Mathematicians look at particle physics Matilde Marcolli Year of Mathematics talk July 2008
We do not do these things because they are easy. We do them because they are hard. (J.F.Kennedy Sept. 12, 1962) 1
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
Parameters of the Standard Model from experiments (particle accelerators) 3
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
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 0 0 2 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 + 1 2 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 0 2 1 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+ ) + 1 2 g2 s w A µ φ 0 (W µ + φ + Wµ φ+ ) + 1 2 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 ) + 2 3 (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 ) + 1 2 igm ( X+ X + φ 0 X X φ 0). 1
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
Geometrization of physics Kaluza Klein theory General Relativity: gravity = metric on 4-dim spacetime Electromagnetism: 5-dimensions Circle bundle over spacetime Gauge theories 6
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
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
Evolution of the Kaluza Klein idea, III Noncommutative geometry (Connes, 1980s) Product of spacetime by a noncommutative space Jackson Pollock Untitled N.3 9
What is a noncommutative space? Example: composition law for spectral lines Compose when target of one is source of next: G = groupoid 10
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
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
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
Example: Composite particles (baryons) Classification in terms of elementary particles (quarks) Mathematics: Lie group SU(3) Linear representations of Lie groups 14
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
Mathematics and reality The test of experiments (different models predict different Higgs masses) Large Hadron Collider (CERN) September 2008 (?) 16