Quantum Gravity Research Los Angeles, U.S.A. Nonification v. 1.0.1 Raymond Aschheim raymond@quantumgravityresearch.org v 1.0.1 presented at AQG V, Los Angeles, October 29th, 2018 Advances in Quantum Gravity V in honor of Piero Truini October 30, 2018
Content 1 Introduction Story 9D coordinates 9D coordinates Group Theoretic View A8 extension of the standard model A8 includes A2+A2+A2 Tensor Network Tensor Network Spin Network Magic star Jordan algebra F 4 action, E 6 action E 7 action, E 8 action The magic star Quantum gravity Induced Fano plane E 8 Quasi-lattice compactification Quasi-lattice action Raymond Aschheim | Nonification Particle Model
Introduction 9D coordinates 2 ◮ This talk is dedicated to the career of Piero Truini which opened the path to an exceptional an magic unification. ◮ We shall see how the partition of the E8 lattice into three A8 lattices, inviting us to use 9 dimensions coordinates, guide also us to an exceptionally symmetric unification... without supersymmetry. ◮ "It is amusing to speculate on the possibility of a theory based on E9." is the conclusion of the chapter V, Exceptional Unification, of Pr. Anthony Zee’s book. REF: 0a Zee, A. Unity of forces in the universe World Scientific. 1982 Raymond Aschheim | Nonification
Introduction 9D coordinates 3 ◮ A n Simplex lattices are naturally expressed in n + 1 dimension coordinates satisfying � n + 1 k = 1 x k = 0 Raymond Aschheim | Nonification
Introduction 9D coordinates 3 ◮ A n Simplex lattices are naturally expressed in n + 1 dimension coordinates satisfying � n + 1 k = 1 x k = 0 ◮ E 8 lattice is the superposition of three A 8 lattices: E 8 = � 2 i = 0 A i 8 ◮ 72 of its roots are permutations of { 3 1 , − 3 1 , 0 7 } , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ 84 of its roots are P ( − 2 3 , 1 6 ) , ∼ = 1 [ 3 ] , ∈ 3 A 1 8 ◮ 84 of its roots are P ( 2 3 , − 1 6 ) , ∼ = 2 [ 3 ] , ∈ 3 A 2 8 Raymond Aschheim | Nonification
Introduction 9D coordinates 3 ◮ A n Simplex lattices are naturally expressed in n + 1 dimension coordinates satisfying � n + 1 k = 1 x k = 0 ◮ E 8 lattice is the superposition of three A 8 lattices: E 8 = � 2 i = 0 A i 8 ◮ 72 of its roots are permutations of { 3 1 , − 3 1 , 0 7 } , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ 84 of its roots are P ( − 2 3 , 1 6 ) , ∼ = 1 [ 3 ] , ∈ 3 A 1 8 ◮ 84 of its roots are P ( 2 3 , − 1 6 ) , ∼ = 2 [ 3 ] , ∈ 3 A 2 8 ◮ E 8 = SU ( 3 ) F + E 6 = A 2 F + E 6 248 \ A 2 F + E 6 = ( 1 , 78 ) + ( 8 , 1 ) + ( 3 , 27 ) + ( 3 , 27 ) (1) Raymond Aschheim | Nonification
Introduction E6 subgroup in 9D 4 ◮ E 6 lattice is the superposition of three A 8 lattices satisfying � 3 k = 1 x k = � 6 k = 4 x k = � 9 k = 7 x k = 0. ◮ 18 of its roots are P ( 3 1 , − 3 1 , 0 7 ) , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ (a) (b) (c) Figure: 3 orthogonal A2 in E6; from left to right: (a) A2L (b) A2C (c) A2R. Raymond Aschheim | Nonification
Introduction E6 subgroup in 9D 4 ◮ E 6 lattice is the superposition of three A 8 lattices satisfying � 3 k = 1 x k = � 6 k = 4 x k = � 9 k = 7 x k = 0. ◮ 18 of its roots are P ( 3 1 , − 3 1 , 0 7 ) , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ 27 of its roots are P ( − 2 3 , 1 6 ) , ∼ = 1 [ 3 ] , ∈ 3 A 1 8 ◮ Figure: 27 lepto-quarks bosons B from E 6 � A 1 8 ; Raymond Aschheim | Nonification
Introduction E6 subgroup in 9D 4 ◮ E 6 lattice is the superposition of three A 8 lattices satisfying � 3 k = 1 x k = � 6 k = 4 x k = � 9 k = 7 x k = 0. ◮ 18 of its roots are P ( 3 1 , − 3 1 , 0 7 ) , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ 27 of its roots are P ( − 2 3 , 1 6 ) , ∼ = 1 [ 3 ] , ∈ 3 A 1 8 ◮ 27 of its roots are P ( 2 3 , − 1 6 ) , ∼ = 2 [ 3 ] , ∈ 3 A 2 8 ◮ B from E 6 � A 2 Figure: 27 anti-lepto-quarks bosons ˆ 8 ; Raymond Aschheim | Nonification
Introduction E6 subgroup in 9D 4 ◮ E 6 lattice is the superposition of three A 8 lattices satisfying � 3 k = 1 x k = � 6 k = 4 x k = � 9 k = 7 x k = 0. ◮ 18 of its roots are P ( 3 1 , − 3 1 , 0 7 ) , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ 27 of its roots are P ( − 2 3 , 1 6 ) , ∼ = 1 [ 3 ] , ∈ 3 A 1 8 ◮ 27 of its roots are P ( 2 3 , − 1 6 ) , ∼ = 2 [ 3 ] , ∈ 3 A 2 8 ◮ E 6 = SU ( 3 ) L + SU ( 3 ) C + SU ( 3 ) R + B + B (2) ◮ 78 = ( 8 , 1 , 1 )+( 1 , 8 , 1 )+( 1 , 1 , 8 )+( 3 , 3 , 3 )+( 3 , 3 , 3 ) = B L + B C + B R + B + B (3) Raymond Aschheim | Nonification
Introduction E6 subgroup in 9D 4 ◮ E 6 lattice is the superposition of three A 8 lattices satisfying � 3 k = 1 x k = � 6 k = 4 x k = � 9 k = 7 x k = 0. ◮ 18 of its roots are P ( 3 1 , − 3 1 , 0 7 ) , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ 27 of its roots are P ( − 2 3 , 1 6 ) , ∼ = 1 [ 3 ] , ∈ 3 A 1 8 ◮ 27 of its roots are P ( 2 3 , − 1 6 ) , ∼ = 2 [ 3 ] , ∈ 3 A 2 8 ◮ E 6 = SU ( 3 ) L + SU ( 3 ) C + SU ( 3 ) R + B + B (2) ◮ 78 = ( 8 , 1 , 1 )+( 1 , 8 , 1 )+( 1 , 1 , 8 )+( 3 , 3 , 3 )+( 3 , 3 , 3 ) = B L + B C + B R + B + B (3) ◮ 27 = ( 3 , 3 , 1 ) + ( 1 , 3 , 3 ) + ( 3 , 1 , 3 ) = q γ q β γ + l α α + ˆ (4) β Raymond Aschheim | Nonification
Introduction E6 subgroup in 9D 4 ◮ E 6 lattice is the superposition of three A 8 lattices satisfying � 3 k = 1 x k = � 6 k = 4 x k = � 9 k = 7 x k = 0. ◮ 18 of its roots are P ( 3 1 , − 3 1 , 0 7 ) , ∼ = 0 [ 3 ] , ∈ 3 A 0 8 ◮ 27 of its roots are P ( − 2 3 , 1 6 ) , ∼ = 1 [ 3 ] , ∈ 3 A 1 8 ◮ 27 of its roots are P ( 2 3 , − 1 6 ) , ∼ = 2 [ 3 ] , ∈ 3 A 2 8 ◮ E 6 = SU ( 3 ) L + SU ( 3 ) C + SU ( 3 ) R + B + B (2) ◮ 78 = ( 8 , 1 , 1 )+( 1 , 8 , 1 )+( 1 , 1 , 8 )+( 3 , 3 , 3 )+( 3 , 3 , 3 ) = B L + B C + B R + B + B (3) ◮ 27 = ( 3 , 3 , 1 ) + ( 1 , 3 , 3 ) + ( 3 , 1 , 3 ) = q γ q β γ + l α α + ˆ (4) β ◮ ˆ ˆ ˆ E − e − u c d c h c u r u g u b N 1 q γ ˆ q β ˆ ˆ l α E + α = γ = ˆ β = ν e d r d g d b u m d m h m N 2 e + ˆ ˆ ν e ˆ ˆ h r h g h b N 3 u y d y h y (5) Raymond Aschheim | Nonification
Introduction Group Theory 5 E 8 = SU ( 9 ) + 84 + 84 (6) ◮ The relationship between the E 8 lattice and the Simplex lattice, E 8 = 3 A 8 , is illustrated and has been extended to exceptional periodicity algebras [ 0 h , 0 i ] . ◮ exceptionally 84 = Λ 3 C 9 3-form and 84 = Λ 6 C 9 6-form in SU ( 9 ) [ 0 j ] . ◮ or generally 84 = 28 + 56 = Λ 2 C 8 � Λ 3 C 8 2-form and 3-form, and 84 = 56 + 28 = Λ 6 C 8 � Λ 5 C 8 6-form and 5-form in Cl ( 8 ) . REF: 0h Marrani, Alessio, and Piero Truini. “Exceptional Lie Algebras at the Very Foundations of Space and Time.” , p-Adic Numbers, Ultrametric Analysis and Applications, 2016, Vol. 8, No. 1, pp. 68–86. 0i Truini, Piero, Michael Rios, and Alessio Marrani. “The Magic Star of Exceptional Periodicity.” ArXiv:1711.07881 . 0j Ferrara, Sergio, Alessio Marrani, and Mario Trigiante. “Super-Ehlers in Any Dimension.” Journal of High Energy Physics 2012, no. 11 (November 2012). Raymond Aschheim | Nonification
A8 extension of the standard model A8 includes A2+A2+A2 6 ◮ A 8 includes A 2 L , A 2 C and A 2 R ◮ 27 = ( 3 , 3 , 1 ) + ( 1 , 3 , 3 ) + ( 3 , 1 , 3 ) (7) ◮ 27 breaks under SU ( 3 ) C xSU ( 2 ) L xU ( 1 ) Y as 27 = 2 ( 1 , 1 , 0 ) + ( 1 , 2 , 1 2 ) + ( 3 , 2 , − 1 3 ) + 2 ( 1 , 2 , − 1 2 ) (8) + 2 ( 3 , 1 , − 1 3 ) + ( 1 , 1 , 1 ) + ( 3 , 1 , − 2 3 ) + ( 3 , 2 , 1 6 ) Raymond Aschheim | Nonification
Tensor Network E8 as a tensor 7 ◮ 248D algebra E 8 is coded by G ± ∈ ℑ ( O ) , H 1 , ... H 7 , H + , H − ∈ Tr 0 ( M 3 8 ) ◮ Its action on J ∈ M 3 O ⊗ O is E 8 ( H 1 , ..., H − )( J ) = δ J = [ H + , ℜ ( J ) , H − ] − � 7 k = 1 e G + e k e G − H k · ℜ ( e k J ) 3 � T = T j i o i z i j j o j z j j i o i z i j j o j z j (9) j i , j j , o i , o j , z i , z j = 1 Or: J 1 J 1 J 1 o 1 o 1 o 1 z 1 z 1 z 1 1 2 3 1 2 3 1 2 3 T = , m j = , o j i j j = J 2 J 2 J 2 J m i o 2 o 2 o 2 z 2 z 2 z 2 (10) 1 2 3 1 2 3 1 2 3 J 3 J 3 J 3 o 3 o 3 o 3 z 3 z 3 z 3 1 2 3 1 2 3 1 2 3 Figure: Fibonacci spaced tensor network projection Raymond Aschheim | Nonification
Spin Network Fermionic quantum tetrahedron 8 ◮ We insert the standard model in a spinfoam by a fermionic quantum tetrahedron whose 4 vertices have SU(3) values coming from the 4 A2 of E8 Figure: Partition function with fermion ◮ Integral on cycles will reduce to SU(3) six-j symbols, when edges SU(2) are embedded in SU(3). Raymond Aschheim | Nonification
Magic star Jordan algebra 9 Magic star [ 0 k ] projected [ 0 l ] from Gosset polytope REF: 0k Truini, Piero. “Exceptional Lie Algebras, SU(3) and Jordan Pairs.” Pacific J. Math. 260, 227 (2012), (arXiv:1112.1258 [math-ph]). 0l Lisi, Garrett. “Elementary Particle Explorer.” http://deferentialgeometry.org/epe/. Raymond Aschheim | Nonification
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