Asymptotically safe Quantum Gravity. Nonperturbative renormalizability and fractal space-times

Transcription

1 p. 1/2 Asymptotically safe Quantum Gravity Nonperturbative renormalizability and fractal space-times Frank Saueressig Institute for Theoretical Physics & Spinoza Institute Utrecht University Rapporteur talk Quantum Gravity in the Americas III 24th to 26th August 2006

2 p. 2/2 Towards a quantum theory of gravity Classical General Relativity: phenomenologically successful at scales l l Pl (laboratory, solar system, galaxies,...) Quantized General Relativity: theory is perturbatively non-renormalizable: need infinite number of counter terms theory has no predictive power belief that General Relativity is an effective theory: not fundamental not valid at arbritary small distances

3 p. 2/2 Towards a quantum theory of gravity Classical General Relativity: phenomenologically successful at scales l l Pl (laboratory, solar system, galaxies,...) Quantized General Relativity: theory is perturbatively non-renormalizable: need infinite number of counter terms theory has no predictive power belief that General Relativity is an effective theory: not fundamental not valid at arbritary small distances there are fundamental theories which are not perturbatively renormalizable so-called non-perturbatively renormalizable theories same predictive power as a perturbative renormalizable theory

4 p. 3/2 Generalized principles of renormalization (K. Wilson) theory is fundamental = infinite UV cutoff limit exists infinite cutoff limit requires fixed point in RG flow Gaussian fixed point (GFP) (u = 0) (e.g. pert. renormalization) non-gaussian fixed point (u 0) fundametal theory is defined through fixed point predictivity linked to how coupling constants approach the fixed point

5 p. 4/2 Weinbergs asymptotic safety conjecture (Weinberg (1979) [hep-th/ ]) conjecture: Euclidean quantum gravity is asymptotically safe RG flow of gravity has non-trivial UV fixed point defining a fundamental theory evidence: non-gaussian fixed point (NGFP) in 2 + ɛ gravity

6 p. 4/2 Weinbergs asymptotic safety conjecture (Weinberg (1979) [hep-th/ ]) conjecture: Euclidean quantum gravity is asymptotically safe RG flow of gravity has non-trivial UV fixed point defining a fundamental theory evidence: non-gaussian fixed point (NGFP) in 2 + ɛ gravity if NGFP also exists in d = 4, it defines a fundamental theory of gravity: Quantum Einstein Gravity (QEG) not clear that fundamental theory is of Einstein-Hilbert form = approach differs form quantizing General Relativity

7 p. 5/2 Outline introduce tool to study non-perturbative RG flows: exact RG equation (ERGE) for effective average action Γ k evidence for NGFP in d = 4 quantum gravity consequences of asymptotic safety: fractal structure of QEG space-times dynamical dimensional reduction to d = 2

8 p. 6/2 The effective average action Γ k Γk is Wilson-type (coarse grained) effective action, based on path integral IR cutoff at momentum k 2 : includes all quantum effects with momenta p 2 > k 2 quantum fluctuations with p 2 < k 2 suppressed by mass 2 = R k (p 2 ) limits: k = bare/classical action S k 0 = ordinary effective action Γ Γ k satisfies exact evolution equation k k Γ k = 1 2 Tr ˆ(δ 2 Γ k + R k ) 1 k k R k nonperturbative approximation scheme: truncating the space of action functionals

9 p. 7/2 Constructing Γ k for gravity Starting point: path integral for euclidean metrics, R Dγ exp[ S grav [γ µν ]] background gauge fixing: decompose quantum metric: γ µν = ḡ µν + h µν add: background gauge fixing S gf [h; ḡ] + S gh [h, C, C; ḡ]

10 p. 7/2 Constructing Γ k for gravity Starting point: path integral for euclidean metrics, R Dγ exp[ S grav [γ µν ]] background gauge fixing: decompose quantum metric: γ µν = ḡ µν + h µν add: background gauge fixing S gf [h; ḡ] + S gh [h, C, C; ḡ] expand h µν in D 2 -eigenmodes and introduce IR cutoff k S modes with D 2 -eigenvalue > k 2 are integrated out modes with D 2 -eigenvalue < k 2 suppressed by mass-term R k

11 p. 7/2 Constructing Γ k for gravity Starting point: path integral for euclidean metrics, R Dγ exp[ S grav [γ µν ]] background gauge fixing: decompose quantum metric: γ µν = ḡ µν + h µν add: background gauge fixing S gf [h; ḡ] + S gh [h, C, C; ḡ] expand h µν in D 2 -eigenmodes and introduce IR cutoff k S modes with D 2 -eigenvalue > k 2 are integrated out modes with D 2 -eigenvalue < k 2 suppressed by mass-term R k adding sources: scale dependent generating function for connected Green s functions W k [sources; ḡ]

12 p. 8/2 Constructing Γ k for gravity adding sources: generating function for connected Green s functions W k [sources; ḡ] Obtaining the effective average action Γ k : introduce classical fields: g µν = γ µν k,... Γ k [g µν, ḡ µν,ghosts] = (modified) Legendre transform of W k

13 p. 8/2 Constructing Γ k for gravity adding sources: generating function for connected Green s functions W k [sources; ḡ] Obtaining the effective average action Γ k : introduce classical fields: g µν = γ µν k,... Γ k [g µν, ḡ µν,ghosts] = (modified) Legendre transform of W k exact RG equation for Γ k : k k Γ k = 1 h`δ2 2 Tr Γ k + R k ( D 2 ) 1 k k R k ( D i 2 ) + ghost contribution

14 p. 8/2 Constructing Γ k for gravity adding sources: generating function for connected Green s functions W k [sources; ḡ] Obtaining the effective average action Γ k : introduce classical fields: g µν = γ µν k,... Γ k [g µν, ḡ µν,ghosts] = (modified) Legendre transform of W k exact RG equation for Γ k : k k Γ k = 1 h`δ2 2 Tr Γ k + R k ( D 2 ) 1 k k R k ( D i 2 ) + ghost contribution diffeomorphism invariant effective action: Γ k [g] = Γ k [g, ḡ = g,ghost = 0] theory: specified by RG trajectory: k Γ k [g]

15 p. 9/2 The Einstein-Hilbert truncation Ansatz for Γ k Γ k = 1 16πG(k) Z d d x g { R + 2Λ(k)} +classical gauge fixing & ghost terms running couplings Newton constant G(k), dimensionless g(k) = k d 2 G(k) cosmological constant Λ(k), dimensionless λ(k) = Λ(k)/k 2

16 p. 9/2 The Einstein-Hilbert truncation Ansatz for Γ k Γ k = 1 16πG(k) Z d d x g { R + 2Λ(k)} +classical gauge fixing & ghost terms running couplings Newton constant G(k), dimensionless g(k) = k d 2 G(k) cosmological constant Λ(k), dimensionless λ(k) = Λ(k)/k 2 substitute ansatz into ERGE and project on subspace: = autonomous β-functions for g, λ k k g = β g (g, λ), k k λ = β λ (g, λ).

17 p. 9/2 The Einstein-Hilbert truncation Ansatz for Γ k Γ k = 1 16πG(k) Z d d x g { R + 2Λ(k)} +classical gauge fixing & ghost terms running couplings Newton constant G(k), dimensionless g(k) = k d 2 G(k) cosmological constant Λ(k), dimensionless λ(k) = Λ(k)/k 2 substitute ansatz into ERGE and project on subspace: = autonomous β-functions for g, λ k k g = β g (g, λ), k k λ = β λ (g, λ). β-functions have a non-gaussian fixed point: exists g > 0, λ > 0 where β g (g, λ) = β λ (g, λ) = 0 IR repulsive in both g, λ

18 p. 10/2 Evidence for asymptotic safety Evidence within the exact RG approach: within the Einstein-Hilbert approximation: test R k -dependence of physical quantities (critical exponents, g λ,... ) = k-independent in exact theory generalized truncation including R 2 -term: NGFP stable under extending truncation space (O. Lauscher, M. Reuter, hep-th/ , hep-th/ )

19 p. 10/2 Evidence for asymptotic safety Evidence within the exact RG approach: within the Einstein-Hilbert approximation: test R k -dependence of physical quantities (critical exponents, g λ,... ) = k-independent in exact theory generalized truncation including R 2 -term: NGFP stable under extending truncation space (O. Lauscher, M. Reuter, hep-th/ , hep-th/ ) NGFP also found via: exact path-integral calculation for metrics with 2 Killing vectors (P. Forgács, M. Niedermaier, hep-th/ ) proper-time RG equation (A. Bonanno, M. Reuter, hep-th/ )

20 p. 11/2 The RG flow of QEG in the Einstein-Hilbert-truncation (M. Reuter, F. Saueressig, Phys. Rev. D 65 (2002) [hep-th/ ]) g Type IIa Type Ia Type IIIa λ 0.25 Type IIIb Type Ib

21 p. 12/2 The RG trajectory realized in Nature (M. Reuter, H. Weyer, JCAP 0412 (2004) 001 [hep-th/ ]) g T P 1 P 2? λ <<1 0.5 originates at NGFP (quantum regime: G(k) = k 2 g,λ(k) = k 2 λ ) linear regime: oscillations around NGFP passing extremely close to the GFP long classical GR regime (classical regime: G(k) = const, Λ(k) = const) λ 1/2: strong IR renormalization effects?

22 p. 13/2 Classical versus Quantum space-time (O. Lauscher and M. Reuter, hep-th/ , hep-th/ ) RG running of Γ k [g] = profound consequences for space-time structure Classical General Relativity, S EH : vacuum Einstein equations: R µν = Λ g µν solution g µν = single metric valid at all length scales

23 p. 13/2 Classical versus Quantum space-time (O. Lauscher and M. Reuter, hep-th/ , hep-th/ ) RG running of Γ k [g] = profound consequences for space-time structure Classical General Relativity, S EH : vacuum Einstein equations: R µν = Λ g µν solution g µν = single metric valid at all length scales effective action Γ k [g]: 1-parameter family of equations of motion: δγ k [ g µν k ] δg µν = 0 solution g µν k = metric seen by physical process with momentum k 2 proper distances calculated from g µν k depend on k 2 = typical fractal behavior (Coastline of England)

24 p. 14/2 Classical versus Quantum space-time effective action Γ k [g]: 1-parameter family of equations of motion: δγ k [ g µν k ] δg µν = 0 solution g µν k = metric seen by physical process with momentum k 2 proper distances calculated from g µν k depend on k 2 = typical fractal behavior (Coastline of England) Classical limit: fractal behavior = linked to k-dependece of Γ k [g µν ] trajectory realized in nature long classical regime : Γ k [g µν ] is k-independent recover classical space-time picture

25 p. 15/2 Relating k to the coarse graining scale l(k) Interpretation: g µν k averaged over proper distances l(k) Algorithm to determining l(k): 1. Recall: Γ k [g] = Γ k [g, ḡ = g,ghost = 0] = k 2 = D 2 = D 2 is last eigenmode integrated out 2. construct on-shell -Laplacians (k) = D 2 ( g µν k ) 3. find eigenfunction ψ(x) with eigenvalue k 2 4. determine typical coordinate distance x µ on which ψ(x) varies 5. calculate proper length l(k) l(k) = q g µν k x µ x ν scaling argument in the quantum regime: l k 1

26 p. 16/2 Quantum regime: QEG space-times are self-similar fractals in the quantum regime (k 2 m 2 Pl ) Λ(k) = λ k 2, G(k) = g k 2 d, k 1/l effective field equations in the Einstein-Hilbert approximation: R µν ( g µν k ) = Λ(k) g µν k radius of curvature r c for a typical solution: r c (k) Λ(k) 1/2 k 1 Radius of curvature at resolution l: r c (l) l = zooming into the space-time structure does not change the image

27 p. 17/2 Dynamical dimensional reduction in QEG space-times Investigate effective graviton propagator 1 Γ k = 16πG(k) Z d d x g R +... expanding around flat space G k (p) G(k) p 2 classical regime: G(k) = const G 1 p 2 = G(x, y) 1 x y d 2

28 p. 17/2 Dynamical dimensional reduction in QEG space-times Investigate effective graviton propagator 1 Γ k = 16πG(k) Z d d x g R +... expanding around flat space G k (p) G(k) p 2 classical regime: G(k) = const G 1 p 2 = G(x, y) 1 x y d 2 fixed point regime: cutoff is p 2 = G(k 2 = p 2 ) p 2 d G k (p) p d = G(x, y) ln(µ x y )

29 p. 17/2 Dynamical dimensional reduction in QEG space-times Investigate effective graviton propagator 1 Γ k = 16πG(k) Z d d x g R +... expanding around flat space G k (p) G(k) p 2 classical regime: G(k) = const G 1 p 2 = G(x, y) 1 x y d 2 fixed point regime: cutoff is p 2 = G(k 2 = p 2 ) p 2 d G k (p) p d = G(x, y) ln(µ x y ) effective graviton propagator reduces dynamically: macroscopically: d = 4 microscopically: d = 2

30 p. 18/2 Spectral dimension for classical manifolds diffusion of scalar test particle on Riemannian manifold with metric g T K g (x, x ; T) = g K g (x, x ; T) g φ g 1/2 µ (g 1/2 g µν ν φ) define average return probability P g (T) 1 V Z d d x p g(x) K g (x, x; T) = 1 V Tr[exp(T g)] = 1 4πT «d/2 X n=0 A n T n asymptotic expansion contains information about space-time dimension d = 2 dln P g(t) d ln T

31 p. 19/2 Spectral dimension of QEG space-times in QEG: metric of manifold is k-dependent = diffusion process with momentum k sees metric g µν k = diffusion equation and return probability will become k-dependent Computation of the spectral dimension: 1. determine k-dependence of (k) 2. solve the k-dependent heat equation 3. evaluate quantum return probability P(T) 4. obtain spectral dimension dln P(T) D s = 2 dln T

32 p. 20/2 Spectral dimension D s of QEG space-times Quantum return probability: P(T) = Z d d p (2π) d exp[ p2 F(p 2 ) T], F(p 2 ) = Λ(p)/Λ(k 0 ) Limits: T : long random walks probe space-time at large distance = classical regime: F(p 2 ) with p 2 0 T 0: short random walks probe space-time at small distance = fixed point regime: F(p 2 ) with p 2 0

33 p. 21/2 Spectral dimension D s of QEG space-times Quantum return probability: P(T) = Z d d p (2π) d exp[ p2 F(p 2 ) T], F(p 2 ) = Λ(p)/Λ(k 0 ) classical regime: no running, F(p 2 ) = 1: P(T) T d/2 = D s = d fixed point regime: Λ(p) p 2 F(p 2 ) p 2 : P(T) T d/4 = D s = d/2

34 p. 21/2 Spectral dimension D s of QEG space-times Quantum return probability: P(T) = Z d d p (2π) d exp[ p2 F(p 2 ) T], F(p 2 ) = Λ(p)/Λ(k 0 ) classical regime: no running, F(p 2 ) = 1: P(T) T d/2 = D s = d fixed point regime: Λ(p) p 2 F(p 2 ) p 2 : P(T) T d/4 = D s = d/2 d = 4: QEG predicts continuous change of fractal dimension D s = 4 macroscopically = D s = 2 microscopically

35 p. 22/2 Spectral dimension D s from Causal dynamical triangulations (J. Ambjørn, et. al. Phys. Rev. Lett. 95 (2005) [hep-th/ ]) CDT Monte Carlo evaluation of Lorentzian Path Integral mesured D s for d = 4 D s (T ) =4.02 ± 0.1 D s (T 0) =1.80 ± 0.25 supports analytic results from exact renormalization group

36 p. 23/2 Summary mounting evidence that Euclidean Quantum Gravity is asymtotically safe exact renormalization group: space-time carries a one-parameter family of metrics g µν k describe metric structure at different length-scales regimes with non-trivial RG running: space-time obtains fractal properties fractal dimension of space-time continuously changes: D s = 4 at macroscopic distances D s = 2 at microscopic scales