Curiosités géométriques et physique de l'univers

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1 Curiosités géométriques et physique de l'univers MidiSciences, Grenoble, 05/2010 String Theory & Invisible Dimensions Ecole Normale Supérieure de Lyon

2 plan fundamental interactions standard model vs. general relativity extra dimensions from Kaluza & Klein to unified theories string theory unification of interactions compactification of extra dimensions

3 fundamental physics particle interactions standard model of particle physics physics at the microscopic scale origin and structure of matter quantum field theory (QED, Yang-Mills) describing the - constituents of matter (fermions) - fundamental interactions (bosons) D µ F µν = J ν A µ = A 0 A 1 A 2 A 3 electro: photon weak: W,Z-boson strong: gluon and gravity..?

4 fundamental physics gravitational interactions general relativity physics at the astronomical scale geometric formulation of the gravitational interactions local deformation (curvature) of space-time g µν = g 00 g 01 g 02 g 03 g 10 g 11 g 12 g 13 g 20 g 21 g 22 g 23 g 30 g 31 g 32 g 33 metric measures distance between nearby points in absence of matter: R µν 1 2 g µν R = 0 - black holes - gravitational waves R µν 1 2 g µν R matter tells space-time how to curve; space-time tells matter how to move (J. Wheeler) T µν

5 fundamental physics gravitational interactions general relativity R µν 1 2 g µν R resplendent marble T µν cheap wood Sie gleicht aber einem Gebäude, dessen einer Flügel aus vorzüglichem Marmor, dessen anderer Flügel aus minderwertigem Holze gebaut ist. (A. Einstein) unification: space-time geometry vs. particle interactions matter obeys quantum theory! so should gravity...

6 fundamental physics unification of interactions standard model of particle physics general relativity GR cannot be cast into standard quantum theory framework non-renormalizable as QFT (graviton is spin-2) standard model / general relativity considered as effective theories, valid on their respective scales search for a fundamental theory! contains GR / SM in the respective limits! explanation of parameters: number of paricles, masses, couplings, symmetries, cosmological constant, etc...

7 fundamental physics unification of interactions large light Compton wavelength l C mc classical physics l S Gm c 2 Schwarzschild radius small standard model general relativity massive black hole physics??? origin of the universe Planck length l PL = G c m

8 extra dimensions from Kaluza & Klein to unified theories

9 Kaluza s fifth dimension How does Einstein s general relativity look like, if the universe has five instead of four dimension? our 4-dimensional universe a 5-dimensional universe Einstein gravity : R µν 1 2 g µν R = 0 now gravitational waves can propagate along the fifth dimension describe a four-dimensional electromagnetic field! (T. Kaluza, 1921)

10 Kaluza s fifth dimension a 5-dimensional geometry g µν = g 00 g 01 g 02 g 03 g 10 g 11 g 12 g 13 g 20 g 21 g 22 g 23 g 30 g 31 g 32 g 33 g 40 g 41 g 42 g 43 g 04 g 14 g 24 g 34 g 44 R µν 1 2 g µν R = 0 geometrical unification gravitational & electromagnetic interactions the fifth dimension must be small (compactified) (O. Klein, 1926) higher frequency waves describe massive particles

11 Kaluza s fifth dimension a 5-dimensional geometry g µν = g 00 g 01 g 02 g 03 g 10 g 11 g 12 g 13 g 20 g 21 g 22 g 23 g 30 g 31 g 32 g 33 g 40 g 41 g 42 g 43 g 04 g 14 g 24 g 34 g 44 photon scalar particle R µν 1 2 g µν R = 0 R µν 1 2 g µν R T µν geometrical unification gravitational & electromagnetic interactions the fifth dimension must be small (compactified) (O. Klein, 1926) higher frequency waves describe massive particles

12 more extra dimensions What if there is more than one extra dimension? one (compact) extra dimension: several (compact) extra dimensions: or or or or... torus sphere Calabi-Yau orbifolds higher-dimensional geometry g 00 g 01 g 02 g 03 g 04 g 05 g 06 g 07 g 10 g 11 g 12 g 13 g 14 g 15 g 16 g 17 g 20 g 21 g 22 g 23 g 24 g 25 g 26 g 27 g µν = g 30 g 31 g 32 g 33 g 34 g 35 g 36 g 37 g 40 g 41 g 42 g 43 g 44 g 45 g 46 g 47 g 50 g 51 g 52 g 53 g 54 g 55 g 56 g 57 g 60 g 61 g 62 g 63 g 64 g 65 g 66 g 67 g 70 g 71 g 72 g 73 g 74 g 75 g 76 g 77 compactification various particles & interactions e.g. weak, strong interactions

13 string theory unification of interactions compactification of extra dimensions

14 string theory: unification of interactions elementary objects are not point particles, but extended strings particles string vibration modes light degrees of freedom open strings gauge fields closed strings gravitational field } string interactions g s

15 string theory: unification of interactions string theory is a consistent theory of quantum gravity however: the geometry of the universe must satisfy Einstein s equation R µν 1 2 g µν R T µν particle interactions must be supersymmetric predicts superpartners of all known particles none of which have been observed yet...! LHC!? the geometry of the universe must be ten-dimensional compactified extra dimensions

16 extra dimensions Calabi/Yau spaces string theory requires to compactify six extra dimensions many choices, leading to different theories... requesting a supersymmetric four-dimensional theory six dimensions compactified on a Calabi-Yau space as we move in the four-dimensional universe the shape of the internal space may change deformation parameters particles of the four-dimensional theory in general, far too much...

17 extra dimensions compactifications / landscape localized anti!d3!branes RR flux mobile, localized D3!brane NSNS flux wrapped D7!branes modern compactification scenarios which also generate sufficiently high masses for extra particles positive cosmological constant supersymmetry breaking

18 so far, we have no experimental hint for the existence of extra dimensions beyond Einstein s general relativity...

19 The idea of a unification of the fundamental interactions in a higherdimensional geometry is essential in modern string theories and remains extremely fascinating!