb.Univ.Press, ,1993.
E.Witten,
Superstring
Theory,
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[11] Bruce Hunt, The Geometry of some Arithmetic Quotients, Lecture notes in Mathematics,1637,Springer,Berlin,Heidelberg,1996. [12] L.Manivel, Configurations of lines and models of Lie algebras, arXiv:math.AGO50711v2.0 [13] R.Slansky, Group theory for unified mmodel building, Reprinted in; Unity of Forces in the Universe, Ed.A.Lee,World Scientific,1992. [14] http:/xahlee.org/surface/clebsch cubic.html. Received: April, 2012
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A standard model algebra u _
s
τ-
μ+
_
d
u e-
+
e
d
d
_
u
_
s
s η
-
μ
τ+
_
_ s
d νμ νe
_
_
u
ντ s
d
d
_
u s
u Fig. 1
Figure 1: The Coxeter Polytope
Figure 2: The Clebsch Cubic
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J. A. de Wet
u
u
u
d
d
d Fig. 3
Figure 3: The Cube in 3-space
