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Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Dilaton Quantum Gravity A Functional Renormalization Group Approach Tobias Henz with Jan Martin Pawlowski & Christof Wetterich Institute for Theoretical Physics, University of Heidelberg TR33 Winter School, December 2012 Summary and Outlook Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Outline 1 Introduction: Asymptotic Safety & Quantum Einstein Gravity Nonrenormalizibility and Asymptotic Safety Beyond Perturbation Theory: Functional Renormalization Challenges & Open Questions 2 Scalar Tensor Theories and Dilatation Symmetry Dynamical Constants Results 3 Summary and Outlook Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Renormalizibility and Quantum Gravity Einstein-Hilbert action 1 √ dd x g (2Λ − R[gµν ]) Γ [gµν ] = 16πGN mass dimensions of the couplings Z dim(Λ) = 2 dim(GN ) = 2 − d =⇒ Perturbatively not renormalizable if d > 2. Summary and Outlook Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Alternative Approaches Introduction of radically new concepts (Super) String Theory, Loop Quantum Gravity, ... What if we give up perturbation theory instead? The Asymptotic Safety Scenario Summary and Outlook Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Alternative Approaches Introduction of radically new concepts (Super) String Theory, Loop Quantum Gravity, ... What if we give up perturbation theory instead? The Asymptotic Safety Scenario Summary and Outlook Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Asymptotic Safety Weinberg, 1979 A theory is said to be asymptotically safe if the essential coupling parameters approach a fixed point as the momentum scale of their renormalization point goes to infinity. working translation Quantum gravity is considered asymptotically safe if the UV-critical surface is finite dimensional and if the dimensionless coupling constants cease to increase if the momentum scale k goes to infinity, but approach an ultraviolet fixed point instead. Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Theory Space and the Functional Renormalization Group Wetterich Equation " # 1 1 ∂ t Γk = STr (2) ∂ t Rk , 2 Γk + Rk t = log(k/k0 ) Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Theory Space and the Functional Renormalization Group Wetterich Equation " # 1 1 ∂ t Γk = STr (2) ∂ t Rk , 2 Γk + Rk t = log(k/k0 ) Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook The Flow Diagram of Quantum Einstein Gravity Reuter, 1998 Coupling Constants approach a nontrivial UV Fixed Point =⇒ Prospect of Gravity being Asymptotically Safe Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook The Flow Diagram of Quantum Einstein Gravity " 0.015 10 gN A 2 Ib 1 0.2 0.1 C 0 0.45 0.5 0.5 Ia C B D 0 !" !2 !0.5 0 0.1 0.2 0.3 0.4 0.45 Christiansen, Litim, Pawlowski, Rodigast, 2012 Stable Infrared Scenarios =⇒ Prospect of UV and IR consistent theory λ 0.5 Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Asymptotically Safe Quantum Gravity FRG Technicalities Coupling to SM Regulator Dependence Yang Mills Theory Background Dependence Background Dependence Truncation Stability Asymptotic Freedom Infrared Limit Hierarchy Problem Trajectory UV → IR MSM ≈ 102 GeV GR as limiting case? MPlanck ≈ 1019 GeV Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Dynamical Constants I: Scalar-Tensor Theories 1 Γk [gµν ] = 16πGN,k ⇓ Z Γk [gµν , χ] = √ Z √ dd x g (2Λk − R[gµν ]) ⇓ ⇓ 1 d x g Vk [χ] − Fk [χ]R[gµν ] + gµν ∂ µ χ∂ ν χ 2 d Assumptions: O(R) truncation no wave function renormalization curved background spacetime & optimized cut-offs Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Dynamical Constants I: Scalar-Tensor Theories 1 Γk [gµν ] = 16πGN,k ⇓ Z Γk [gµν , χ] = √ Z √ dd x g (2Λk − R[gµν ]) ⇓ ⇓ 1 d x g Vk [χ] − Fk [χ]R[gµν ] + gµν ∂ µ χ∂ ν χ 2 d Assumptions: O(R) truncation no wave function renormalization curved background spacetime & optimized cut-offs Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Dynamical Constants II: Dilatation Symmetry Dilatations ↔ Conformal Transformations gµν (x ) 7→ Ω(x )gµν (x ) with Ω = const. Dilatation ↔ global resetting of the physical scale Dilatation Symmetry ↔ Physical Scale is introduced only by expectation value of the scalar field Arising Goldstone Boson: Dilaton Dilatation Symmetric Actions Γ is invariant under dilatations m all couplings have scaling dimension 0. (d = 4 : F ∝ χ2 , V ∝ χ4 ) Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Limiting Cases dimensionless field χ̃ (d = 4 : χ̃ = k −2 χ) Ultraviolet: χ̃ → 0 χ→0 Infrared: χ̃ → ∞ χ→∞ k→0 k→∞ V , F power series in Percacci, 2009 χ̃2 V , F power series in χ̃−2 current project, 2012 Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook The Infrared Limit I: Expansions Dilatation Symmetric Couplings have vanishing β-functions Closed set of flow equations to each order in χ̃−2 Fixed point value only in constant order nontrivial ⇒ stable Einstein-Hilbert infrared limit Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook The Infrared Limit II: Dilatation Symmetry Dilatation Symmetric Infrared Scenario V = 0 and F = ξχ2 m √ 1 Γk→0 = d x g gµν ∂ µ χ∂ ν χ − ξχ2 R 2 Flow diverges at conformal coupling parameters Z d =⇒ Weyl Anomaly is realized Introduction: Asymptotic Safety Current Research: Dilatation Symmetry Summary and Outlook Multifaceted evidence for asymptotic safety scenario Dilatation symmetric infrared limit Stability surveys Continuation past asymptotic cases Thank you very much! Summary and Outlook