Mathematicians look at particle physics. Matilde Marcolli

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1 Mathematicians look at particle physics Matilde Marcolli Year of Mathematics talk July 2008

2 We do not do these things because they are easy. We do them because they are hard. (J.F.Kennedy Sept. 12, 1962) 1

3 Elementary particle physics Constituents of all known matters and forces (except gravity) Is there new physics beyond? (massive neutrinos; supersymmetry? dark matter? dark energy?) Unification with gravity? (loops? strings? branes? noncommutative spaces?) 2

(massive neutrinos; supersymmetry? dark matter? dark energy?

4 Parameters of the Standard Model from experiments (particle accelerators) 3

5 Particle accelerators are giant microscopes Higher energies = smaller scales Theory: perform calculations that predict results of events that can be seen in accelerators Formula: Standard Model Lagrangian 4

calculations that predict results of events that

6 L SM = 1 2 νgµ a ν gµ a g s f abc µ gνg a µg b ν c 1 4 g2 sf abc f ade gµg b νg c µg d ν e ν W µ + ν Wµ M 2 W µ + Wµ 1 2 νzµ 0 ν Zµ 0 1 2cwM 2 ZµZ µ 1 2 µa ν µ A ν igc w ( ν Zµ(W 0 µ + Wν W ν + W µ ) Z0 ν (W µ + νwµ W µ νw µ + ) + Z0 µ (W ν + νwµ W ν νw µ + )) igs w ( ν A µ (W µ + W ν W ν + W µ ) A ν(w µ + νwµ W µ νw µ + ) + A µ(w ν + νwµ Wν νw µ + )) 1 2 g2 W µ + W µ W ν + W ν g2 W µ + W ν W µ + W ν + g2 c 2 w (Z0 µ W µ + Z0 ν W ν ZµZ 0 µw 0 ν + Wν ) + g 2 s 2 w(a µ W µ + A ν Wν A µ A µ W ν + Wν ) + g 2 s w c w (A µ Zν(W 0 µ + Wν W ν + Wµ ) 2A µ Z ( µw 0 ν + Wν ) 1 2 µh µ H 2M 2 α h H 2 ) µ φ + µ φ 1 2 µφ 0 µ φ 0 2M β 2 h + 2M H + 1 g 2 g 2 (H2 + φ 0 φ 0 + 2φ + φ ) + 2M4 α g 2 h gα h M (H 3 + Hφ 0 φ 0 + 2Hφ + φ ) 1 8 g2 α h (H 4 + (φ 0 ) 4 + 4(φ + φ ) 2 + 4(φ 0 ) 2 φ + φ + 4H 2 φ + φ + 2(φ 0 ) 2 H 2 ) gmw µ + Wµ H 1 g M 2 cwzµz 0 µh ig ( W + 2 µ (φ 0 µ φ φ µ φ 0 ) Wµ (φ 0 µ φ + φ + µ φ 0 ) ) + 1 g ( W + 2 µ (H µφ φ µ H) + Wµ (H µφ + φ + µ H) ) + 1g 1 2 c w (Zµ 0 (H µφ 0 φ 0 µ H) + M ( 1 c w Zµ 0 µφ 0 +W µ + µφ +Wµ µφ + ) ig s2 w cw MZµ 0 (W µ + φ Wµ φ+ )+igs w MA µ (W µ + φ Wµ φ+ ) ig 1 2c2 w 2c w Zµ 0 (φ+ µ φ φ µ φ + ) + igs w A µ (φ + µ φ φ µ φ + ) 1 4 g2 W µ + W µ (H2 + (φ 0 ) 2 + 2φ + φ ) 1 8 g2 1 cwz 0 2 µ Z0 µ (H2 + (φ 0 ) 2 + 2(2s 2 w 1)2 φ + φ ) 1 2 g2 s 2 w cw Zµ 0 φ0 (W µ + φ + Wµ φ+ ) 1 2 ig2 s 2 w cw Zµ 0 H(W µ + φ Wµ φ+ ) g2 s w A µ φ 0 (W µ + φ + Wµ φ+ ) ig2 s w A µ H(W µ + φ Wµ φ+ ) g 2 sw c w (2c 2 w 1)Z0 µ A µφ + φ g 2 s 2 wa µ A µ φ + φ + 1 ig 2 s λ a ij( q i σ γ µ qj σ )gµ a ē λ (γ + m λ e)e λ ν λ (γ + m λ ν)ν λ ū λ j (γ + m λ u )uλ j d λ j (γ + mλ d )dλ j + igs ( wa µ (ē λ γ µ e λ ) (ūλ j γµ u λ j ) 1( d λ 3 j γµ d λ j )) + ig 4c w Zµ{( ν 0 λ γ µ (1 + γ 5 )ν λ ) + (ē λ γ µ (4s 2 w 1 γ 5 )e λ ) + ( d λ jγ µ ( 4 3 s2 w 1 γ 5 )d λ j ) + (ū λ j γµ (1 8 3 s2 w + γ5 )u λ ig j )} + 2 W ( + 2 µ ( ν λ γ µ (1 + γ 5 )U lep λκe κ ) + (ū λ j γµ (1 + γ 5 )C λκ d κ j )) + ( ig (ē κ U lep κλγ µ (1 + γ 5 )ν λ ) + ( d ) κ jc κλ γµ (1 + γ 5 )u λ j) + ( ig 2M 2 φ 2 W 2 µ ig ( 2M 2 φ+ m κ e ( νλ U lep λκ(1 γ 5 )e κ ) + m λ ν ( νλ U lep λκ(1 + γ 5 )e κ) + ) m λ e(ē λ U lep λκ(1 + γ 5 )ν κ ) m κ ν(ē λ U lep λκ(1 γ 5 )ν κ g 2 m λ ν M H( νλ ν λ ) g m λ e 2 M H(ēλ e λ ) + ig m λ ν 2 M φ0 ( ν λ γ 5 ν λ ) ig m λ e 2 M φ0 (ē λ γ 5 e λ ) 1 ν 4 λ Mλκ R (1 γ 5)ˆν κ 1 ν 4 λ Mλκ R (1 γ 5)ˆν κ + ( ) ig 2M 2 φ+ m κ d (ūλ jc λκ (1 γ 5 )d κ j) + m λ u(ū λ jc λκ (1 + γ 5 )d κ j + ig (m 2M 2 φ λ d ( d λ j C λκ (1 + γ5 )u κ j ) mκ u ( d ) λ j C λκ (1 γ5 )u κ j g m λ u 2 M H(ūλ j uλ j ) g m λ d 2 m λ u H( d λ M j dλ j ) + ig 2 M φ0 (ū λ j γ5 u λ j ) ig 2 X + ( 2 M 2 )X + + X ( 2 M 2 )X + X 0 ( 2 M2 c 2 w m λ d M φ0 ( d λ j γ5 d λ j ) + Ḡa 2 G a + g s f abc µ Ḡ a G b gµ c + )X 0 + Ȳ 2 Y + igc w W µ + ( X0 µ X X+ µ X 0 )+igs w W µ + ( µȳ X X+ µ Y ) + igc w Wµ ( X µ X 0 X0 µ X + )+igs w Wµ ( X µ Y µ Ȳ X + ) + igc w Zµ( 0 X+ µ X + X µ X )+igs w A µ ( X+ µ X + µ X X ) 1 2 gm ( X+ X + H + X X H + 1 X 0 X 0 H ) + 1 2c2 w 2c w igm ( X+ X 0 φ + X X 0 φ ) + c 2 w 1 2c w igm ( X0 X φ + X 0 X + φ ) ( +igms X0 w X φ + X 0 X + φ ) igm ( X+ X + φ 0 X X φ 0). 1

+ W ν W µ + W ν + g2 c 2 w (Z0 µ W µ + Z0 ν W ν ZµZ 0 µw 0 ν + Wν ) + g 2 s 2 w(a µ W µ + A ν Wν A µ A µ W ν + Wν ) + g 2 s w c w (A µ Zν(W 0 µ + Wν W ν + Wµ ) 2A µ Z ( µw 0 ν + Wν ) 1 2 µh µ H 2M 2

7 We have a formula: does it mean we understand? The task of mathematics: Is there a simple principle behind? Does the formula follow? What does it mean? Geometry: guiding principle for tackling complexity Collision II Dawn N. Meson, San Francisco artist 5

8 Geometrization of physics Kaluza Klein theory General Relativity: gravity = metric on 4-dim spacetime Electromagnetism: 5-dimensions Circle bundle over spacetime Gauge theories 6

9 Evolution of the Kaluza Klein idea, I Gauge theories: vector bundles connections and curvatures (gauge potentials, force fields) sections (matter particles/fields) bundle symmetries (gauge symmetries) 7

10 Evolution of the Kaluza Klein idea, II String theory: fibration of Calabi-Yau manifolds over 4-dim spacetime Kaluza-Klein (Invisible Architecture III) Dawn N.Meson 8

11 Evolution of the Kaluza Klein idea, III Noncommutative geometry (Connes, 1980s) Product of spacetime by a noncommutative space Jackson Pollock Untitled N.3 9

12 What is a noncommutative space? Example: composition law for spectral lines Compose when target of one is source of next: G = groupoid 10

13 First instance of noncommutive variables in Quantum Mechanics ( ) ( ) ( ) a b u v au + bx av + by = c d x y cu + dx cv + dy ( ) ( ) ( ) au + cv bu + dv u v a b = ax + cy bx + dy x y c d Observables in quantum mechanics usually don t commute Uncertainty principle NCG: Geometry of Quantum Mechanics 11

dv u v a b = ax + cy bx + dy x y c d Observables in quantum mechanics

14 The main idea: There are more dimensions than the 4-dimensions of space and time The extra dimensions account for forces and particles and their interactions (internal symmetries) 12

15 But when is a mathematical model a good model of the physical world? Simplicity: difficult computations follow from simple principles Predictive power: new insight on physics, new testable calculations Elegance: entia non sunt multiplicanda praeter necessitatem (Ockham s razor) More than one mathematical model may be needed to explain different aspects of the same physical phenomenon 13

physics, new testable calculations Elegance: entia non sunt multiplicanda praeter necessitatem

16 Example: Composite particles (baryons) Classification in terms of elementary particles (quarks) Mathematics: Lie group SU(3) Linear representations of Lie groups 14

17 Example: Noncommutative Geometry: Standard Model Lagrangian computed from simple input Matrix algebras and quaternions A = C H M 3 (C) Predictions: Higgs mass, mass relation Mathematics: Spectral triples, spectral action 15

and quaternions A = C H M 3 (C) Predictions: Higgs

18 Mathematics and reality The test of experiments (different models predict different Higgs masses) Large Hadron Collider (CERN) September 2008 (?) 16

Theoretical Particle Physics FYTN04: Oral Exam Questions, version ht15

Theoretical Particle Physics FYTN04: Oral Exam Questions, version ht15 Examples of The questions are roughly ordered by chapter but are often connected across the different chapters. Ordering is as in