ds/cft Contents Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, Lecture Lecture 2 5

Transcription

1 ds/cft Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, 2011 Contents 1 Lecture Lecture 2 5 1

2 ds/cft Lecture 1 1 Lecture 1 We will first review calculation of quantum field theory in de-sitter space. This is a calculation that you should do yourself and you should understand, and can even do it when waking in the middle of the night. The action for a massless scalar field is S = ( φ) 2 (1) The metric for de-sitter space is taken as the conformal slicing ds 2 = dη2 + dx 2 η 2 (2) If we take two observers staying at constant x, their distance will be growing because of the growing of η. It is good to use the Fourier modes of the fields φ = d 3 xe ik x φ k (η) (3) The action is then like S = dη 1 η 2 ( η φ k 2 k 2 φ k 2) (4) This is like a simple harmonic oscillator action. When η then the field becomes massless at early time. When η 0 at late time the field becomes infinitely heavy. The conjugate momentum is π = L ( η φ) = 1 η 2 ηφ (5) We can write φ = a f + af, and by requiring [a, a ] = 1 and the canonical commutation relations we can normalize f. For very early time, we can ignore η term in the equation of motion and the solution will just be a harmonic oscillator f e ikη. And compare to late time we can get f (1 ikη)e ikη (6) And we will work out the normalization constant to be 1/k 3/2. At very early times the two observer will be very close to each other compare to the size of the space R, so the early time vacuum will be almost free vacuum. Take the vacuum defined by a 0 = 0 then the vacuum expectation value for two point correlation function at equal time is φ(η)φ(η) = f(η) 2 = 1 k 3 (1 + k2 η 2 ) (7) As a consistency check we can see that for early times η we have η 2 /k and for late time η 0 we have 1/k 3. The interesting point is at ηk 1 and that is when the distance of the observers cross the causal lightcone. We can get the position space two point function by integration and we get a logarithmic infinity φ(x)φ(x) log(x) = 2 d 3 k eikx k 3 (8)

3 ds/cft Lecture 1 This is due to the infrared divergence of the massless scalar field. Now let s look at interacting field theories. Let s add a φ 3 interaction to the Lagrangian and compute the 3-point function φ(η)φ(η)φ(η). We will need to choose appropriate boundary condition at early times, similar to the positive frequency condition for the massless scalar field. The proper way to evaluate the above expression is φ(η)φ(η)φ(η) = 0 U 1 φ 3 (η)u 0 (9) The evolution operator U is from to η, and the integral takes the contour from to η and back to. By boundary condition at early times we have φ e ikη and this is an oscillating function for large η. The trick to make it converge is to give time a little imaginary part η η + iε η. This is for the top contour from to η, and we shift it upwards a bit. This mathematical language is equivalent to choosing a vacuum, which means that the vacuum is Minkowski at very early times. For the lower contour we shift it downwards a bit. For massive fields and interacting fields this will lead to a finite answer, but for massless fields there is subtlety because of infrared divergences. What happens if we go to another coordinate? The above prescription is de-sitter invariant, and give us a two-point function which is defined everywhere on the de-sitter space. The coordinate of the global slicing is ds 2 = dτ 2 + cosh 2 τdω 2 3 (10) We calculate correlation functions in halfsphere of S 4 instead and analytically continue it to de-sitter space, and define it on the whole space. So let s consider a massless field on S 4. However it can t be well defined. This fact is intimately related to the infrared divergence. This prescription is choosing a vacuum in the middle τ = 0 of the de-sitter space. This is different from choosing a vacuum at earlier time and evolve it to this time. This is a peculiarity of this vacuum. Let s try to understand this calculation in the wave function point of view. If we calculate the two point function then we can calculate the square of the wave function. For late times we have φ 2 (x) = 1/k 3 so we expect that the wave function will be Φ 2 = e k3 φ 2 (11) When we are saying that the wave function distribution is gaussian, this is what we mean. Let s leave quantum field theory and come to some gravity calculation. The action for gravity is g(m S = 2 R 2Λ) = M 2 g(r 6) (12) The metric will be and the correlation function for graviton is H 2 ds 2 = dη2 + dx i dx j h ij η 2 (13) h h H2 M 2 1 k 3 (14) Now we need to do some calculation for the wave function. For tree level Feyman diagrams they are contained in the solution of the classical equation of motion. The question is which solution to use. One of the boundary condition is that φ(x, η) at late time should be known as φ b (x), which we measure. For early time boundary condition we assume that φ e ikη when η. The way we define the wave function will be to take the inner product Φ[φ b ] = φ b (x) BH (15) 3

4 ds/cft Lecture 1 We would like to calculate it using the solution of the classical equation Φ = e is[φ cl(x,η)] (16) The classical solution we want is φ φ b (x)(1 ikη)e ikη. We can plug this into the classical action S = dη( φ) 2 /η 2. The trick we want to use is integration by parts and turn it into a term like φ η φ. In the end the wave function is like Φ = e i 1 η φ b(x) 2 k 2 e φ b 2 k 3 (17) 4

5 ds/cft Lecture 2 2 Lecture 2 We start with considering the condition when the fluctuation can be thought of as classical. compute that So by the canonical commutation relation we have We can π = 1 η 2 ηφ = 1 η 3 tφ (18) [π, φ] = i = [φ, t φ] = η 3 (19) So at late times the commutator becomes small and can be neglected, and this is when the fluctuation can be thought of as classical. The next point we want to emphasize is scale invariance. The metric of de-sitter space in conformal slicing is invariant under the transformation x λx, η λη (20) This can be easier understood using diagrams. This is useful because if we want to calculate something time-independent, then we d better have it to be scale independent. De-Sitter space have special conformal symmetry which can be better thought of using the hyperbloid This is invariant under SO(1, 4) inside R 1,4. These are transformations X X X 2 4 = R 2 ds (21) δx i = x 2 b i ± (x j b j )x i, δx i = a i, δx i = λx i (22) Now let s come back to the main topic. We defined QFT on de-sitter space, and the Bunch Davies vacuum, which is the adiabatic vacuum in the far past. We defined the wave functional as Φ[φ(x)] = φ(x) BD = e is (23) And the expectation value or the two point correlation function is φ φ = Dφ φ 2 Φ 2. The integration contour for the propagator was chosen as a tilted line in the complex η plane. We can rotate it to the imaginary axis by analytic continuation and write η = iz, and the metric is ( dz ds 2 = RdS dx 2 ) (24) This is the Euclidean Anti-deSitter space. The Euclidean AdS space is described by the same equation as the ds space but with the RdS 2 changed to R2 AdS. This is the result of analytic continuation. The classical action of gravity with fixed boundaries is [ ] e S L = e ( M 2 R g(r AdS 2 + 2) 2K (25) z 2 Let s calcute the euclidean acion demanding that z = υ. ( ) [ e S E = e (MR AdS gb(x)d 3 ε 3 x + 1 ] gb R (1) B ε + F inite (26) 5

6 ds/cft Lecture 2 We want to calculate the partition function of the metric. There will be divergent piece and a renormalizable piece Z = Z div Z ren (g). We define this partition function as [ ] gdx Z[g ij ] = D[ψ, A] exp 3 (ψdψ + F 2 ) (27) For small perpurbation of the metric g ij = η ij +h ij, then we can write the exponential as exp[s 0 + T ij h ij ] where S 0 is the action with Minkowski metric. We know that this is also the generating functional for the correlation functions, so we can get the correlation of the stress tensor by T ij (x)t kl (y) CFT = δ δ δh ij (x) δh kl (y) Z h ij =0 (28) The AdS/CFT correspondence tells us that the wave functional for gravity in the AdS space Φ is equal to the generating functional of a CFT that is defined on the boundary. Φ = Z CF T. This equality only holds for the finite piece of the partition function. The partition function has similar structure as above which can be written as [ Z = exp (M R AdS ) 2 F 0 + F ] ( ) F 2 (29) where if we consider a gauge theory of U(N) then in the large N expansion the fist coefficient is identified as N 2 and the third coefficient is identified as 1/N 2. As a concrete example let s consider AdS 4 S 7 gravity which is dual to a gauge theory in 3D with gauge group U(N) U(N). The action is defined by the Chern-Simons term, and it has the form exp [ ik ] h(f A). Let s now consider the case when we have an S 3 boundary for the Euclidean AdS space at ρ =. The metric is ds 2 = dρ 2 + sinh 2 ρ dω 2 3 (30) and the on-shell action is S E = ρc 0 dρ sinh ρ 3 e 3ρc + e ρc + constant (31) People have calculated the same thing for the dual field theory and they found exactly the same result. We discussed that in de-sitter space we need to use a contour which has a part in S 4. We can also do this on the contour which is τ = ρ + iπ/2, which is equivalent to doing the integration in Euclidean AdS. The constant that we get from integrating on the S 4 part is just e isds/2 where S ds is the entropy of the de-sitter space. This constant is also related to the constant that appear in the above equation. In what sense is Φ a wave function? We can choose a gauge where the metric is ds 2 = dt 2 +g ij dx i dx j. There are a few constraints on the wave function, one of which is ( ) δ i Φ[g ij ] = 0 (32) δg ij This means that the wave function should be invariant under reparametrization. The other constraint is the Hamiltonian constraint. 6

7 ds/cft Lecture 3 3 Lecture 3 We start with local quantum field theory. At each point of space we have a few degrees of freedom, and we construct quantum theory that is unitary and Lorentz invariant. To say that the theory is local means that the theory is causal, and that locally it doesn t know the metric change at other places. This manifests as an operator T µν (x) defined at every point in space, and it is coupled locally with the metric perturbation δg µν. We want to ask whether we can have emergent graviton from a local quantum field theory. It is easy to show that this is not possible. This is called the Weinberg-Witten theorem. We start with an energymomentum tensor for the particles T µν (x) and we assume that at low energies we have a massless spin 2 particle. We also assume that this theory is Lorentz invariant. Let s consider the amplitude of the graph of a incoming particle with momentum q and two graviton legs with helicity +. We can boost the particle momentum to q = (0, 0, 0, q z ). But then the graviton changes its spin from +2 to 2, and the particle can t change its spin by 4, which is not possible because the particle has to have spin 2 or less. This gives rise to a contradiction. There are other handwaving arguments which are given by entropy bounds. We know that the entropy is bounded by the area, but if it s a local theory it would be bound by volume. Let s look at an example of emergent theory. Let s look at a system with N fermions with a harmonic oscillator potential. We have a Fermi surface in phase space lice a circle, and we have ripples on the radial direction. This corresponds to particle-hole excitations. But this emergent theory is not UV complete, because to a high enough energy the weak coupling approximation breaks down. Ripples will not interact with another far away, and we have approximate locality here. An emergent theory of gravity should have a Fock space, where we have 0 where E = 0 a 0 where E = ε 1 (p) a a 0 where E = ε(p 1 ) + ε(p 2 ) + small interaction We want to comment on how this is realized in string theory, where there is one kind of excitation of spin 2 and mass 0. This ensures that gravity has to emerge from string theory. We just argued that we can have emergent gravity in the same spacetime as the local QFT, but we can have D dimensional gravity emerging from d dimensional QFT. This is what happens in AdS/CFT. We have conformal symmetry in CFT as the local QFT on the left hand side, and diffeomorphism symmetry on the right hand side for AdS where ds 2 = dz2 + dx 2 d z 2 (33) Before we go on we need to explain the large N limit of SU(N) gauge theory and its correspondence to string theory. This argument was given first by t Hooft. For example we have the fied theory with Lagrangian L = 1 g 2 Tr [ F 2 + (Dϕ) 2 + ϕ 3 + ϕ ] (34) where all the fields are N N matrices. Because fields like A carry two indices A j i we use a double-line graph to draw the processes. There are planar diagrams. Every loop will give a factor of g 2 N because loops need to be summed over flavor index. However all the diagrams which can t be drawn on the plane will be suppressed by factors of N. Therefore we can take the limit where N whereas g 2 N is fixed. In this limit only planar diagrams dominate, and it looks similar to a free string theory. 7

8 ds/cft Lecture 3 Now let s come back to the conformal theory. We will be considering a gauge theory in the large N limit. The stress energy tensor will look like T µν = Tr[F µδ Fµ δ. And at weak coupling we will have only 2 gluons in this local operator, but at higher coupling we have a whole bunch of gluons. Now an operator like Tr[ϕ 2 ] will have scaling dimension which is given by = (mr) 2. Giving an anomalous dimension to the operators is like giving mass to the graviton in the bulk. 8