Introduction: Noncommutative Geometry Models for Particle Physics and Cosmology. Matilde Marcolli

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1 Introduction: Noncommutative Geometry Models for Particle Physics and Cosmology Matilde Marcolli Ma148b: Winter 2016

2 Focus of the class: Geometrization of physics: general relativity, gauge theories, extra dimensions... Focus on: Standard Model of elementary particle physics, and its extensions (right handed neutrinos, supersymmetry, Pati-Salaam,...) Also focus on Matter coupled to gravity: curved backgrounds Gravity and cosmology: modified gravity models, cosmic topology, multifractal structures in cosmology, gravitational instantons Mathematical perspective: spectral and noncommutative geometry 1

3 What is a good geometric model of physics? Simplicity: difficult computations follow from simple principles Predictive power: recovers known physical properties and provides new insight on physics, from which new testable calculations Elegance: entia non sunt multiplicanda praeter necessitatem (Ockham s razor) 2

4 Two Standard Models: Standard Model of Elementary Particles Standard Cosmological Model For both theories looking for possible extensions: for particles right-handed-neutrino sector, supersymmetry, dark matter,...; for cosmology modified gravity, brane-cosmology, other dark matter and dark energy models, inflation scenarios,... Both theories depend on parameters that are not fixed by the theory Particle physics and cosmology interact (early universe models) What input from geometry on these models? 3

5 Basic Mathematical Toolkit Smooth Manifolds: M (with M = ) locally like R n : smooth atlas M = i U i, φ i : U i R n homeomorphisms, local coordinates x i = (x µ i ), on U i U j change of coordinates φ ij = φ j φ 1 i C -diffeomorphisms 4

6 Vector Bundles: E M vector bundle rank N over manifold of dim n projection π : E M M = i U i with φ i : U i R N π 1 (U i ) homeomorphisms with π φ i (x, v) = x transition functions: φ 1 j φ i : (U i U j ) R N (U i U j ) R N with φ 1 j φ i (x, v) = (x, φ ij (x)v) φ ij : U i U j GL N (C) satisfy cocyle property: φ ii (x) = id and φ ij (x)φ jk (x)φ ki (x) = id Sections: s Γ(U, E) open U M maps s : U E with π s(x) = x s(x) = φ i (x, s i (x)) and s i (x) = φ ij (x)s j (x) 5

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8 Tensors and differential forms as sections of vector bundles Tangent bundle T M (tangent vectors), cotangent bundle T M (1-forms) Vector field: section V = (v µ ) Γ(M, T M) metric tensor g µν symmetric tensor section of T M T M (p, q)-tensors: T = (T i 1...i p j 1...j q ) section in Γ(M, T M p T M q ) 1-form: section α = (α µ ) Γ(M, T M) k-form ω Γ(M, k (T M)) 7

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10 Connections Linear map : Γ(M, E) Γ(M, E T M) with Leibniz rule: (fs) = f (s) + s df for f C (M) and s Γ(M, E) local form: on U i R N section s i (x) = s α (x)e α (x) with e α (x) local frame with ω α βe α = e α s i = (ds α i + ωα βs β i )e α ω = (ω α β) is an N N-matrix of 1-forms: 1-form with values in End(E) in local coordinates x µ on U i : ω α β = ω α β µ dx µ V s contraction of s with vector field V 9

11 Curvature of a connection F Γ(M, End(E) Λ 2 (T M)) F (V, W )(s) = V W s W V s [V,W ] s F α β µν curvature 2-form: Ω = dω + ω ω F α β = dω α β + ω α γ ω γ β 10

12 Action Functionals Classical mechanics: equations of motion describe a path that is minimizing (or at least stationary) for the action functional... variational principle S(q(t)) = t1 t 0 Lagrangian L(q(t), q(t), t) δs = t1 t 0 ɛ L q + ɛ L q = t1 t 0 L(q(t), q(t), t) dt ɛ( L q d L dt q ) after integration by parts + boundary conditions ɛ(t 0 ) = ɛ(t 1 ) = 0 Euler Lagrange equations: δs = 0 L q d dt equations of motion L q = 0 11

13 Symmetries and conservation laws: Noether s theorem Example: Lagrangian invariant under translational symmetries in one direction q k p k = L q k L q k = 0 d L dt q k = 0 momentum conservation Important conceptual step in the geometrization of physics program: physical conserved quantity have geometric meaning (symmetries) 12

14 Action Functionals General Relativity: Einstein field equations are variational equation δs = 0 for Einstein Hilbert action 1 S(g µν ) = M 2κ R g d 4 x M = 4-dimensional Lorentzian manifold g µν = metric tensor signature (, +, +, +) g = det(g µν ) κ = 8πGc 4 G = gravitational constant, c = speed of light in vacuum R = Ricci scalar 13

15 Ricci scalar R: Riemannian curvature R ρ σµν R ρ σµν = µ Γ ρ νσ ν Γ ρ µσ + Γ ρ µλ Γλ νσ Γ ρ νλ Γλ µσ Levi-Civita connection (Christoffel symbols) Γ ρ νσ = 1 2 gρµ ( σ g µν + ν g µσ µ g νσ ) convention of summation over repeated indices for tensor calculus Ricci curvature tensor: contraction of Riemannian curvature R µν = R ρ µρν Ricci scalar: further contraction (trace) R = g µν R µν = R µ µ 14

16 Variation Variation of Riemannian curvature δr ρ σµν = µ δγ ρ νσ ν δγ ρ µσ + δγ ρ µλ Γλ νσ + Γ ρ µλ δγλ νσ δγ ρ νλ Γλ µσ Γ ρ νλ δγλ µσ δγ ρ µλ is a tensor (difference of connections) Rewrite variation δr ρ σµν as δr ρ σµν = µ (δγ ρ νσ) ν (δγ ρ µσ) with covariant derivative λ (δγ ρ νµ) = λ (δγ ρ νµ) + Γ ρ σλ δγσ νµ Γ σ νλ δγρ σµ after contracting indices: Ricci tensor δr µν = ρ (δγ ρ νµ) ν (δγ ρ ρµ) Ricci scalar variation δr = R µν δg µν +g µν δr µν δr = R µν δg µν + σ (g µν δγ σ νµ g µσ δγ ρ ρµ) 15

17 so get variation δr δg µν = R µν also have total derivative: g µ A µ = µ ( ga µ ) so M σ(g µν δγ σ νµ g µσ δγ ρ ρµ) g = 0 Variation of determinant g = det(g µν ): Jacobi formula d dt log det A(t) = Tr(A 1 (t) d dt A(t)) for invertible matrix A, so get δg = δ det(g µν ) = g g µν δg µν δ g = 1 2 g δg = 1 2 gg µν δg µν δg µν = g µσ (δg σλ )g λν for inverse matrix δ g δg µν = 1 2 g µν g 16

18 Variation equation for Einstein Hilbert action δs = (R µν 1 M 2 g µνr) δg µν g d 4 x stationary equation δs = 0 R µν 1 2 g µνr = 0 Einstein equation in vacuum Other variants: Gravity coupled to matter L M Lagrangian S = ( 1 M 2κ R + L M) g d 4 x energy momentum tensor T µν = 2κ g δ( gl M ) δg µν gives variational equations δs = 0: R µν 1 2 g µνr = 8πG c 4 T µν = matter 17

19 Action with cosmological constant: S = ( 1 M 2κ (R 2Λ) + L M) g d 4 x gives variational equation δs = 0: R µν 1 2 g µνr + Λg µν = 8πG c 4 T µν For more details see: Sean Carroll, Spacetime and Geometry, Addison Wesley, 2004 Note: Euclidean gravity when g µν Riemannian: signature (+, +, +, +) used in quantum gravity and quantum cosmology Not all geometries admit Wick rotations between Riemannian and Loretzian signature (there are, for example, topological obstructions) 18

20 Action Functionals Yang Mills: gauge theories SU(N) Hermitian vector bundle E Lie algebra su(n) generators T a = (T a α β) Tr(T a T b ) = 1 2 δab, [T a, T b ] = if abc T c structure constants f abc Connections: ω α β µ = A a µ T a α β gauge potentials A a µ Curvature: Fµν a = µ A a ν ν A a µ + gf abc A b µ A c ν covariant derivative µ = µ igt a A a µ [ µ, ν ] = igt a Fµν a 19

21 Yang Mills Lagrangian: L(A) = 1 2 Tr(F 2 ) = 1 4 F a µν F a µν Yang Mills action: S(A) = M L(A) g d 4 x Equations of motion: µ F a µν + gf abc A b µ F c µν = 0 or equivalently ( µ F µν ) a = 0 where F µν = T a F a µν case with N = 1: electromagnetism U(1) equations give Maxwell s equations in vacuum unlike Maxwell in general Yang-Mills equations are non-linear 20

22 Lagrangian formalism in perturbative QFT For simplicity consider example of scalar field theory in dim D L(φ) = 1 2 ( φ)2 m2 2 φ2 L int (φ) Lorentzian signature with ( φ) 2 = g µν µ φ ν φ; interaction part L int (φ) polynomial; action: S(φ) = L(φ) gd D x M Classical solutions: stationary points of action; quantum case: sum over all confiurations weights by the action: oscillatory integral around classical Z = e is(φ) D[φ] Observables: O(φ) function of the classical field; expectation value: O = Z 1 O(φ)e is(φ) D[φ] This -dim integral not well defined: replace by a formal series expansion (perturbative QFT) 21

23 Euclidean QFT: metric g µν Riemannian L(φ) = 1 2 ( φ)2 + m2 2 φ2 + L int (φ) action S(φ) = M L(φ) gd D x Z = e S(φ) D[φ] O = Z 1 O(φ)e S(φ) D[φ] free-field part and interaction part S(φ) = S 0 (φ) + S int (φ) interaction part as perturbation: free-field as Gaussian integral (quadratic form) integration of polynomials under Gaussians (repeated integration by parts): bookkeeping of terms, labelled by graphs (Feynman graphs) 22

24 Perturbative expansion: Feynman rules and Feynman diagrams S eff (φ) = S 0 (φ)+ Γ Γ(φ) = 1 N! i p i=0 U(Γ(p 1,..., p N )) = l = b 1 (Γ) loops Γ(φ) #Aut(Γ) (1PI graphs) ˆφ(p 1 ) ˆφ(p N )U(Γ(p 1,..., p N ))dp 1 dp N I Γ (k 1,..., k l, p 1,..., p N )d D k 1 d D k l Renormalization Problem: Integrals U(Γ(p 1,..., p N )) often divergent: Renormalizable theory: finitely many counterterms (expressed also as perturbative series) can be added to the Lagrangian to simultaneously remove all divergences from all Feynman integrals in the expansion 23

25 The problem with gravity: Gravity is not a renormalizable theory! Effective field theory, up to some energy scale below Planck scale (beyond, expect a different theory, like string theory) Observation: some forms of modified gravity (higher derivatives) are renormalizable (but unitarity can fail...) The model we will consider has some higher derivative terms in the gravity sector 24

26 Standard Model of Elementary Particles Forces: electromagnetic, weak, strong (gauge bosons: photon, W ±, Z, gluon) Fermions 3 generations: leptons (e,µ,τ and neutrinos) and quarks (up/down, charm/strange, top/bottom) Higgs boson 25

27 Parameters of the Standard Model Minimal Standard Model: 19 3 coupling constants 6 quark masses, 3 mixing angles, 1 complex phase 3 charged lepton masses 1 QCD vacuum angle 1 Higgs vacuum expectation value; 1 Higgs mass Extensions for neutrino mixing: 37 3 neutrino masses 3 lepton mixing angles, 1 complex phase 11 Majorana mass matrix parameters Constraints and relations? Values and constraints from experiments: but a priori theoretical reasons? Geometric space... Note: parameters run with energy scale (renormalization group flow) so relations at certain scales versus relations at all scales 26

28 Experimental values of masses 27

29 Constraints on mixing angles (CKM matrix) 28

30 Standard Model Langrangian L SM = 1 2 νgµ a ν gµ a g s f abc µ gνg a µg b ν c 1 4 g2 s f 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 M 2 Z 2c µz 0 2 µ 0 1 w 2 µa ν µ A ν igc w ( ν Zµ(W 0 µ + Wν W ν + Wµ ) Zν 0 (W µ + ν Wµ Wµ ν W µ + ) + Z 0 µ(w + ν ν W µ W ν ν W + µ )) igs w ( ν A µ (W + µ W ν W + ν W µ ) A ν (W µ + ν Wµ Wµ ν W µ + ) + A µ (W ν + ν Wµ Wν ν W µ + )) 1 2 g2 W µ + Wµ W ν + Wν g2 W µ + Wν W µ + Wν + g 2 c 2 w(zµw 0 µ + Zν 0 Wν 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 0 (W µ + Wν W ν + Wµ ) 2A µ ZµW 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 ( ) β 2M 2 h + 2M g 2 g H (H2 + φ 0 φ 0 + 2φ + φ ) + 2M 4 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 W µ H 1 2 g M c 2 w Z 0 µz 0 µh 1 2 ig ( W + µ (φ 0 µ φ φ µ φ 0 ) W µ (φ 0 µ φ + φ + µ φ 0 ) ) g ( W + µ (H µ φ φ µ H) + W µ (H µ φ + φ + µ H) ) g 1 c w (Z 0 µ(h µ φ 0 φ 0 µ H) + M ( 1 c w Z 0 µ µ φ 0 + W + µ µ φ + W µ µ φ + ) ig s2 w cw MZ 0 µ(w + µ φ W µ φ + ) + igs w MA µ (W + µ φ W µ φ + ) ig 1 2c2 w 2c w Z 0 µ(φ + µ φ φ µ φ + ) + igs w A µ (φ + µ φ φ µ φ + ) 1 ( H 2 + (φ 0 ) 2 + 2φ + φ ) 4 g2 W µ + Wµ ( 1 8 g2 1 Z c µz µ H 2 + (φ 0 ) 2 + 2(2s 2 w 1) 2 φ + φ ) 1 w 2 g2 s 2 w cw Zµφ 0 0 (W µ + φ + W µ φ + ) 1 2 ig2 s 2 w cw Z 0 µh(w + µ φ W µ φ + ) g2 s w A µ φ 0 (W + µ φ + W µ φ + ) ig2 s w A µ H(W + µ φ W µ φ + ) g 2 s w cw (2c 2 w 1)Z 0 µa µ φ + φ g 2 s 2 wa µ A µ φ + φ ig s λ a ij ( qσ i γµ q σ j )ga µ ē λ (γ + m λ e )e λ ν λ (γ + m λ ν)ν λ ū λ j (γ + mλ u)u λ j d λ j (γ + mλ d )dλ j + igs w A µ ( (ē λ γ µ e λ ) (ūλ j γµ u λ j ) 1 3 ( d λ j γµ d λ j )) + ig 4c w Z 0 µ{( ν λ γ µ (1 + 29

31 γ 5 )ν λ )+(ē λ γ µ (4s 2 w 1 γ 5 )e λ )+( d λ j γµ ( 4 3 s2 w 1 γ 5 )d λ j )+(ūλ j γµ (1 8 3 s2 w+ γ 5 )u λ j )} + ig 2 W + 2 µ ig 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 )) + ( (ē κ U lep κλ γµ (1 + γ 5 )ν λ ) + ( d κ j C κλ γµ (1 + γ 5 )u λ j )) + ( ig 2M 2 φ+ ( m κ e( ν λ U lep λκ(1 γ 5 )e κ ) + m λ ν( ν λ U lep λκ(1 + γ 5 )e κ) + ig 2M 2 φ m λ e (ē λ U lep λκ (1 + γ5 )ν κ ) m κ ν(ē λ U lep λκ (1 γ5 )ν κ) g 2 m λ ν M H( νλ ν λ ) g 2 m λ e M H(ēλ e λ ) + ig 2 m λ ν M φ0 ( ν λ γ 5 ν λ ) ig ν λ M R λκ (1 γ 5)ˆν κ 1 4 ν λ M R λκ (1 γ 5)ˆν κ + m λ e M φ0 (ē λ γ 5 e λ ) ( ) ig 2M 2 φ+ ( m κ d (ūλ j C λκ(1 γ 5 )d κ j ) + mλ u(ū λ j C λκ(1 + γ 5 )d κ j + ) ig 2M 2 φ m 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 H( d λ j dλ j ) + ig m λ u 2 M φ0 (ū λ j γ5 u λ j ) ig m λ d 2 M φ0 ( d λ j γ5 d λ j ) + Ḡ a 2 G a + g s f abc µ Ḡ a G b g c µ + X + ( 2 M 2 )X + + X ( 2 M 2 )X + X 0 ( 2 M 2 )X 0 + Ȳ 2 Y + igc c 2 w W µ + ( µ X 0 X w µ X + X 0 )+igs w W µ + ( µ Ȳ X µ X + Y ) + igc w Wµ ( µ X X 0 µ X 0 X + )+igs w Wµ ( µ X Y µ Ȳ X + ) + igc w Zµ( 0 µ X + X + µ X ( X )+igs w A µ ( µ X + X + ) µ X X ) 1 2 gm X + X + H + X X H + 1 X 0 X 0 H + c 2 w ( 1 2c 2 ( w 2c w igm X + X 0 φ + X X 0 φ ) + 1 2c w igm X 0 X φ + X 0 X + φ ) ( ( +igms w X 0 X φ + X 0 X + φ ) igm X + X + φ 0 X X φ 0).

32 Fundamental ideas of NCG models Derive the full Lagrangian from a simple geometric input by calculation Machine that inputs a (simple) geometry and produces a uniquely associated Lagrangian Very constrained: only certain theories can be obtained (only certain extensions of the minimal standard model) Simple action functional (spectral action) that reduces to SM + gravity in asymptotic expansion in energy scale Effective field theory: preferred energy scale (at unification energy) 30

33 The role of gravity: What if other forces were just gravity but seen from the perspective of a different geometry? This idea occurs in physics in different forms: holography AdS/CFT has field theory on boundary equivalent to gravity on bulk in NCG models action functional for gravity (spectral action) on an almost-commutative geometry gives gravity + SM on spacetime manifold... What is Noncommutative Geometry? 31