J. Ambjørn QG from CDT

Transcription

1 Lattice gravity Jan Ambjørn 1 1 Niels Bohr Institute, Copenhagen, Denmark Lectures, Strongnet, Bielefeld, June 15-16, 2011

2 4d QG regularized by CDT Main goal (at least in 80ties) for QG Obtain the background geometry ( g µν ) we observe Study the fluctuations around the background geometry What lattice gravity (CDT) offers: A non-perturbative QFT definition of QG A background independent formulation An emergent background geometry ( g µν ) The possibility to study the quantum fluctuations around this emergent background geometry.

3 Problems to confront for a lattice theory (1) How to face the non-renormalizability of quantum gravity (this is a problem for any field theory of quantum gravity, not only lattice theories) (2) Provide evidence of a continuum limit (where the continuum field theory has the desired properties) (3) If rotation is performed to Euclidean signature, how does one deal with the unboundedness of the Euclidean Einstein-Hilbert action? (4) If there exists no continuum field theory of gravity, can a lattice theory be of any use?

4 (1) Facing the non-renormalizability of gravity Effective QFT of gravity We believe gravity exists as an effective QFT for E 2 1/G. True for other non-renormalizable theories Weak interactions L = ψ ψ + G F ψ( )ψ ψ( )ψ Nonlinear sigma model L = ( π) F 2 π Good for E 2 1/G F and E 2 F 2 π. (π π) 2 1 π 2 /F 2 π

5 Effective QFT of gravity Lowest order quantum correction to the gravitational potential of a point particle: G G(r) Gr, G(r) = G (1 ω + ), ω = 167 r r 2 30π. The gravitational coupling constant becomes scale dependent and transferring from distance to energy we have G(E) = G(1 ωge 2 + ) G 1 + ωge 2.

6 Effective QFT of the electric charge Same calculation in QED e 2 r ( e2 (r), e 2 (r) = e 2 r 1 e2 6π ) ln(m r) + 2, m r 1. The electric charge is also scale dependent and has a Landau pole ( ) e 2 (E) = e e2 6π 2 ln(e/m) + e 2 1 e2 6π 2 ln(e/m).

7 BUT GE 2 1 G(E)E 2 1 G(E)E 2 < 1 ( 1?). Suddenly seems as if quantum gravity has become an (almost) reliable quantum theory at all energy scales. The behavior can be described the β function for QG. For the dimensionless coupling constant G(E) = G(E)E 2 E d G de = β( G), β( G) = 2 G 2ω G 2. Two fixed points (β( G) = 0): G = 0 and G = 1/ω.

8 β(g) β(g) g g Generic situation for asymptotic free theories in d dimensions, extended to d + ε dimensions. β(g) εg + β(g)

9 The four-fermi action, the nonlinear sigma model and QG are all renormalizable theories in 2d, with a negative β-function and have a 2 + ε expansion. For QG first explored by Kawai et al. Alternatively one can use the exact renormalization group approach. (Reuter et al., Litim,...). Philosophy: asymptotic safety (Weinberg).

10 (2) Continuum limit? C A P 0 B Triple point K 0

11 Defining the continuum limit in lattice field theory Let the lattice coordinate be x n = a n, a being the lattice spacing and O(x n ) an observable. log O(x n )O(x m ) n m /ξ(g 0 ) + o( n m ). ξ(g 0 ) 1 g 0 g c 0 ν, a(g 0) g 0 g c 0 ν. m ph a(g 0 ) = 1/ξ(g 0 ), e n m /ξ(g 0) = e m ph x n x m O(x n )O(y m ) falls off exponentially like e m ph x n y m for g 0 g c 0 when x n y m, but not n m, is kept fixed in the limit g 0 g c 0.

12 How to define the equivalent of O(x n )O(y m ) in a diffeomorphism invariant theory φφ(r) D[g µν ] e S[gµν] g(x) [g g(y) φ(x)φ(y) µν] matter δ(r d g µν (x, y)). φ(x)φ(y) [gµν] matter denotes the correlator of the matter fields calculated for a fixed geometry, defined by the metric g µν (x). It works in 2d Euclidean QG (Liouville gravity)

13 (3) Unboundedness of the Euclidean action Already the discussion about continuum limit of the lattice theories hinted a rotation to Euclidean signature. The Einstein-Hilbert action is unbounded from below, caused by the conformal factor: g µν = Ω 2 g µν S[g, Λ, G] = 1 16πG S[ g, Λ, G] = 1 16πG How is this dealt with?. d 4 ξ ( ) g R 2Λ. d 4 ξ ( g Ω 2 R + 6 µ Ω µ Ω 2ΛΩ 4).

14 Using the lattice regularization called dynamical triangulations (DT) the Euclidean action is bounded for a fixed lattice spacing a and a fixed four-volume V 4 = N 4 a 4. However, for a 0 the unboundedness re-emerges. S[T] = κ 2 N 2 (T) + κ 4 N 4 (T), c 1 < N 2 N 4 (= x) < c 2. The unbounded configurations corresponds to x c 2. But are they important in the non-perturbative path integral? Z = T e S[T] = N 4 e k 4N 4 N(N 2, N 4 )e κ2n2 N(N 2, N 4 ) e κ 2N 2 = P N4 (x), N 2 P N4 (x) = f(n 4 ) e κc 4 (κ 2)N 4 x

15 P N4 (x) small κ2 large κ 2 x=n 2/N 4 ) ) (κ c4 α(x x 0) ( κ 2 c4 α(x x 0) 2 P N4 (x) A e N 4 + Ã en 4. k 2 κ 2 + κ 2, κ c 4 k 4 c + κ 2x 0, κ c 4 k 4 c + κ 2 x 0 Phase transition when κ c 4 = κc 4.

16 The A-C transition e+06 2e e+06 3e e+06 4e e+06 5e N0/N e+06 2e e+06 3e e+06 4e e+06 5e N0/N

17 Do we know examples of such entropy driven phase transitions? Yes, the Kosterlitz-Thouless transition in the XY model. This Abelian 2d spin model has vortices with energy E = κ ln(r/a) Saturating the partition function by single vortex configurations: Z e F/k BT = spin configurations e E[spin]/k BT ( ) R 2 e [κ ln(r/a)]/kbt. a S = k B ln(number of configurations) has the same functional form as the vortex energy. Thus F = E ST = (κ 2k B T) ln(r/a)

18 (4) No continuum limit? Examples Lattice compact U(1) gauge theory in 3 dimensions has confinement for all values of the coupling constant, due to lattice monopoles. It describes perfectly the non-perturbative physics of the Georgi-Glashow model, i.e. the physics below the scale of Higgs and the W-particle. The formula for the string tension is the same expressed in terms of lattice monopoles masses and continuum monopole masses. Lattice compact U(1) gauge theory in 4 dimensions at the phase transition point describes the low energy physics of certain broken N = 1, 2 supersymmetric field theories. In fact, one can use the supersymmetric symmetry breaking technology of Seiberg et al. scale matching to post-dict (unfortunately) the lattice critical exponents.

19 Lattice gravity: causal dynamical triangulations (CDT) Basic tool: The path integral Text-book example: non-relativistic particle in one dimension. x f x a x(t) = x(t) + y(t) y /mω In QG we want x(t) x i t i t f t y G Transition amplitude as a weighted sum over all possible trajectories. On the plot: time is discretized in steps a, trajectories are piecewise linear.

20 In a continuum limit a 0 G(x i, x f, t) := trajectories: x i x f e is[x(t)] where S[x(t)] is a classical action. The QG amplitude between the two geometric states G(g i, g f, t) := geometries: g i g f e is[gµν(t )] To define this path integral we need a geometric cut-off a and a definition of the class of geometries entering.

21 showcasing piecewise linear geometries via building blocks: 2d 3d 4d Piecewise linear geometry is defined without coordinates

22 a b d b b d c b d c a c a a a

23 t+1 t+1 t t (4,1) (3,2) t 1 CDT slicing in proper time. Topology of space preserved. a 2 t = αa 2 s, is L [α] = S E [ α] ( ) ( ) S E [ α] = (κ 0 +6 )N 0 +κ 4 N (2,3) 4 +N (1,4) 4 + N (2,3) 4 +2N (1,4) 4

24 The way this comes about: 1 S E (Λ, G) = 16πGa 2 σ d 2 T d 2 Introducing dimensionless quantities: ( ) 2ε σ d 2 V σ d 2 2Λ V(σ d 2 ), V(σ d 2 ) = a d V(σ d 2 ), V σ d 2a 2 d, κ = ad 2 16πG, λ = 2Λad 16πG, and the action becomes S E (λ, κ; T) = σ d 2 T d 2 ( ) κ 2ε σ d 2 V σ d 2 λ V(σ d 2 ) Using building blocks the action can be expressed in terms of the number of building blocks (and the number of subsimplices)

25 G(g i, g f, t) := e is[gµν(t )] = lim a 0 geometries: g i g f T:T (3) i T (3) f 1 C T e is T G E (g i, g f, t, κ 0, κ 4, ) = lim a 0 T:T (3) i T (3) f 1 C T e S E[T] x f e iĥt x i x f e Ĥτ x i

26 Scaling in the IR limit? Z(κ 0, κ 4 ) = N 4 e κ 4N 4 Z N4 (κ 0 ), where Z N4 (κ 0 ) is the partition function for a fixed number N 4 of four-simplices (we ignore for simplicity), namely, Z N4 (κ 0 ) = e k c 4 N 4 f(n 4, κ 0 ), Z(κ 0, κ 4 ) = N 4 e (κ 4 κ c 4 )N 4 f(n 4, κ 0 ) We want to consider the limit N 4, and fine-tune κ 4 κ c 4 for fixed κ 0. We expect the physical cosmological constant Λ to be defined by the approach to the critical point according to κ 4 = κ c 4 + Λ 16πG a4, (κ 4 κ c 4 ) N 4 = Λ 16πG V 4, V 4 = N 4 a 4,

27 How can one imagine obtaining an interesting continuum behavior as a function of κ 0? Assume f(n 4, κ 0 ) has the form (numerical evidence) f(n 4, κ 0 ) = e k 1(κ 0 ) N 4, e 1 G RV gr 4 = e c V4 G. Z(κ 4, κ 0 ) = N 4 e (κ 4 κ c 4 )N 4+k 1 (κ 0 ) N 4. Search for κ c 0 with k 1(κ c 0 ) = 0, with the approach to this point governed by k 1 (κ 0 ) a2 G, i.e. k 1(κ 0 ) N 4 ( k 2 Z(κ 4, κ 0 ) exp 1 (κ 0 ) ) 4(κ 4 κ c 4 ) = exp V4 G. ( c GΛ as one would naïvely expect from Einstein s equations, with the partition function being dominated by a typical instanton contribution, for a suitable constant c. ),

28 UV scaling limit? If we are close to the UV fixed point, we know that G will not be constant when we change scale, but Ĝ(a) will. Writing G(a) = a 2Ĝ(a) a2ĝ, κ 4 κ c 4 = Λ G(a) a4 Λ Ĝ a2, k 1 (κ c 0 ) = a2 G(a) 1 Ĝ. The first of these relations now looks two-dimensional because of the anomalous scaling of G(a)! Nevertheless, the expectation value of the four-volume is still finite: V 4 = N 4 a 4 κ2 1 (κc 0 ) (κ 4 κ c a4 4 )2

29 Relation to asymptotic freedom Assume now that we have a fixed point for gravity. The gravitational coupling constant is dimensionful, and we can write for the bare coupling constant G(a) = a 2 Ĝ(a), a dĝ da = β(ĝ), β(ĝ) = 2Ĝ cĝ3 +. The putative non-gaussian fixed point corresponds to Ĝ Ĝ, i.e. G(a) Ĝ a 2. In our case it is tempting to identify our dimensionless constant k 1 with 1/Ĝ, up to the constant of proportionality. Close to the UV fixed point we have Ĝ(a) = Ĝ Ka c, k 1 = k 1 + Ka c, c = β (Ĝ ). Usually one relates the lattice spacing near the fixed point to the bare coupling constants with the help of some correlation length ξ.

30 line of constant V 4 IR UV κ 0 Consider V 4 = N 4 a 4 as fixed. It requires the fine-tuning of coupling constants. k 1 (N 4 ) = k c 1 K N c/4 4. How to determine k 1 (N 4 )?

31 Phase diagram of CDT C A P B K 0 Triple point Lifshitz-like diagram... Phase C: constant magnetization (constant 4d geometry) Phase B: zero magnetization (no 4d geometry) Phase A: oscillating magnetization (conformal mode?)

32 Volume distribution in (imaginary) time Phase A. The universe oscillating in time direction. The oscillation maybe reflecting the dominance of the conformal mode. Phase B. Compactification into a 3d Euclidean DT. Only minimal extension in the time direction. Phase C. Extended de Sitter phase. d H = 4.

33 S Lifshitz [φ] = d D x ( ) µ i ( i φ) 2 + ( φ) νφ 2 + φ 4 + φ > 0 [g] > 0 C dφ_ > 0 dt dg _ > 0 dt φ = 0 A, [g] = 0 B

34 Snapshot of a typical configuration A typical configuration. Distribution of a spatial volume N 3 (t) as a function of (imaginary) time t. Quantum fluctuation around a semiclassical background? Configuration consists of a stalk of the cut-off size and a blob. Center of the blob can shift. We fix the center of mass to be at zero time.

35 N3(i) N 3 (i) N 3/4 4 cos 3 i ( i s 0 N 1/4 4 )

36 Minisuperspace model The semiclassical distribution can be obtained from the minisuperspace effective action of Hartle and Hawking S eff = 1 24πG dt g tt ( g tt V ) 2 3 (t) + k 2 V 1/3 V 3 (t) 3 (t) λv 3 (t), The discretization of this action is (and we have reconstructed it from the date (the 3-volume 3-volume correlations)) ( (N3 (i + 1) N 3 (i)) 2 S discr = k 1 N 3 (i) i ) + k 2 N 1/3 3 (i) λn 3 (i), G = a2 C4 s0 2 k

37 Quantum fluctuations The classical solution to the minisuperspace action is ( ) gtt V3 cl (t) = V 3 t 4 4B cos3 B where τ = g tt t, V 4 = 8π 2 R 4 /3 and g tt = R/B. Writing V 3 (t) = V3 cl (t) + x(t) we can expand the action around this solution S(V 3 ) = S(V3 cl ) + 1 B dt 18πG x(t)ĥx(t). where the Hermitian operator Ĥ is: V 4 Ĥ = d 1 d dt cos 3 (t/b) dt 4 B 2 cos 5 (t/b),

38 In the quadratic approximation the volume fluctuations are: C(t, t ) := x(t)x(t ) Ĥ 1 (t, t ). Ĉ and Ĥ have the same eigenfunctions. C(t, t ) can be measured as ( )( ) C(i, i ) = N 3 (i) N 3 (i) N 3 (i ) N 3 (i ), and its eigenfunctions can be found and compared to the ones calculated from Ĥ.

39 e (2) (t) 0.1 e (4) (t) t t No parameters are put in! (expect t i /B = i/s 0 N 1/4 4 ) We conclude that the quadratic approximation to the minisuperspace action describes the measured quantum fluctuations well.

40 Size of our Quantum universe For a specific value of the bare coupling constants (κ 0 = 2.2, = 0.6) we have high-statistics measurements for N 4 ranging from to four-simplices. Largest universe corresponds to approx hyper-cubes. We have G = const. a 2 /k 1 and we have measured k 1. G 0.23a 2, l P 0.48a, l P G. From V 4 = 8π 2 R 4 /3 = C 4 N 4 a 4, we obtain that R = 3.1a The linear size πr of the quantum de Sitter universes studied here lies in the range of l P for the N 4 used.

41 Trans-Planckian? l P = G a k1 (κ 0, ) i.e. k 1 (κ 0, ) 0. BUT IS IT POSSIBLE? k κ 0

42 Summary and perspectives We have obtained the (Euclidean) minisuperspace action from first principles. (The self-organized de Sitter space) We have an effective field theory of (something we call) QG down to a few Planck scales. Investigate a possible UV fixed point (points, the B-C line). Possibly Hořava-Lifshitz gravity. couple matter to the system and investigate cosmological implications. Measure the wave function of the universe x e tĥ y Ψ 0 (y)ψ 0 (x) e te 0

43 Monte Carlo simulations of lattice gravity Z = φ e S[φ], O(φ) = Z 1 φ O(φ) e S[φ]. The purpose of the Monte Carlo simulations is to generate a sequence of statistically independent field configurations φ(n), n = 1,...,N with the probability distribution Then P(φ(n)) = Z 1 e S[φ(n)]. O(φ) N = 1 N N O(φ(n)). n=1 serves as an estimator of the expectation value and one has O(φ) N O(φ) for N.

44 In a Monte Carlo simulation a change φ φ of the field configuration is usually generated by a stochastic process T, a Markow chain. The field will perform a random walk in the space of field configurations with a transition function, or transition probability, T (φ φ ). Thus, if we after a certain number n of steps (changes of the field configuration) have arrived at a field configuration φ(n), T (φ(n) φ(n + 1)) is the probability of changing φ(n) to φ(n + 1) in the next step. We have φ T (φ φ ) = 1 for all φ.

45 The transition probability should be chosen such that (i) Any field configuration φ can be reached in a finite number of steps ( ergodicity) (ii) The probability distribution of field configurations converges, as the number of steps goes to infinity, to the Boltzmann distribution. The convergence of the Markov chain is usually ensured by choosing T to satisfy the so-called rule of detailed balance P(φ) T (φ φ ) = P(φ )T (φ φ). Thus T (φ φ ) = T (φ φ) = 0 or T (φ φ ) T (φ φ) = P(φ ) P(φ).

46 Usually decomposes the transition probability T (φ φ ) into a selection probability g(φ φ ) and an acceptance ratio A(φ φ ). We then have: P(φ ) P(φ) = T (φ φ ) T (φ φ) = g(φ φ )A(φ φ ) g(φ φ)a(φ φ). The selection probability g(φ φ ) is now designed to select the configurations φ, φ where T (φ φ ) is different from zero and assign a weight of our own choice to the transition φ φ. The acceptance ratio A(φ φ ) should then be chosen to ensure detailed balance. A general choice, used in many Monte Carlo simulations, is the so-called Metropolis algorithm: ( A(φ φ ) = min 1, g(φ φ) g(φ φ ) ( A(φ φ) = min 1, g(φ φ ) g(φ φ) P(φ ) P(φ) P(φ) P(φ ) ), ).

47 DT and CDT implenetation of MC Unlabeled versus labeled triangulations: Z = T 1 C(T) e S(T) = T l 1 N 0 (T l )! e S(T l) Two labeled triangulations will be identical if a mapping (a relabeling) i σ(i) maps neighbors to neigbors. In the computer we use labeled triangulations. We fix topology. We need moves to get ergodically arround in the class of d-dimensions triangulations of a fixed topology. A minimal set of moves are the Pachner moves. (here for CDT in the case of three dimensions)

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49 In 2d gr = 2πχ, i.e. a topological invariant for fixed topology. Thus S[T] = λn 2 (T) = λ(2n 0 χ), and P(T) = 1 Z 1 N 0 (T)! e 2λN 0(T).

50 The (2,4) move: If the old vertices are labeled from 1 to N 0, we assign the label N to the new vertex (and new links and new triangles are defined by pairs and triples of vertex labels, the triples also defining the correct orientation). Given the labeled triangulation T N0 we can in this way get to N 0 labeled triangulations T N0 +1 by chosing different vertices and performing the (2,4)-move. We define the selection probability g(t N0 T N0 +1) to be the same for all triangulations T N0 +1 which can be reached in this way and to be zero for all other labeled T N0 +1 triangulations. Thus for the labeled triangulations which can be reached we have: g(t N0 T N0 +1) = 1 N 0, and we implement this in the computer program by choosing randomly with uniform probability a vertex in T N0.

51 The (4,2) move: Given a labeled triangulation T N0 +1 we perform the inverse move by the following procedure: we select the vertex labeled N Let us assume it is of order 4. Then we delete it from the list, in this way creating a labeled triangulation T N0. If the vertex labeled N is not of order 4 we do not perform the move. Thus, for the triangulations T N0 +1 where the move can be performed we can only reach one triangulation T N0 and the selection probability g(t N0 +1 T N0 ) defined by this procedure is one. Finally we choose the acceptance ratios A(T T ) in accordance with the Metropolis algorithm ( N 0 A(T N0 T N0 +1) = min 1, N e 2λ), ( A(T N0 +1 T N0 ) = min 1, N e 2λ). N 0

52 Data structure: natural pointer oriented Label the vertices of (a 2d) triangulation by i, i = 1,...,n. Denote the neighbors to vertex i by k(i, m), m = 1,...,o(i), ordered cyclicly in accordance with the orientation, and where o(i) denotes the order of vertex i i 1 l(i,1) l(i,3) k(i,3) l(i,4) l(i,1) k(i,1) l(i,2) k(i,4) i k(i,3) k(i,2) n l(i,2) k(i,2) l(i,3) k(i,5) k(i,1)