Duality between shapes and spectra The music of shapes

Transcription

1 Duality between shapes and spectra The music of shapes A. Connes How to specify our location in space. Vibrations of shapes. Hearing simple shapes. Isospectral shapes and CKM invariant. Towards a musical shape. 1

2 Where are we? It gives the position of the sun wrt 14 pulsars and the center of the galaxy. 2

3 Two questions Find complete invariants of geometric spaces, of shapes How can we invariantly specify a point in a geometric space? 3

4 It is well known since a famous one page paper of John Milnor that the spectrum of operators, such as the Laplacian, does not suffice to characterize a compact Riemannian space. But it turns out that the missing information is encoded by the relative position of two abelian algebras of operators in Hilbert space. Due to a theorem of von Neumann the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner, independent of the geometry one starts with. Its relative position with respect to the other abelian algebra given by all functions of the Laplacian suffices to recover the full geometry, provided one knows the spectrum of the Laplacian. For some reason which has to do with the inverse problem, it is better to work with the Dirac operator. 4

5 Vibrations du disque (3, 2) 5

6 Vibrations of the disk (5, 3) 6

7 Eigenfrequencies of the disk

8 Spectrum of disk , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

9 Eigenfrequencies of the disk

10 High frequencies

11 The square 11

12 Vibrations of the square 12

13 Spectrum of square { π 2 n 2 + m 2 n, m N, nm 0} 13

14 Eigenfrequencies of square

15 High frequencies

16

17 The sphere 16

18 Spectrum of sphere { j(j + 1) j 0} 17

19 Eigenfrequencies of sphere

20 High frequencies of sphere

21 The unitary (CKM) invariant of Riemannian manifolds The invariants are : The spectrum Spec(D). The relative spectrum Spec N (M) (N = {f(d)}). 20

22 Flavor changing weak decays ig 2 2 W µ + ig 2 2 W µ ) + (ūλ j γ µ (1 + γ 5 )C λκ d κ j ( d κ j C κλ γµ (1 + γ 5 )u λ ) j 21

23 Cabibbo-Kobayashi-Maskawa C = [ cosθc sinθ c sinθ c cosθ c ] C = C ud C us C ub C cd C cs C cb C td C ts C tb C = c 1 s 1 c 3 s 1 s 3 s 1 c 2 c 1 c 2 c 3 s 2 s 3 e δ c 1 c 2 s 3 + s 2 c 3 e δ s 1 s 2 c 1 s 2 c 3 + c 2 s 3 e δ c 1 s 2 s 3 c 2 c 3 e δ c i = cos θ i, s i = sin θ i, and e δ = exp(iδ) 22

24 Points Once we know the spectrum Λ of D, the missing information is contained in Spec N (M). It should be interpreted as giving the probability for correlations between the possible frequencies, while a point of the geometric space X can be thought of as a correlation, i.e. a specific positive hermitian matrix ρ λκ (up to scale) in the support of ν. 23

25 Redshift Thanks to great telescopes like Hubble we now have an eye that allows us to look out in space, but what strikes me is that all the information we get is of spectral nature. It is for instance thanks to the red-shift (which is a rescaling factor of spectra and can be as large as 10 and possibly a thousand) that we can estimate cosmic distances. 24

26 Redshift Radiations emitted in ultraviolet (10 14 cycles per second) are observed in infrared (10 12 cycles per second) : factor 100 (!) 25

27 It is rather striking also that our faith in outer space is based on the strong correlations that exist between different frequencies, as encoded by the matrix g λµ, so that the picture in infrared of the milky way is not that different from its visible light counterpart, which can be seen with a bare eye on a clear night. 26

28 27

29 Towards a musical shape {q n n N}, q = /12 = , 3 1/19 =

30 Dimension =

31 The quantum sphere S 2 q (Poddles, Brain and Landi) { qj q j j N} avec multiplicité 2j + 1 q q 1 30

32 Milnor, John (1964), Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences of the United States of America 51 Kac, Mark (1966), Can one hear the shape of a drum?, American Mathematical Monthly 73 (4, part 2) : 1 23 Sunada, T. (1985), Riemannian coverings and isospectral manifolds, Ann. Of Math. (2) 121 (1) : Gordon, C. ; Webb, D. ; Wolpert, S. (1992), Isospectral plane domains and surfaces via Riemannian orbifolds, Inventiones mathematicae 110 (1) : 1 2 Zelditch, S. (2000), Spectral determination of analytic bi-axisymmetric plane domains, Geometric and Functional Analysis 10 (3) :

33 Space-Time Our knowledge of spacetime is described by two existing theories : General Relativity The Standard Model of particle physics Curved Space, gravitational potential g µν ds 2 = g µν dx µ dx ν Action principle S E [ g µν ] = 1 G M r g d 4 x S = S E + S SM 32

34 33

35 Standard Model L SM = 1 2 νg a µ νg a µ g sf abc µ g a ν gb µ gc ν 1 4 g2 s f abc f ade g b µ gc ν gd µ ge ν 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 Z 0 µ(w + µ W ν W + ν W µ ) Z 0 ν (W + µ ν W µ W µ 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 W µ W µ ν W + µ ) + A µ (W + ν ν W µ W ν ν W + µ )) 1 2 g2 W + µ W µ W + ν W ν g2 W + µ W ν W + µ W ν +g 2 c 2 w(z 0 µw + µ Z 0 ν W ν Z 0 µz 0 µw + ν W ν )+g 2 s 2 w(a µ W + µ A ν W ν A µ A µ W + ν W ν ) +g 2 s w c w (A µ Z 0 ν (W + µ W ν W + ν W µ ) 2A µ Z 0 µw + ν W ν ) 1 2 µh µ H 1 2 m2 h H2 µ ϕ + µ ϕ M 2 ϕ + ϕ 1 2 µϕ 0 µ ϕ 0 1 β h ( 2M 2 2c 2 w g 2 + 2M g H (H2 + ϕ 0 ϕ 0 + 2ϕ + ϕ ) M 2 ϕ 0 ϕ 0 ) + 2M 4 g α 2 h 34

36 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 gm c 2 w Z 0 µ Z0 µ H 1 2 ig ( W + µ (ϕ0 µ ϕ ϕ µ ϕ 0 ) W µ (ϕ0 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) ig s2 w c w 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 4 g2 W + µ W µ 1 8 g2 1 c 2 w Z 0 µz 0 µ ( H 2 + (ϕ 0 ) 2 + 2ϕ + ϕ ) ( H 2 + (ϕ 0 ) 2 + 2(2s 2 w 1) 2 ϕ + ϕ ) 1 2 g2s2 w c w Z 0 µϕ 0 (W + µ ϕ + W µ ϕ + ) 1 2 ig2s2 w c w 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 2s w c w (2c 2 w 1)Z 0 µa µ ϕ + ϕ g 2 s 2 wa µ A µ ϕ + ϕ 35

37 g ig sλ a ij ( qσ i γµ q σ j )ga µ ē λ (γ + m λ e )e λ ν λ γ ν λ ū λ j (γ + mλ u)u λ 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 3 ( d λ j γµ d λ j ) ) + ig 4c w Z 0 µ{( ν λ γ µ (1 + 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 2 W + µ + ig 2 2 W µ + ig 2 m λ e 2 M ( ( ν λ γ µ (1 + γ 5 )e λ ) + (ū λ j γµ (1 + γ 5 )C λκ d κ j )) ( (ē λ γ µ (1 + γ 5 )ν λ ) + ( d κ j C κλ γµ (1 + γ 5 )u λ j )) ( ϕ + ( ν λ (1 γ 5 )e λ ) + ϕ (ē λ (1 + γ 5 )ν λ ) ) g 2 m λ e M ( H(ē λ e λ ) + iϕ 0 (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 m λ u 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 ) ) ) 36

38 Symmetries of S = S E + S SM Question : G = U(1) SU(2) SU(3) G = Map(M, G) Diff(M) Is there a space X whose group of diffeomorphisms is directly of that form? Answer : No : for ordinary commutative spaces Yes : for noncommutative spaces 37

39 Let us consider the simplest example A = C (M, M n (C)) = C (M) M n (C) Algebra of n n matrices of smooth functions on manifold M. The group Inn(A) of inner automorphisms is locally isomorphic to the group G of smooth maps from M to the small gauge group SU(n) 1 Inn(A) Aut(A) Out(A) 1 becomes identical to 1 Map(M, G) G Diff(M) 1. 38

40 We have shown that the study of pure gravity on this space yields Einstein gravity on M minimally coupled with Yang-Mills theory for the gauge group SU(n). The Yang-Mills gauge potential appears as the inner part of the metric, in the same way as the group of gauge transformations (for the gauge group SU(n)) appears as the group of inner diffeomorphisms. 39

41 What is a metric in spectral geometry d(a, B) = Inf γ g µ ν dx µ dx ν 40

42 Dirac s square root of the Laplacian 41

43 (A, H, D), ds = D 1, d(a, B) = Sup { f(a) f(b) ; f A, [D, f] 1 } Meter Wave length (Krypton (1967) spectrum of 86Kr then Caesium (1984) hyperfine levels of C133) 42

44 Kinematic relations The algebra A is commutative. The commutator [[D, a], b] = 0 for a, b A. The following holds, for some a α j A : α a α 0 [[D, aα 1 ], [D, aα 2 ],..., [D, aα n ]] = 1 [T 1, T 2,..., T n ] = σ ϵ(σ) T σ(1) T σ(2) T σ(n) Even case, 1 γ. 43

45 The meaning of the second kinematic condition is that the operator D is differential of order one. The meaning of the third kinematic condition is that the determinant of the metric g µν does not vanish, and in fact more precisely that its square root multiplied by the volume form α a α 0 daα 1 daα 2 daα n gives 1. 44

46 Spectral conditions The n-th characteristic value of the resolvent of D is O(n 1/p ). Regularity for the geodesic flow e it D. Absolute continuity. 45

47 Let (A, H, D) be a spectral geometry fulfilling the kinematic relations and the spectral conditions. Assume that the multiplicity of the action of A in H is 2 p/2. Then there exists a smooth oriented compact (spin c ) manifold M such that A = C (M). Moreover D is a Dirac operator. The Einstein action is obtained by the spectral action Tr(f(D/Λ)) which counts the number of eigenvalues of D of size < Λ. 46

48 The restriction to spin manifolds is obtained by requiring a real structure i.e. an antilinear unitary operator J acting in H which plays the same role and has the same algebraic properties as the charge conjugation operator in physics. 47

49 The following further relations hold for D, J and γ J 2 = ε, DJ = ε JD, J γ = ε γj, Dγ = γd The values of the three signs ε, ε, ε depend only, in the classical case of spin manifolds, upon the value of the dimension n modulo 8 and are given in the following table : n ε ε ε

50 In the classical case of spin manifolds there is thus a relation between the metric (or spectral) dimension given by the rate of growth of the spectrum of D and the integer modulo 8 which appears in the above table. For more general spaces however the two notions of dimension (the dimension modulo 8 is called the KO-dimension because of its origin in K-theory) become independent since there are spaces F of metric dimension 0 but of arbitrary KO-dimension. 49

51 Starting with an ordinary spin geometry M of dimension n and taking the product M F, one obtains a space whose metric dimension is still n but whose KO-dimension is the sum of n with the KO-dimension of F. As it turns out the Standard Model with neutrino mixing favors the shift of dimension from the 4 of our familiar space-time picture to 10 = = 2 modulo 8. 50

52 The shift from 4 to 10 is a recurrent idea in string theory compactifications, where the 6 is the dimension of the Calabi-Yau manifold used to compactify. The difference of this approach with ours is that, in the string compactifications, the metric dimension of the full space-time is now 10 which can only be reconciled with what we experience by requiring that the Calabi-Yau fiber remains unnaturally small. 51

53 In order to learn how to perform the above shift of dimension using a 0-dimensional space F, it is important to classify such spaces. This was done in joint work with A. Chamseddine. We classified there the finite spaces F of given KO-dimension. A space F is finite when the algebra A F of coordinates on F is finite dimensional. We no longer require that this algebra is commutative. 52

54 We classified the irreducible (A, H, J) and found out that the solutions fall into two classes. Let A C be the complex linear space generated by A in L(H), the algebra of operators in H. By construction A C is a complex algebra and one only has two cases : 1. The center Z (A C ) is C, in which case A C = M k (C) for some k. 2. The center Z (A C ) is C C and A C = M k (C) M k (C) for some k. 53

55 Moreover the knowledge of A C = M k (C) shows that A is either M k (C) (unitary case), M k (R) (real case) or, when k = 2l is even, M l (H), where H is the field of quaternions (symplectic case). This first case is a minor variant of the Einstein-Yang-Mills case described above. It turns out by studying their Z/2 gradings γ, that these cases are incompatible with KOdimension 6 which is only possible in case (2). 54

56 If one assumes that one is in the symplectic unitary case and that the grading is given by a grading of the vector space over H, one can show that the dimension of H which is 2k 2 in case (2) is at least 2 16 while the simplest solution is given by the algebra A = M 2 (H) M 4 (C). This is an important variant of the Einstein-Yang-Mills case because, as the center Z (A C ) is C C, the product of this finite geometry F by a manifold M appears, from the commutative standpoint, as two distinct copies of M. 55

57 We showed that requiring that these two copies of M stay a finite distance apart reduces the symmetries from the group SU(2) SU(2) SU(4) of inner automorphisms to the symmetries U(1) SU(2) SU(3) of the Standard Model. This reduction of the gauge symmetry occurs because of the second kinematical condition [[D, a], b] = 0 which in the general case becomes : [[D, a], b 0 ] = 0, a, b A. of the even part of the algebra 56

58 Spectral Model Let M be a Riemannian spin 4-manifold and F the finite noncommutative geometry of KOdimension 6 described above. Let M F be endowed with the product metric. 1. The unimodular subgroup of the unitary group acting by the adjoint representation Ad(u) in H is the group of gauge transformations of SM. 2. The unimodular inner fluctuations of the metric give the gauge bosons of SM. 3. The full standard model (with neutrino mixing and seesaw mechanism) minimally coupled to Einstein gravity is given in Euclidean form by the action functional S = Tr(f(D A /Λ))+ 1 2 J ξ, D A ξ, ξ H + cl, where D A is the Dirac operator with the unimodular inner fluctuations. 57

59 Standard Model Spectral Action Higgs Boson Inner metric (0,1) Gauge bosons Inner metric (1,0) Fermion masses u, ν Dirac (0,1) in CKM matrix Dirac (0,1) in ( 3) Masses down Lepton mixing Dirac (0,1) in ( 1) Masses leptons e Majorana mass matrix Gauge couplings Higgs scattering parameter Tadpole constant Dirac (0,1) on E R J F E R Fixed at unification Fixed at unification µ 2 0 H 2 58

60 Inner fluctuations of metric One defines a right A-module structure on H by ξ b = b 0 ξ, ξ H, b A The unitary group of the algebra A then acts by the adjoint representation in H in the form ξ H Ad(u) ξ = u ξ u, ξ H The inner fluctuations of the metric are given by D D A = D + A + ε J A J 1 where A is a self-adjoint operator of the form A = a j [D, b j ], a j, b j A. 59

61 The bosonic part of the spectral action is S = 1 π 2(48 f 4 Λ 4 f 2 Λ 2 c + f 0 g 4 d) d 4 x(1) + 96 f 2 Λ 2 f 0 c 24π 2 + f 0 10 π 2 + ( 2 a f 2 Λ 2 + e f 0 ) + f 0 2 π 2 f 0 12 π 2 + f 0 2 π 2 R g d 4 x ( 11 6 R R 3 C µνρσ C µνρσ ) g d 4 x π 2 a D µ φ 2 g d 4 x a R φ 2 g d 4 x φ 2 g d 4 x (g 2 3 Gi µν Gµνi + g 2 2 F α µν F µνα g2 1 B µν B µν ) g d 4 x + f 0 2 π 2 where b φ 4 g d 4 x R R = 1 4 ϵµνρσ ϵ αβγδ R αβ µν Rγδ ρσ is the Euler characteristic. 60

62 We first perform a trivial rescaling of the Higgs field φ so that kinetic terms are normalized. To normalize the Higgs fields kinetic energy we have to rescale φ to : a f0 H = φ, π so that the kinetic term becomes 1 2 D µh 2 g d 4 x The normalization of the kinetic terms imposes a relation between the coupling constants g 1, g 2, g 3 and the coefficient f 0, of the form g 2 3 f 0 2π 2 = 1 4, g2 3 = g2 2 = 5 3 g

63 The bosonic action then takes the form S = ( 1 2κ 2 R + α 0 C µνρσ C µνρσ + γ 0 + τ 0 R R Gi µν Gµνi F µν α F µνα B µν B µν D µ H 2 µ 2 0 H 2 ξ 0 R H 2 + λ 0 H 4 ) g d 4 x, where 1 κ 2 0 = 96 f 2 Λ 2 f 0 c 12 π 2 µ 2 0 = 2 f 2 Λ 2 f 0 e a α 0 = 3 f 0 10 π 2 τ 0 = 11 f 0 60 π 2 γ 0 = 1 π 2(48 f 4 Λ 4 f 2 Λ 2 c + f 0 4 d) λ 0 = 2 π2 b f 0 a 2 ξ 0 =