Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry.
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- Εὐκλείδης Παπακωνσταντίνου
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1 Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry. Shane Farnsworth, joint work with Latham Boyle. Perimeter Institute for Theoretical Physics arxiv: , arxiv: , arxiv: November 02, 2014
2 Overview 1. Introduction: - The NCG Standard particle model
3 Overview 1. Introduction: - The NCG Standard particle model 2. Reformulation: - {m,g} {A,H,D} B 3. Applications: - Constraining D F - Fixing the Higgs Mass - Non-associative geometry 4. Open problems
4 Introduction: The standard model S SM = d 4 xψ /Dψ + ψφψ +F 2 +(Dφ) 2 m 2 φ φ2 + λ φ φ 4
5 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
6 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
7 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
8 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
9 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
10 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
11 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
12 Introduction: The Standard model. SU(3) SU(2) U(1) u R /3 d R 3 1 1/3 q L /6 ν R e R l L 1 2 1/2 h /2 A F = M 3 (C) H C π {H F,J F,γ F } D F S = S SM S = Tr[f (D/Λ)]+ < ψ Dψ >
13 Reformulation: WHY 1. Long obscure list of axioms
14 Reformulation: WHY 1. Long obscure list of axioms A is a real, associative, involutive, unital algebra D = D J is a unitary anti-linear operator on H γ = γ 1 = γ [a,jb J ] = 0 for all a,b A [[D,a],Jb J ] = 0 for all a,b A Ra = JL aj {D,γ} = 0...
15 Reformulation: WHY 1. Long obscure list of axioms A is a real, associative, involutive, unital algebra D = D J is a unitary anti-linear operator on H γ = γ 1 = γ [a,jb J ] = 0 for all a,b A [[D,a],Jb J ] = 0 for all a,b A Ra = JL aj {D,γ} = Too Many Higgs elds
16 Reformulation: WHY 1. Long obscure list of axioms A is a real, associative, involutive, unital algebra D = D J is a unitary anti-linear operator on H γ = γ 1 = γ [a,jb J ] = 0 for all a,b A [[D,a],Jb J ] = 0 for all a,b A Ra = JL aj {D,γ} = Too Many Higgs elds 3. Incorrect Higgs mass predicted
17 Reformulation: {A,H,J} B 0 B 0 = A H bb = (a +h)(a +h ) = aa +ah +h a + 0 b = (a +h) = a +Jh
18 Reformulation: {A,H,J} B 0 B 0 = A H bb = (a +h)(a +h ) = aa +ah +h a + 0 b = (a +h) = a +Jh
19 Reformulation: {A,H,J} B 0 B 0 = A H bb = (a +h)(a +h ) = aa +ah +h a + 0 b = (a +h) = a +Jh
20 Reformulation: {A,H,J} B 0 B 0 = A H bb = (a +h)(a +h ) = aa +ah +h a + 0 b = (a +h) = a +Jh
21 Reformulation: {A,H,J} B 0 Associativity: [b 1,b 0,b 2 ] = (b 1 b 0 )b 2 b 1 (b 0 b 2 ) = 0 (a 1 h)a 2 a 1 (ha 2 ) = [R a2,l a1 ]h = 0 [R a2,l a1 ] = 0 a i A,h H,b i B 0
22 Reformulation: {A,H,J} B 0 Associativity: [b 1,b 0,b 2 ] = (b 1 b 0 )b 2 b 1 (b 0 b 2 ) = 0 (a 1 h)a 2 a 1 (ha 2 ) = [R a2,l a1 ]h = 0 [R a2,l a1 ] = 0 a i A,h H,b i B 0 Involution: (b 1 b 0 ) = b 0 b 1 JL a1 h = R a 1 Jh R a1 = JL a 1 J a 1 A,h H,b i B 0
23 Reformulation: {A,H,D,J} B A ΩA = A Ω 1 A Ω 2 A... B = ΩA H Associativity: [ω 1,h,ω 2 ] = [R ω2,l ω1 ]h = 0 [[D,a],Jb J ] = 0 [[D,a],J[D,b ]J ] = 0...
24 Reformulation: {A,H,D,J} B A ΩA = A Ω 1 A Ω 2 A... B = ΩA H Associativity: [ω 1,h,ω 2 ] = [R ω2,l ω1 ] = 0 [[D,a],Jb J ] = 0 [[D,a],J[D,b ]J ] = 0...
25 Reformulation: {A,H,D,J} B A ΩA = A Ω 1 A Ω 2 A... B = ΩA H Associativity: [ω 1,h,ω 2 ] = [R ω2,l ω1 ] = 0 [[D,a],Jb J ] = 0 [[D,a],J[D,b ]J ] = 0...
26 Reformulation: {A,H,D,J} B A ΩA = A Ω 1 A Ω 2 A... B = ΩA H Associativity: [ω 1,h,ω 2 ] = [R ω2,l ω1 ] = 0 [[D,a],Jb J ] = 0 [[D,a],J[D,b ]J ] = 0...
27 Reformulation: {A,H,D,J} B A ΩA = A Ω 1 A Ω 2 A... B = ΩA H Associativity: [ω 1,h,ω 2 ] = [R ω2,l ω1 ] = 0 [[D,a],Jb J ] = 0 [[D,a],J[D,b] J ] = 0...
28 Applications:Too many Higgs Fields Constraints on D: D = D,{D,γ} = 0,DJ = ɛjd,[[d,a],jb J ] = 0 D 11 D 12 D 13 D 14 D 15 D 16 D 17 D 18 D 21 D 22 D 23 D 24 D 25 D 26 D 27 D 28 D 31 D 32 D 33 D 34 D 35 D 36 D 37 D 38 D 41 D 42 D 43 D 44 D 45 D 46 D 47 D 48 D F = D 51 D 52 D 53 D 54 D 55 D 56 D 57 D 58 D 61 D 62 D 63 D 64 D 65 D 66 D 67 D 68 D 71 D 72 D 73 D 74 D 75 D 76 D 77 D 78 D 81 D 82 D 83 D 84 D 85 D 86 D 87 D 88 Basis:{l R,q R,l L,q L,l R,q R,l L,q L }
29 Applications:Too many Higgs Fields Constraints on D: D = D,{D,γ} = 0,DJ = ɛjd,[[d,a],jb J ] = 0 D F = Basis:{l R,q R,l L,q L,l R,q R,l L,q L } 0 0 y l 0 m n y q n y l y q m n T yl T 0 n yq T y l y q 0 0
30 Too Many Higgs Fields [ω m,h,ω n ] = 0 [[D,a],J[D,b] J ] = 0 m = ( ) ( a b c, n = ) d b 0 0 0
31 Applications: Incorrect Higgs mass D F = 0 0 y l 0 m y q y l y q m yl T yq T y l y q 0 0 m = ( ) a 0, y l = 0 0 ( Yν α Y e β Y ν β Y e α ), y q = ( Yu α ) Y d β Y u β Y d α
32 Applications: Incorrect Higgs mass Traditional approach: symmetries = Aut(A) Symmetry Generators: δ = L a R a, where a = a A.
33 Applications: Incorrect Higgs mass Traditional approach: symmetries = Aut(A) Symmetry Generators: δ = L a R a, where a = a A. D D A = D +A+JAJ ( ) ( ) Yν α Y y l = e β Yν φ 1 Y e φ 2 Y ν β Y e α Y ν φ 2 Y e φ 1 ( ) ( ) Yu α Y y q = d β Yν φ 1 Y e φ 2 Y u β Y d α Y ν φ 2 Y e φ 1 m = ( ) ( ) a 0 a
34 Applications: Incorrect Higgs mass
35 Applications: Incorrect Higgs mass Fused algebra approach: symmetries = Aut(B) Symmetry Generators: δ = L a R a +t, where a = a A, [t,l x ] = [t,r x ] = [t,j] = [t,γ] = 0, and t = t End(H).
36 Applications: Incorrect Higgs mass Fused algebra approach: symmetries = Aut(B) Symmetry Generators: δ = L a R a +t, where a = a A, [t,l x ] = [t,r x ] = [t,j] = [t,γ] = 0, and t = t End(H). D D B = D +A+JAJ +T ( ) ( ) Yν α Y y l = e β Yν φ 1 Y e φ 2 Y ν β Y e α Y ν φ 2 Y e φ 1 ( ) ( ) Yu α Y y q = d β Yν φ 1 Y e φ 2 Y u β Y d α Y ν φ 2 Y e φ 1 m = ( ) ( ) a 0 YM σ(x)
37 Applications: Non-associative geometry [b 1,b 0,b 2 ] 0
38 Summary: {A,H,D} B
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