The world as a hologram: News from string theory
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The world as a hologram: News from string theory Prof. Dr. Jan Plefka Humboldt-Universität zu Berlin Integrability in Gauge and String Theory 2015 Public Lecture The problem Two keystones of fundamental physics: 1. Einstein’s theory of gravity [1915] 2. Quantum theory: [1920-1930] ⇒ Standard model of elementary particle physics Electromagnetic, weak and strong forces Not known how to combine the two! The most promising ansatz: ⇒ String theory [1950-75] “unified theory” [since 1984] Both keystones are intimately connected ⇒ Holographic principle of quantum gravity [1/25] [since 1997] The world as a holgram [2/25] Die Allgemeine Relativitätstheorie Das Prinzip: Die Raumzeit bestimmt die Bewegung der Materie, die Materie Gravity bestimmt die Krümmung der Raumzeit. Einsteinsche Feldgleichungen: Rµν − 12 gµν R + gµν Λ = 8π GN c4 Tµν Geometrie (Mathematik) = Materie (Physik) Krümmung R Kosmologische Konstante Λ Energie-Impuls-Tensor Tµν Gravity according to Newton Sir Isaac Newton, 1642-1727 Two bodies moving through empty space in the absence of forces FA = mA d2 x =0 dt2 ⇒ x(t) = vt+c Asbodies have masses mA und mB there is an attractive gravitational force acting between them F = GN mA · mB (xA − xB )2 Newton’s law of gravity Time is absolute: t “ticks” the same for A and B. Forces are mediated instantaneously Space is flat and infinitely extended (Euclidean): R3 [3/25] Gravity according to Einstein Albert Einstein, 1879-1955 The attraction of two masses arises through the curvature of space (and time) Bodies A and B continue to move on straight paths albeit in a curved space! There is no force acting between them. Gravity is a fictitious force similar to the centrifugal force known from everyday life. Space-time is dynamic: Gets curved by the matter moving through it and influences the motion of matter via its curvature. A highly coupled system. [4/25] A Gedankenexperiment Let us remove the sun from our solar system! Newton’s World: The gravitational force acts instantaneous: The “removal” of the sun is immediately felt by the earth Einstein’s World: Changes in the curvature of space and time propagate with c the velocity of light Gravitational waves = ˆ ripples of space-time! Predicted alrady in 1915, presently searched for in graviational wave detectors. [5/25] A Gedankenexperiment Let us remove the sun from our solar system! Newton’s World: The gravitational force acts instantaneous: The “removal” of the sun is immediately felt by the earth Einstein’s World: Changes in the curvature of space and time propagate with c the velocity of light Gravitational waves = ˆ ripples of space-time! Predicted alrady in 1915, presently searched for in graviational wave detectors. [5/25] The theory of general relativity The principle: Space-time dictates the movement of matter, matter dictates the curvature of space-time. Einstein’s field equations: Rµν − 12 gµν R + gµν Λ = 8π GN c4 Tµν Geometry (Math) = Matter (Physics) Curvature R Cosmological constant Λ [6/25] Energy-momentum tensor Tµν The theory of general relativity The principle: Space-time dictates the movement of matter, matter dictates the curvature of space-time. Einstein’s field equations: Rµν − 12 gµν R + gµν Λ = 8π GN c4 Tµν Geometry (Math) = Matter (Physics) Curvature R Cosmological constant Λ [6/25] Energy-momentum tensor Tµν Beispiel Quantenelektrodynamik: Elementarteilchenph Physikal. Theorie der Elektronen und P (g:“Kopplungskonstante”) • Beschrieben durch Quantenfeldtheor Quantum mechanics e− e− = e− Zeit e− g2 · Ph Streuung von Elektronen Streuprozesse Störun Renormierung: g → g(E) • Renormierung: g → g(E) Drei Naturkräfte beschrieben durch Ei Quantum mechanics M.Planck W.Heisenberg E.Schrödinger A particle starts it motion from point A at time tA . Will it reach the point B at tb ? Classical physics: Prediction (from Newton’s law): Depending on its intitial velocity and the forces acting upon it, the particle will either reach B at tb or not: Deterministic prediction: yes/no Quantum physics: There is no definite answer to this question! Only the prediction for a probability of observing the particle at space-time point B at the time tB is possible. W(A,ta )→(B,tb ) = 0.73 [7/25] Quantum mechanics M.Planck W.Heisenberg E.Schrödinger A particle starts it motion from point A at time tA . Will it reach the point B at tb ? Classical physics: Prediction (from Newton’s law): Depending on its intitial velocity and the forces acting upon it, the particle will either reach B at tb or not: Deterministic prediction: yes/no Quantum physics: There is no definite answer to this question! Only the prediction for a probability of observing the particle at space-time point B at the time tB is possible. W(A,ta )→(B,tb ) = 0.73 [7/25] Feynman’s path integral Richard Feynman, 1918-1988 Feynman’s approach to quantum mechanics allows for the computation of this probability: Consider all possible paths from A to B. Every path is weigthed by a factor (the action) and the total probability follows from the sum of all possible paths X 2 W(A,ta )→(B,tb ) = eaction/~ all paths The classical path is the path with minimal action ⇒ typically gives a dominant contribution to W . Time is still an absolute quantity here ⇒ “non-relativistic” quantum mechanics [8/25] Quantum field theory or relativistic quantum mechanics Absolutely surprising effect of the constancy of the speed of light: γ Creation and annihilation of particles γ Photon e− Elektron Photon e − e− Elektron Photon radiation e− e− e− Forces are transmitted via the exchange of elementary particles e− e− Carrier particle of the electro− e− magnetic force:ePhoton (“light”) Photon e− Photon e− [9/25] [1950s-70s] Quantum field theory Example: Quantum electrodynamics: Elementarteilchenphysik (ohne Gravitation) Theoretical description of electrons and photons and their interactions (g: charge or “coupling constant”) • Beschrieben durch Quantenfeldtheorie: (hier QED) e− e− − − = e Zeit e g2 · Photon + g4 · [1950-1975] + g 6(. . .) + . . . ScatteringStreuprozesse of electrons Perturbative in g 1 Störungsreihe in g �series 1 g:“Kopplungskonstante” Renormalization: g → g(E) • Renormierung: → g(E) But what happens gwhen g ∼ 1? ⇒ non-perturbative quantum field theory Three fundamental forces described via gauge field theory • Drei Naturkräfte beschrieben durch Eichfeldtheorien • Was passiert bei g ∼ 1? ⇒ nichtperturbative Quantenfeldtheorie [10/25] [1955,1971] [1955,1971] Standard model of particle physics Forces: Gauge bosons SU(N) gauge fields N × N matrices Matter: Quarks & leptons Higgs boson: Scalar particle Standard Model of Elementary Particles 18.02.15 09:56 [11/25] Photon The strong force: Quantum chromodynamicse Gluon − e− e− Photon SU(3) gauge theory: gluons and quarks (q) q q q g Interaction: e− q e− g Responsible for the staq bility of the proton and nuclei Gluon e− q Gluon q q g Gluons have selfg interactions g g g Gluon g q g g g g g2 ∼ g∼ g Strong force: q g (at LHC energies) g g g 1000 = 10g3 times stronger thatg electromagnetic force Gluon Gluon 100000 = 105 times stronger than the weak force 10000 . . . 000 = 1038 times stronger than gravity!! [12/25] g Quantum chromodynamics: The inverse giant Coupling strength depends on energy [Gross, Wilczeck, Pollitzer] g → g(E) g E Low Energies (g 1): Confinement No free quarks and gluons are being observed. Instead: Bound states (hadrons) Mesons: q̄ q L q̄q-potential: V = “Jim Knopf”, M. Ende [13/25] 1 2 lS L “color flux tube” The strong force as a string theory Color flux tube reminds of a microscopic string: Hadron String picture Meson (e.g. pion) = ˆ Glueball = ˆ q̄ q q Baryon (e.g. Proton) [’t Hooft, 1974] q = ˆ q Vision: Strings are adequate description for strongly coupled gauge fields. But: Strings describe quantum gravity and the strong force does not contain gravity! Puzzle resolved by holographic principle... [14/25] ! Quantum gravity andeiner string theorySchwerpunktsb • Quantenmechanik Quantum mechanics of a“Saite”: relativistic string • Quantenmechanik einer “Saite”: Schwerpunk Graviton Gauge bosonS Stringwechselwirkungen: Vibrationsspektrum Spektrum Eichteilche der Graviton = Oscillation spectrum ˆ =ˆspectrum of “Elemen “elem • Verallgemeinerung Teilchengraphen: [198 Vibrationsspektrum = ˆ Spektrum der “Elem Extended structurevon’softens’ divergences: Gravitation ? Gravity as a quantum theory? Treat Einstein’s theory of gravity b y the rules of quantum field theory: • Behandle Einsteins Gravitationstheorie als Quantenfeldtheorie: Gravitons: Small curvature fluctuations about a given space-time structure Gravitonstreuung: gScattering κ · hµν (x) Gedankenexperiment: of+gravitons µν (x) = ηµν h h = h (Gedankenexperiment) Zeit κ2 · Graviton + κ4 · + ... h =∞ NOT “renormalizable” ⇒ Forced to extend Einstein’s theory NICHT “renormierbar” ⇒ Quantenfeldtheorie der Gravitation existiert nicht ! Quantum field theory of gravity requires measurements of infinitely many empirically determined parameters Kopplungskonstante der Gravitation: [κ] = M 1 = ˆ 10−33cm Planck Very limited predictability . . . 4 √ Coupling constant κ = GN of gravitons extremely weak: Becomes relevant only at length scales 10−33 cm [15/25] String theory How can we reach predictability within quantum gravity? Stringtheorie • Simple Idee: Ersetze Teilchen durch by ausgedehntes Objekt: “String” idea: Replace particles extended 1d1d object: “String” o� −33 lS ∼ cm cm lS 10 ∼ 10−33 Quantum mechanics of a relativistic string: • Quantenmechanik einer “Saite”: Schwerpunktsbewegung + Eigenschwingung Centre of mass movement + internal oscillations: Graviton Eichteilchen Materieteilchen Vibrationsspektrum = ˆ Spektrum der “Elementarteilchen” [16/25] 7 String interactions: Perturbative series Stringwechselwirkungen: Störungsreihe Generalization of particles interactions: [1984-1995] • Verallgemeinerung von Teilchengraphen: + gS2 · [1984-1995] + gS4 · + gS6 · (. . .) + . . . Zeit gS : Stringkopplungskonstante gS : string coupling constant Es treten keine Divergenzen mehr auf! Wechselwirkung ist “weich” There are no divergencies! Interaction is “soft” as not localized to a point • Gravitonstreuung: Graviton scattering: gS2 · = ˆ Einsteins Gravitationstheorie = ˆ Quantenkorrekturen zu Einsteins Theorie = ˆ Einstein’s theory of gravity 7 gS2 · = ˆ Quantum corrections to Einstein’s theory [17/25] Properties and predictions of string theory Higher dimensions: As a quantum theory strings are only consistent in 9 space dimensions! ⇒ 6 extra space dimensions or Graviton Materieteilchen Geometry of hidden 6 dimensions predicts particle spectrum 3 dimensional world Prediction of supersymmetry: bosons ⇔ fermions Puzzle: There exist a gigantic number of possible compactifications from 9 → 3 [18/25] The world as a hologram Source: Scientific american Quantum gravity in negatively curved space-times Since 1997 revolutionary progress in our understanding of quantum gravity in anti-de-Sitter space (AdSd ) = ˆ constant neg. curvature [Willem de Sitter, 1872-1934] AdS5 is (4+1)-dimensional space-time with a boundary: R3 × time “Gravity in a box” String theory well defined on AdS5 × M5 , e.g. choose M5 = S 5 5d-sphere. [19/25] Maldacena’s String-Gauge Duality [1997] Holographic principle: Strings in the bulk of space-time (Anti-de-Sitter space), quantum particles (gluons) on the boundary String theory in higher dimensional space Gauge field theory on 4d boundary Two dual descriptions of one physical entity: Gauge theory = ˆ String theory in AdS [20/25] The world as a hologram: Resolution of dimensions 4d particle theory ⇔ (5+5)d gravity theory ⇔ 2d string theory [21/25] Integrability in gauge and string theory Novel insights discussed at the IGST Conference allow computation of exact quantum properties in both theories. ⇒ No approximation in coupling constant g any longer!! Probability for the scattering of two gluons into two gluons in supersymemtric QCD Example 1: g g = ? g g ( 73 6 4 6 2 8 8g 2 − 38 π 2 g 4 + 88 45 π g − 16( 630 π + 4 ζ(3) ) g + . . . = 27 ζ(3) 1 2 K 1 4g − 3 log − 4π 2 g−3 log 2/4π − 29 π 3 g 2 − . . . π for for g 1o g1 × (Simple functions of the involved momenta) Exactly known function of g and gluon-momenta [22/25] [Beisert,Eden,Staudacher] Stringtheorie Theory as a consistent theory of quantum gravity pling asymptotics. For this purpose Stringtheorie we solve numerically the integral form of the em equations for the exact energies of excited states proposed by us and A. Kozak. Integrability in gauge and string theory ee: Ersetze ausgedehntes1d1d Objekt: “String” Idee: ErsetzeTeilchen Teilchen durch durch ausgedehntes Objekt: “String” : Example Replace2: particle by extended 1d object: string Exact internal excitation energy of a closed string in AdS-space. !! −33 −33 lS ∼ lS 10 ∼ 10cm cm UCTION ills theories are at the heart cs, describing all fundamenty. Nevertheless, in spite of First excitation mode of the most 40mechanics years, we stilleiner don’t Quantenmechanik Schwerpunktsbewegung + Eigenschwingung: ntum of a“Saite”: relativistic string in flat space-time: string uantenmechanik einer tive description of the most“Saite”: Schwerpunktsbewegung + Eigenschwingung: h as QCD, in the region of plings. The low energy quanis mostly known only from 7 ice YM theories. A few imned the topological, BPS secGraviton Gauge bosonStörungsreihe Matter particle tained. Stringwechselwirkungen: on complete exact 4D soluVibrationsspektrum Spektrum der “Elementarteilchen” Figure 1: Numerical solutionstring of exact finite size integral YGraviton Eichteilchen Materieteilchen ∆K= = Energy stored in vibrating lation spectrum ˆ(λ)=ˆspectrum of “elementary particles” uantities given by nontrivial system equations for the Konishi dimension ∆K (λ) in a wide 4 to the asymptotic o start waning, N=4 superrange of ’tder Hooft couplings λ, compared Curvature radius of AdS-space allgemeinerung von Teilchengraphen: [1984-1995] Vibrationsspektrum = ˆλ = Spektrum “Elementarteilchen” 5 g 2 λ asymptotics y gave structure us serious hopes for nded ’softens’ divergences: Bethe ansatz curve and to the predicted = large String length 1/4 1/4 ∆K (λ) ≃ 2λ + 2/λ obtained by fit. he dynamics of strongly inDue to the AdS/CFT cor[Gromov,Kazakov,Vieira] 2 4 6 o the quantum integrability + g · + [23/25] g · + g · (. . .) + . . . Summary Quantum gravity (= ˆ string theory) in (d+1)-dimensions is equivalent to quantum particle theory (gauge field theory) in d-dimensions w/o gravity Novel tools for solution of the quantum particle theory in 3 space dimensions [24/25] Outlook Prospect: Use (2+1)d particle theory to describe quantum gravity in (3+1)d Black holes! Our universe? [25/25] Thank you for your attention
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