UNIVERSITY OF AMSTERDAM

UNIVERSITY OF AMSTERDAM MSc Physics Theoretical Physics MASTER THESIS Investigating the ER=EPR proposal A field theory study of long wormholes by Bernardo Zan 10513752 May 2015 54 ECTS Supervisor: Prof. dr. Jan de Boer Examiner: dr. Ben Freivogel Institute for Theoretical Physics

Abstract The ER=EPR correspondence states that two entangled systems are always connected by an Einstein Rosen bridge. This proposal might solve the question of black hole evaporation without requiring the presence of firewalls. However, it has been argued that the conditions necessary to have a smooth geometrical description cannot be satisfied by the randomness of a typical state. It was shown that correlators between two entangled theories are weak for typical states and depend on the spectrum of the field theory; this is taken as evidence for the absence of a semiclassical wormhole. We study a non typical state which presents a long semiclassical wormhole description from the field theory point of view. We find that, under certain conditions, a random matrix approach is able to model this state: the correlator between the two theories is small and it does not depend on the spectrum of the theories.

Contents Introduction 1 1 Quantum field theory at finite temperature 5 1.1 KMS condition and periodicity of fields........................ 5 1.1.1 Path integral for the partition function..................... 6 1.2 Thermofield dynamics.................................. 7 1.3 Real time formalism................................... 8 1.4 Schwinger Keldysh propagators............................ 9 1.5 Free bosonic field..................................... 11 1.6 Bosonic propagators in position space......................... 12 1.6.1 Operators on the same theory.......................... 13 1.6.2 Operators on different theories......................... 15 1.6.3 Massless limit................................... 16 2 The ER=EPR proposal 19 2.1 AdS space and the AdS/CFT correspondence..................... 19 2.2 The BTZ black hole.................................... 20 2.2.1 Propagators for the BTZ black hole....................... 22 2.2.2 Holography for the BTZ black hole....................... 23 2.3 Hawking radiation.................................... 24 2.4 The information paradox and the firewall proposal................. 26 2.5 The ER=EPR proposal.................................. 28 2.5.1 Reactions to the ER=EPR proposal....................... 31 2.6 Long semiclassical wormholes............................. 32 2.7 Motivation......................................... 35 3 Quenching the system 37 3.1 Response theory to second order............................ 37 3.2 A first toy model..................................... 38 3.3 A second toy model.................................... 39 3.3.1 On a circle..................................... 40 III

4 Random matrices 43 4.1 Why random matrices?.................................. 43 4.2 Ensembles and states................................... 44 4.2.1 Non energy changing operators........................ 45 4.2.2 Energy changing operators........................... 45 4.3 Random matrices and wormholes........................... 46 4.4 Random matrices and time dependence........................ 48 4.5 Unitary matrices...................................... 49 4.6 Numerical results..................................... 52 4.6.1 Triangular matrices................................ 53 4.6.2 Matrices with no diagonal elements...................... 53 4.7 Comparison with the bulk description......................... 56 Conclusion 59 Appendices 61 A Second order quench 61 A.1 On the line......................................... 61 A.2 On the circle........................................ 63 B Weingarten function 65 B.1 Weingarten function................................... 65 B.2 Unitary matrices on the microcanonical state..................... 66 B.3 Unitary matrices on the TFD state............................ 66 Bibliography 69 IV

Introduction Many aspects of the physics of black hole are still not understood or subject to debate. Since the discovery of Hawking radiation [1] in 1975, combining gravitation and quantum effects has proven to be a challenging task. The fate of the information that enters a black hole after this evaporates through Hawking radiation was initially not well understood. This radiation appears thermal to a distant static observer. Unitarity, one of the well established foundations of quantum mechanics, appeared to be violated by the evolution of a black hole from a pure state to such a mixed state. Most of the scientific community agrees now that the evaporation of a black hole is a unitary process. The information leaks out through entanglement between the quanta being emitted through Hawking radiation at different times. The AdS/CFT correspondence [2] has proven important in answering this question. This correspondence relates a gravitational theory on a manifold which is asymptotically AdS, a spacetime with a negative cosmological constant, in d + 1 dimensions to a conformal field theory, a quantum field theory that is conformally invariant, living on the d dimensional boundary of the manifold. When one of the two theories is strongly coupled, the other one is weakly coupled. This correspondence has many applications. One of these, for example, is to carry out computations in strongly coupled theories that would not be possible otherwise. It turns out to be useful also when considering the process of evaporation of a black hole. Non unitary of this process would not be consistent with the AdS/CFT correspondence. Since some types of black holes are dual to conformal field theories which evolve unitarily, their evaporation must be unitary as well. However, the solution of the information paradox has generated other apparent inconsistencies. The quanta emitted through Hawking radiation are entangled with quanta inside the black hole. Information starts leaking out after half of the black hole has evaporated. If effective field theory is a valid description of the physics outside the horizon, this information must be carried by the entanglement between early and late Hawking radiation. These modes are approximately maximally entangled: monogamy of entanglement, the principle for which one system cannot be maximally entangled with two independent systems, is violated. In 2012 it was proposed [3] that the modes of Hawking radiation are not entangled with modes inside the black hole, and monogamy of entanglement is not violated. However, there is not a clear mechanism which would allow this breaking of entanglement. Besides, it would mean that just behind the horizon there are many energetic particles, reason for which this proposal is called 1

Introduction 2 the firewall proposal. The horizon of the black hole would then be a special place for a freely falling observer, hence violating Einstein equivalence principle. In 2013 a different solution of the paradox was proposed [4]. It conjectures a correspondence between entanglement and Einsten Rosen bridges, and goes under the name of ER=EPR. According to the proposal, two entangled systems are always connected by a wormhole. There are cases for which this is known to be true. The BTZ black hole [5] is dual to an entangled state, the thermofield double state (TFD) [6]. However, according to the ER=EPR proposal, the duality holds for all entangled systems. For example, even simpler systems, such as two entangled qubits, should be connected by a wormhole; however, this wormhole will have a highly quantum nature, and cannot be studied given our current understanding of quantum gravity. If correct, the ER=EPR proposal could imply the absence of firewalls at the black hole horizon, consistently with Einstein equivalence principle. An observer trying to measure the violation of the monogamy of entanglement would first measure a mode of the early and one of the late Hawking radiation. Since information is not lost when a black hole evaporates, the observer would see that these two modes are entangled. Then he would enter the black hole to check the entanglement between the late radiation mode and its Hawking partner mode inside the black hole. He would however meet an energetic particle on the horizon, making the measurement impossible. This energetic particle would be created by the action of the observer when he first made a measurement on the early Hawking radiation. This action at a distance would be possible only because, according to the ER=EPR proposal, two entangled systems are connected by a wormhole. This small violation of locality might explain the process of evaporation of black holes without requiring the presence of firewalls. Investigating the ER=EPR proposal turns out to be a difficult task. Different works [7, 8] have argued that, while the TFD is dual to a semiclassical wormhole, typical states are not. Correlations between the two entangled systems suppressed by factors of e S, dependent on the spectrum of the field theory, are taken as evidence of the absence of a wormhole, since the geometry should arise from the coarse graining of the field theory. However, it was shown in [9] and [10] by Shenker and Stanford that it is possible to construct long semiclassical wormholes, for which correlators can be made of order e S. In [8] random matrices were used to study states more general than the TFD. Under the assumption that a correlator that depends on the number of states of a theory is evidence for the absence of a wormhole, the authors claimed that typical states are not connected by wormholes. The SS construction is, instead, highly non typical. The main questions we will try to answer in this work concerns the validity of a free field or a random matrix approach for this latter SS state and whether this state presents correlations that depend on the number of sates of the field theory. If e S correlations follow only from an ad hoc construction, it would look like the claim of [7, 8] is correct and typical states are not dual to smooth geometries. The structure of this work is as follows. In chapter 1 we review some properties of quantum field theory at finite temperature, and focus on the bosonic 1 + 1 dimensional case. This will play a role in the context of the AdS/CFT correspondence. In chapter 2, the BTZ black hole, which provides an example of a wormhole with an entangled dual, is reviewed. We also

3 Introduction explain the process of black hole evaporation and the firewall as well as the ER=EPR proposal. In chapter 3 we try to model the SS construction, which consists in introducing matter on the boundary of the black hole, using a quench in a free field theory. Eventually, in chapter 4, we try to model the same process using random matrices drawn from different ensembles.

1 Quantum field theory at finite temperature Quantum field theory was initially developed at zero temperature; much work has been devoted to generalizing its results to finite temperature. Different approaches, allowing to keep some tools of T = 0 QFT, exist; in the following section we will review the thermofield formalism and the real time formalism. The first makes it possible to express the statistical value of an operator as the expectation value of the operator in a quantum field theory, and has a direct application when talking about black holes in the context of the AdS/CFT correspondence; the second achieves the generalization of Feynman diagrams to finite temperature. We will show that the latter approach is equivalent to the first one, and proves useful in computations in the same context. Later, we will specialize in the case of a free boson in 1 + 1 dimensions, which will be useful in the context of the AdS/CFT correspondence, as will be explained in chapter 2. We begin by reviewing some general features of finite temperature field theory. At finite temperature T, the expectation value of an operator O is given by O = Tr(ρ eq O) (1.1) where ρ eq is the thermal density matrix, ρ eq = e βh /Z and β = 1 T (in this work we set Boltzmann s constant k B to 1). This density matrix represents the case of a canonical ensemble. In the presence of a Noether conserved charge Q and a chemical potential µ, we need to consider the grand canonical density matrix, ρ = 1 Z e β(h+µq). However, for our purposes it will be enough to consider the canonical ensemble. 1.1 KMS condition and periodicity of fields Lorentz invariance is broken at finite temperature. It is possible to see this from the Kubo- Martin-Schwinger (KMS) condition. If we consider the expectation value of two operators A(t)B(t ) at finite temperature, we 5

KMS condition and periodicity of fields 6 can use the cyclicity of the trace to see Tr(ρ eq A(t)B(t )) = 1 Z Tr(e βh A(t)B(t )) = 1 Z Tr(e βh B(t )e βh A(t)e βh ) (1.2) = 1 Z Tr(e βh B(t )A(t + iβ)). We are working in the Heisenberg picture, A(t) = e ith A(0)e ith. We have therefore what is known as the KMS condition: A(t)B(t ) = B(t )A(t + iβ) (1.3) If we define (expressing only the time dependence for simplicity) C > (t t ) = C < (t t) = φ(t)φ(t ), then we have C > (t iβ) = φ(t iβ)φ(0) = φ(0)φ(t) = C < (t). (1.4) Fourier transforming the two sides of (1.4) we obtain φ k φ k = e βk 0 φ k φ k, (1.5) and we see that, because of the term e βk 0, this is not Lorentz invariant. An equivalent way of stating that finite temperature breaks Lorentz invariance is that, when inserting a system in a heat bath, we have a preferred frame, the rest frame of the heat bath. 1.1.1 Path integral for the partition function The partition function Z is defined as the trace of the density matrix Z = Tr ρ eq. (1.6) This quantity plays a crucial role in statistical mechanics, since most of the interesting thermodynamic quantities can be obtained from it. It has been long known that statistical quantum mechanics is connected to quantum field theory in Euclidean space. In fact, it is possible to express the partition function Z as a path integral in Euclidean space, with some periodic boundary conditions. The lagrangian for a real scalar field φ is L = 1 2 µφ µ φ V(φ) (1.7) where V(φ) is a potential. The transition amplitude can be expressed as (see for example [11]) φ b (x) e iht φ a (x) = [ t Dφ exp i dt 0 ] d d 1 xl (x) (1.8)

7 Quantum field theory at finite temperature with the condition φ(x 0 = 0, x) = φ a (x) and φ(x 0 = t, x) = φ b (x). We can use this to compute the partition function so that Z = Tr e βh = Z = N [ iβ Dφ exp i dt 0 Dφ φ e βh φ (1.9) ] d d 1 xl [φ]. (1.10) where the fields φ satisfy the periodic condition φ(0) = φ( iβ). It is convenient now to go to Euclidean time, performing the change of variable τ = it. This gives [ β Z = N Dφ exp dτ periodic 0 ] d d 1 xl E [φ] (1.11) where L E = 1 2 ( τφ) 2 + 1 2 ( iφ) 2 + V(φ) is the Euclidean lagrangian density. The identification of imaginary time τ = τ + β shows as well how Lorentz invariance is broken at finite temperature. 1.2 Thermofield dynamics Thermofield dynamics was developed in [12]. Its aim is to express statistical averages at a finite temperature as the expectations value in a quantum field theory. The idea is to find a temperature dependent state Ψ β for which Ψ β A Ψ β = Tr(e βh A) Tr(e βh ) = 1 Z n e βe n n A n (1.12) It is not possible to create such a state using only one Hilbert space. However, if we consider another (fictitious) Hilbert space, we can build the following state, which will hold the desired property. Ψ β = 1 Z e βen/2 n ñ (1.13) n where Z = n e βe n. This is usually called thermofield double state (TFD), and we will write it as Ψ, dropping the β dependence. The state n belongs to the first Hilbert space, while ñ to the doubled Hilbert space. The usual relations ñ m = δ nm hold for the latter states as well. If we compute the expectation value of an operator living in the first Hilbert space, then we find the required relation (1.13). This is because, if we start from the density matrix ρ = Ψ Ψ, the reduced density matrix obtained by tracing out the second Hilbert space is the thermal density matrix ρ eq = e βh /Z. It is also possible to consider operators Õ living in the second Hilbert space. Inside the context of thermofield dynamics it might not be immediately clear what these fictitious operators represent physically. However, it will be explained in chapter 2 that in the context of the AdS/CFT correspondence these operators have a straightforward interpretation. We are interested in finding propagators between these two Hilbert spaces. In order to

Real time formalism 8 t i Im t O 1 (t) t f Re t t i iσ O 2 (t ) t f iσ t i iβ Figure 1.1: Schwinger Keldysh time contour. have a Lagrangian formulation we turn to the real time formalism, which we will show to be equivalent to the thermofield dynamics approach. 1.3 Real time formalism Another formalism allowing the computation of expectation values of physical observables at finite temperature is the real time, or time path, formalism. This method was developed by Schwinger and Keldysh [13, 14] and it allows the generalization of Feynman rules to finite temperature. Similarly to the previous approach, this formalism doubles the degrees of freedom: besides the ordinary physical fields, living at real times, there are also ghost fields which live at imaginary times. The time path is shown in figure 1.1. It extends into the complex plane: it runs on the real axis from a time t i to a time t f, then runs in the imaginary time direction to t f iσ, runs back parallel to the real axis to t i iσ 1, and eventually runs downward to t i iβ. This latter point is identified with t i, since the fields are periodic in imaginary time with period β. The limits t i and t f are taken. Boundary conditions are imposed so that fields vanish at t = ±, and the contributions of the vertical segments of the contour vanish. Operators living on the real time contour are denoted as O 1, while the ghost operators on the imaginary time segment are denoted as O 2. They are defined as O 1 (t) = O(t) = e iht Oe iht (1.14) O 2 (t) = O(t iσ) = e ih(t iσ) Oe ih(t iσ) While σ can take any value, provided 0 < σ < β, the most convenient choice in our case is to take σ = β/2. The reason for this is that such a symmetric choice makes this formalism closely related to the TFD state, and it will be important for our future applications. This approach is equivalent to the thermofield formalism. In order to see this, we set σ = 1 Actually, the horizontal segments are slightly tilted in order to have the correct pole prescription. The upper line runs around the poles in the usual Feynman prescription, while the lower in the opposite way.

9 Quantum field theory at finite temperature β/2 and we look for an operator B acting on the second Hilbert space such that Tr(ρ eq A(t)B(t + iβ/2)) = Ψ A(t) B(t ) Ψ. (1.15) Remembering that B(t + iβ/2) = e βh/2 B(t )e βh/2, the previous equation becomes 1 Z e βen/2 e βem/2 n A(t) m m B(t ) n = n,m = 1 Z n,m e βe n/2 e βe m/2 n A(t) m n B(t ) m. (1.16) From this equation we can see that we require B = B T. Therefore an operator living on the second Hilbert space in the thermofield formalism is related to an operator living on the imaginary part of the Schwinger-Keldysh contour. The previous formula can be generalized to the insertion of many operators B in the same manner. 1.4 Schwinger Keldysh propagators The TFD state appears in the context of the AdS/CFT correspondance. We will be interested in computing the propagator between the two different Hilbert space; having shown that the TFD formalism is the same as the real time formalism, we turn to this last one to compute the propagators. We define the Schwinger Keldysh propagators as id 11 (x, t) = TO 1 (x, t)o 1 (0) id 12 (x, t) = O 2 (0)O 1 (x, t) id 21 (x, t) = O 2 (x, t)o 1 (0) id 22 (x, t) = TO 2 (x, t)o 2 (0). (1.17) T denotes time ordering, while T denotes anti time ordering (the second branch of the Schwinger Keldysh time contour has a different pole prescription from that of the first branch). We define also the retarded propagator id R (x, t) = θ(t) [O 1 (x, t), O 1 (0)]. (1.18) This turns out to be a useful quantity, since it allows us to express the propagators (1.17) in momentum space. Besides, we will later focus on a free boson. For this theory, the retarded propagator is state independent, therefore it is possible to compute it at T = 0, where Lorentz invariance is still present. Omitting for the moment any spatial coordinate, which remains trivial, we can express D 11 as id 11 (t) = 1 Z a a e βh[ θ(t)o 1 (t)o 1 (0) + θ( t)o 1 (0)O 1 (t) ] a. (1.19)

Schwinger Keldysh propagators 10 Inserting a complete set of states and expressing O 1 (t) = e iht O(0)e iht we find id 11 (t) = 1 Z e i(e a E b )t ( O ab O ba θ(t)e βe a + θ( t)e βe ) b. (1.20) a,b We now Fourier transform to momentum space id 11 (ω) = id 11 (t)e iωt ε t. (1.21) The ε factor makes the integral convergent. After solving the integral we find D 11 (ω) = 1 ( Z e βe ) a O ab O ba ω + (E a,b a E b ) + iε e βeb. (1.22) ω + (E a E b ) iε The same procedure can be applied to D R. We find and, in momentum space, Using the Dirac identity id R (t) = θ(t) Z a,b e i(e a E b )t O ab O ba (e βe a e βe b ) (1.23) D R (ω) = 1 ( Z e βe ) a O ab O ba ω + (E a,b a E b ) + iε e βeb. (1.24) ω + (E a E b ) + iε we can find the following relations 1 lim ε 0 + x iε = P 1 ± iπδ(x), (1.25) x Re D 11 (ω) = Re D R (ω) (1.26) Im D 11 (ω) = eβω + 1 e βω 1 Im DR (ω) = coth βω 2 Im DR (ω). (1.27) Repeating the same for D 21, we find D 12 (ω) = 1 ( ) Z e βea/2 e βeb/2 1 O ab O ba w + (E a,b b E a ) iε 1 w + (E b E a ) + iε (1.28) and therefore D 12 (ω) = 2ie βω/2 1 e βω Im DR (ω). (1.29) The same analysis can be repeated for D 21 and D 22. D 21 is the same as D 12 because of the symmetric choice σ = β/2, and D 22 has the same imaginary part as D 11 but opposite real part, due to the reverse time ordering T D 21 (ω) = D 12 (ω) (1.30)

11 Quantum field theory at finite temperature D 22 (ω) = D 11 (ω) = Re D R (ω) + i coth βω 2 Im DR (ω) (1.31) The Schwinger Keldysh formalism makes it possible to generalize Feynman rules to finite temperature. The propagator for the field φ is now represented by the 2 2 matrix D ij. Since the field living on imaginary times φ 2 plays the role of an unphysical or ghost field, the external legs of all Feynman diagrams will be the physical field φ 1. The interaction between the two types of fields is possible thanks to the off diagonal elements of the matrix D, and in the internal lines of the Feynman diagram both φ 1 and φ 2 will appear. There are two types of interaction vertex, one for each kind of field. In this work we will treat free theory at finite temperature, therefore we will not use Fenyman diagrams. We are only interested in the D ij matrix in the bosonic case. A more detailed overview of Fenyman diagrams in the Schwinger Keldysh formalism, as well as a treatment of the fermionic case, can be found in [15]. 1.5 Free bosonic field Given the lagrangian for a free bosonic field in d dimensions the mode expansion for the field is L = 1 2 µφ µ φ 1 2 m2 φ 2 (1.32) φ(x, t) = d d 1 k 1 (2π) d 1 [a 2ω k e ik x + a k e ik x ] (1.33) k where ω k = k 2 + m 2. In thermal equilibrium we have [16] a k a k = (2π)d 1 2ω k [N(ω k ) + 1]δ(k k ) a k a k = (2π)d 1 2ω k N(ω k )δ(k k ) (1.34) with N(ω k ) = (e βω k 1) 1. We define C > (x, y) = C < (y, x) = φ(x)φ(y), and the spectral density as ρ(k) = C > (k) C < (k). (1.35) The spectral density turns out to be an useful quantity, since it allows us to find the retarded and advanced propagator using the relations D R (k 0, k) = D A (k 0, k) = dk 0 2π dk 0 2π ρ(k 0, k) k 0 k 0 iε ρ(k 0, k) k 0 k 0 + iε. (1.36)

Bosonic propagators in position space 12 In the case of a free bosonic field we find ρ(k) = 2π 2ω k ( δ(k0 ω k ) δ(k 0 + ω k ) ) = 2π sgn(k 0 )δ(k 2 + m 2 ) (1.37) where sgn(k 0 ) = θ(k 0 ) θ( k 0 ). We have D R (k) = 1 (k 0 + iε) 2 + ω 2 k (1.38) This quantity is independent of the temperature. The temperature dependence appears only when we turn to the time ordered correlator. Now, let s consider the propagators D ab for O = φ. Using Re D R 1 (k) = P k 2 0 + ω2 k Im D R (k) = π sgn(k 0 )δ(k 2 + m 2 ) (1.39) we determine D 11 (k) = 1 k 2 + m 2 iε + 2πi e β k0 1 δ(k2 + m 2 ) D 12 (k) = D 21 (k) = 2πie β k 0 /2 δ(k 2 + m 2 ) 1 e β k 0 1 D 22 (k) = k 2 + m 2 iε + 2πi e β k0 1 δ(k2 + m 2 ). (1.40) We see clearly that these quantities are not Lorentz invariant. If we take the limit T 0, however, we find the usual correlator for a free boson. The off diagonal elements D 12 and D 21 vanish at zero temperature, because the two theories in the TFD state are not entangled anymore, or because the two horizontal parts of the Schwinger Keldysh contour are infinitely far apart. We find the usual rules for Feynman diagrams. Since the first and second field cannot interact anymore, and in the outer legs of the Feynman diagrams only the physical field φ 1 appears, the whole Feynman diagram will only consist φ 1 and the ghost field φ 2 will not play a role. 1.6 Bosonic propagators in position space Throughout this work we will be interested in the 1 + 1 dimensional case. We compute now the Fourier transform of the Schwinger Keldysh propagators. D ab (x) = d 2 k (2π) 2 eik x D ab (k) (1.41)

13 Quantum field theory at finite temperature 1.6.1 Operators on the same theory The first term of D 11 in (1.40) gives 1 (2π) 2 e ik x dkdk 0 k 2 0 + ω2 k iɛ = 1 (2π) 2 dke ikx e ik 0t dk 0 k 2 0 ω2 k + iɛ = i 2π = i 4π θ(t) = i 4π θ(t) dk eikx 2ω k ( θ(t)e iω k t + θ( t)e iω kt ) dk e ikx iωkt + ( t t ) ω k dye im(t cosh y x sinh y) + ( t t ). (1.42) We first carried on the usual contour integration around the two poles and then made the change of variables k = m sinh y. = i 4π θ(t) dye m x 2 t 2 cosh(y+a) + ( t t ) = i 2π θ(t)k 0(m x 2 t 2 ) + ( t t ) = i 2π K 0(m x 2 t 2 ). (1.43) We used d cosh(y + a) = d cosh a cosh y + d sinh a sinh y = it cosh y ix sinh y, to find that d = x 2 t 2 and a = tanh 1 ( x/t). Eventually we made the shift y y a. This is the usual Feynman correlator for T = 0. The second term is 1 (2π) 2 = i 2π = i 2π dkdk 0 e ik x 2πi e β k0 1 δ(k2 + m 2 ) dkdk0 e ik x e β k0 ( δ(k0 + ω 2ω k 1 e β k k ) + δ(k 0 ω k ) ) 0 dk e βωk 2ω k 1 e βω e ikx( e iωkt + e iω kt ). k Now, using the expansion 1 1 x = xn, we can express the integral as i 2π n=0 = i 4π n=0 (1.44) dk e (n+1)βω k e ikx( e iωkt + e iω kt ) 2ω k dye m[(n+1)β+it] cosh y+ix sinh y + ( t t ) (1.45) where we made the substitution k = m sinh y. Now we repeat the same trick as before. Using d n cosh(y + a n ) = d n cosh y cosh a n + d n sinh y sinh a n, we can rewrite the previous integral as i 4π dye md n cosh(y+a n ) n=0 (1.46)

Bosonic propagators in position space 14 where d n and a n are given by d n cosh a n = (n + 1)β + it d n sinh a n = ix (1.47) We find therefore the explicit values d 2 n = ((n + 1)β + it) 2 + x 2 ( ) a n = tanh 1 ix (n + 1)β + it (1.48) Since the imaginary part of a n is given by ( ) Im a n = Re tan 1 x, (1.49) (n + 1)β + it then Im a n [ π 2, π 2 ]. Since the integral does not present singularities, and Re cosh(x + ib) > 0 if x R and b [ π 2, π 2 ], we can make the shift y y a n = i 4π = i 2π = i 2π n=0 n=0 ( 2 n= dye m (β(n+1)+it) 2 +x 2 cosh y + ( t t ) (K 0 ( m (β(n + 1) + it) 2 + x 2) + K 0 ( m (β(n + 1) it) 2 + x 2)) + n=0 )K 0 ( m (β(n + 1) + it) 2 + x 2). (1.50) Summing the two terms together we obtain D 11 (x, t) = i 2π n= = i 2π K 0 (m n= ) K 0 (m (β(n + 1) + it) 2 + x 2 ) (βn + it) 2 + x 2. (1.51) Using the relation in momentum space D 22 (k) = D 11 (k), we can find D 22 (x) in positions space: dk D 22 (x) = (2π) 2 ( D 11(k)) e ik x [ dk = (2π) 2 D 11(k)e ik x = D 11 ( x) = D 11 (x) ] (1.52)

15 Quantum field theory at finite temperature 1.6.2 Operators on different theories To find D 12 in position space we follow the same method. D 12 (x, t) = i 2π = i 2π = i 2π dkdk 0 e β k 0 /2 1 e β k 0 e ik 0t+ikx δ(k 2 + m 2 ) dkdk0 2 k 0 e β k 0 /2 1 e β k 0 e ik 0t+ikx [δ(k 0 + ω k ) + δ(k 0 ω k )] dk e βωk/2 ( ) 2ω k 1 e βω e +ikx e iωkt + e +iω kt k (1.53) Let us consider the first term only for the moment. Using the series expansion (1 x) 1 = n=0 x n i 2π dk 2ω k 1 e βω e +ikx e iωkt = k e βωk/2 i 2π = i 4π n=0 n=0 dk 2ω k e (n+ 1 2 )βω k e +ikx e iω kt dye ((n+ 1 2 )β+it)m cosh y e imx sinh y (1.54) where in the last line we have made the usual change of variables k = m sinh y, for which ω k = m cosh y and dk = ω k dy. Using again d cosh(y + a) = d cosh y cosh a + d sinh y sinh a, we can rewrite the previous integral as where d n and a n are given this time by i 4π dye md n cosh(y+a n ) n=0 (1.55) d 2 n = [ (n + 1 2 )β + it] 2 + x 2 ( ) a n = tanh 1 ix (n + 1 2 )β + it (1.56) Once again, we have no problem with the shift y y a n i 4π dye md n cosh(y) = n=0 i 2π Considering also the second part of the integral, we obtain D 12 (x, t) = i 2π = i 2π [ K 0 (m (n + 1 n=0 [ (n + 1 K 0 (m n= )β + it 2 )β + it 2 [ K 0 (m (n + 1 ] 2 ) )β + it + x n=0 2 2. (1.57) ] 2 ) + x 2 + i [ 2π K 0 (m (n + 1 ] 2 ) )β it + x n=0 2 2 ] 2 ) + x 2. (1.58)

Bosonic propagators in position space 16 We note that both D 11 and D 12 are periodic in t t + iβ, as required by the periodic identification of the complex time coordinate. We also note that D 11 (x, t + i β 2 ) = D 12(x, t). 1.6.3 Massless limit Since we will turn our attention to the AdS/CFT correspondence, we are interested in conformal field theories. A free CFT is given by the lagrangian (1.7) with m = 0, since a mass term would introduce a length scale that would make the theory not scale invariant. For m 0, both D 11 and D 12 diverge. The behavior of the K 0 function, for small argument, is K 0 (x) log( x 2 ) γ E + O(x 2 ) as x 0. (1.59) Therefore D11 0 (x, t) = i 2π = i 2π n= n= [ [ ( 2 ) γ E + log 2 log(m nβ + it) ] + x 2 γ E + log(2) log(m) 1 2 log ((nβ + it) 2 + x 2 )]. (1.60) We get rid of the γ E + log(2) log(m) term, to obtain, as a massless propagator, D11 0 i (( 2 (x, t) = 4π log nβ + it) ) + x 2. (1.61) n= This sum clearly diverges, so we need to renormalize it. In order to do so we first derive twice with respect to the x coordinate; this will make the sum convergent. After the sum is taken care of, we integrate back. This gives [ D11 0 i (x, t) = 4π = i 4π log log [ sinh ( π(t x) )] [ ( π(t + x) )] ] + log sinh β β [ 1 2πt cosh 2 β 1 ] 2πx cosh. 2 β (1.62) The same procedure can be carried out for D 12, and we obtain as a final result [ D12 0 i (x, t) = 4π = i 4π log log [ cosh ( π(t x) )] [ ( π(t + x) )] ] + log cosh β β [ 1 2πt cosh 2 β + 1 ] 2πx cosh. 2 β (1.63) In a two dimensional conformal field theory, the field φ does not have a definite scaling dimension. Instead, the operator φ has weight (1, 0), and φ has weight (0, 1). and are the shorthand notation for = 1 2 ( x t ) = 1 2 ( x + t ). (1.64)

17 Quantum field theory at finite temperature Therefore for these fields we have the following propagators 2 φ 1 (x, t) φ 1 (0) = φ 1 (x, t) φ 1 (0) = φ 1 (x, t) φ 2 (0) = φ 1 (x, t) φ 2 (0) = π 1 4β 2 [sinh π β (x t)]2 π 1 4β 2 [sinh π β (x + t)]2 π 1 4β 2 [cosh π β (x t)]2 π 1 4β 2 [cosh π β (x + t)]2 (1.65) φ i (x, t) φ j (0) = 0. Now let s consider the operator O i = : φ i φi :, which has weight (1, 1). We have O 1 (x, t)o 1 (0) = π2 16β 4 1 [ 1 2 cosh 2πx O 1 (x, t)o 2 (0) = π2 16β 4 1 [ 1 2 β cosh 2πx β 1 2 + 1 2 ] 2 2πt cosh β ] 2. 2πt cosh β (1.66) We can also consider the operator V j α = :e iαφ j :, which has weight ( α2 8π, α2 8π ). Since :eiαφ i(x,t) : :e iαφ j(0) : = e α2 φ i (x,t)φ j (0) we have V 1 α (x, t)v 1 α (0) = V 1 α (x, t)v 2 α (0) = [ 1 2 [ 1 2 cosh 2πx β cosh 2πx β 1 1 2 1 + 1 2 ] α 2 2πt 4π cosh β. ] α 2 2πt 4π cosh β (1.67) It is worth noting that these quantities can also be obtained by conformal mapping the plane onto the cylinder. 2 There is a small sublety here: the (massive) field on the second boundary is quantized as φ 2 (x, t) = dk 2π2ω k ã k e ik x + ã k eik x. Therefore, the φ has an overall minus sign when compared to its counterpart φ. This explains why in third and fourth formula of (1.65) there is no minus sign.

2 The ER=EPR proposal Black holes are solutions of General Relativity. Their peculiarity is the presence of an event horizon, which prevents anything, including light, that is inside the black hole to escape, therefore making it black. They are characterized only by mass, angular momentum and electric charge. The attempt to obtain a theory of quantum gravity is focused mostly on the study of black holes, since quantum effects become important in this situation, while in the vast majority of gravitational situations they can be neglected. However, black holes still represent an open issue for theoretical physics, since the presence of these quantum effects leads to apparent inconsistencies in the theory. Studying and understanding the origin of these paradoxes can help us moving closer to a theory of quantum gravity. In the following chapter we will explain the ER=EPR proposal. First, we want to have a simple situation we can study. In order to do so, we will briefly review the AdS/CFT correspondence [2]. We will then talk about the BTZ black hole [5], a black hole solution in AdS 2+1, and some of its properties, for which the AdS/CFT correspondance has proven useful [6]. Since gravity in 2 + 1 dimensions does not present any propagating degree of freedom, i.e. gravitons, and its field theory dual is a two dimensional CFT, which has the advantage of having an infinite dimensional symmetry goup, it provides an accessible example to investigate the physics of black holes and has been intensively studied in the last twenty years. The BTZ black hole will serve as the simplest example of a black hole we can study. We will finally introduce the information paradox and some of the solutions that have been proposed, such as the firewall [3] and the ER=EPR proposal [4]. 2.1 AdS space and the AdS/CFT correspondence Anti de Sitter space is one of the maximally symmetric spaces. This means that AdS d has d(d + 1)/2 killing vectors, the largest number possible for a d dimensional spacetime. Contrary to the other maximally symmetric spaces, it has a negative cosmological constant. AdS n can be defined as an embedded hyperboloid of radius l in a n + 1 dimensional flat space. X 2 1 X2 0 + X 2 1 +... + X2 n 1 = l2 (2.1) 19

The BTZ black hole 20 The metric can be expressed in global coordinates ds 2 = l 2 ( cosh 2 ρdt 2 + dρ 2 + sinh 2 ρdω d 2 ) (2.2) with t [0, 2π], and ρ R +. This choice of coordinates covers the whole AdS space. is The Poincaré patch instead covers only half of AdS. The metric in this choice of coordinates ds 2 = l2 z 2 (dz2 dt 2 d 2 + i=1 dxi 2 ). (2.3) Using the substitution z = 1 u we find the metric ds 2 = l 2 ( du 2 u 2 u2 (dt 2 d 2 + i=1 dx 2 i ) ). (2.4) The space has a conformal boundary at u = (or z = 0). This boundary is of vital interest for the AdS/CFT correspondence [2], which relates a theory of gravity living on AdS d+1 to a conformal theory living on this boundary. More specifically, the AdS/CFT correspondence states that d e d xφ 0 (x)o(x) CFT = Z bulk [φ(x, z)] (2.5) φ(x,0)=φ0 (x). φ 0 (x) represents the boundary value of the field in the bulk φ(x, z). Both the right and left hand sides of the last formula might present divergences, and in general it is necessary to carry out a renormalization procedure. See for example [17]. For our purposes, it will be enough to use the following formula. When taking a n-point function in the bulk and sending it to the boundary, one has the relation [18] O(x 1 )... O(x n ) CFT = lim z 0 z n φ(x 1, z)... φ(x n, z) bulk. (2.6) The mass m of the field φ is related to the scaling dimension of the CFT operator O by = d 2 + d 2 4 + m2 l 2. (2.7) 2.2 The BTZ black hole The BTZ black hole can be obtained as a quotient of AdS 2+1 [19]. This turns out to be useful, since some quantities, e.g. the geodesic length between two points, will be the same in the AdS and BTZ case. We will focus only on the non-rotating chargeless solution. In this case the metric can be written as ds 2 = r2 r 2 l 2 dt 2 + l2 r 2 r 2 dr 2 + r 2 dφ 2. (2.8) The radius at which the event horizon is situated is r, while l is the radius of AdS. For sim-

21 The ER=EPR proposal u v Figure 2.1: Kruskal diagram for the BTZ black hole. The thick lines represent the boundaries and the zigzag lines represent the future and past singularity. B C A D Figure 2.2: Penrose diagram for the BTZ black hole. A and C are the right and left exterior. plicity, from here on we will set l to 1. The φ coordinate is periodically identified, φ = φ + 2π. The mass of the black hole M is related to the black hole radius as M = r2 8, and the temperature of the black hole is β 1 = r 2π. It is possible to find the maximally extended solution of the BTZ space time by switching to Kruskal coordinates. The metric becomes [6] ds 2 = 4dudv (1 + uv) 2 + (1 uv) 2 r2 (1 + uv) 2 dφ2. (2.9) The future and past singularities are located at uv = 1. We have two boundaries of AdS, located at uv = 1. In the form (2.8) the boundaries are located at r =. In this limit the metric reduces (ignoring a conformal factor) to Therefore the CFTs on the boundaries are defined on R S 1. ds 2 dt 2 + dφ 2. (2.10) The Penrose diagram of the BTZ black hole is shown in figure 2.2. Region B is the inside of

The BTZ black hole 22 the black hole. A and C can be interpreted as two disconnected spaces, or as distant regions of the same space. It is possible to have a path that goes from region A to region C; this is called Einstein Rosen bridge, or wormhole. It would appear at first sight that this is a shortcut through space time, and could therefore violate causality. It is easy to see from figure 2.2 that no signal can travel from A to C; it is normally said that the length of the bridge increases so that no signal can possibly go through. 2.2.1 Propagators for the BTZ black hole This discussion follows the one from [19]. In order to obtain the bulk to boundary propagator for the BTZ black hole, we can exploit the fact that it is a quotient of AdS 3. Therefore we just need to add a sum over images to the bulk to boundary propagator of the latter, to take into account the periodic identification of φ. Let us consider the bulk to boundary propagator for a scalar field of mass m in region A. The point x = (t, φ, r) = (x, r) is located in the right bulk, while the point z = (t, φ ) is on the boundary of region A. Then the propagator will be K AA (x, r; z) [ r 2 r 2 n= r 2 cosh r t + r ] 2 cosh r ( φ + 2πn) (2.11) r where = 1 + 1 + m 2, t = t t and φ = φ φ. If we now want to move one point to region C, it is sufficient to shift t t + iβ/2. If the point x is in region C, we have K CA (x, r; z) [ r 2 r 2 n= r 2 cosh r t + r ] 2 cosh r ( φ + 2πn). (2.12) r We can see that, while K AA might diverge for some values of r, φ and t, K CA is always non singular. This is consistent with the fact that, in general, propagators diverge on light like geodesics. There is no such geodesic that connects region 1 with region 3. We can obtain the boundary to boundary propagator by sending the bulk point x to the boundary. According to the AdS/CFT correspondence, this will be obtained by the limit P(x, z) = C lim r r 2 K(x, r; z) (2.13) with C being a r independent factor. Applying this limit we obtain P AA (x, z) P CA (x, z) n= n= [ ] 2 cosh r t + cosh r ( φ + 2πn) [ ] 2 (2.14) cosh r t + cosh r ( φ + 2πn) There is a subtlety when it comes to P CA : generally the time in the second boundary is seen as

23 The ER=EPR proposal u v Figure 2.3: Constant t slice in Kruskal coordinates decreasing towards the future. In this case, t is to be intended as t = t + t. The reason for this is that the ratio between the Kruskal coordinates u and v is only dependent on time (see [20] for the explicit form of u and v) u v = f (t) (2.15) therefore a slice with t constant is represented by a straight line passing through the point u = v = 0, as shown in figure 2.3. 2.2.2 Holography for the BTZ black hole It was shown in [6] that performing the path integral on the boundary CFT gives a state Ψ = 1 Z e βei/2 i L i R. (2.16) i As seen in chapter 1, this is the thermofield state. The propagators obtained from the bulk, (2.14), are the same as the one obtained with the Schwinger Keldysh formalism in the boundary CFTs in section 1.6.3 for a field with scaling dimension (, ), if we include also a sum over images due to the periodic identification of the spatial coordinate. This is consistent with the claim (2.6). The BTZ black hole is dual to the TFD state. The fact that the insertion of two operators on two independent and non interacting theories has a non vanishing correlator can be understood as the effect of the entangled state of the theories, or, equivalently, of the presence of the wormhole connecting the two theories. We considered only the case with a non rotating and chargeless black hole. In the case of a conserved charge Q, the state (2.16) will also have a chemical potential µ Ψ = 1 Z e βe i/2 βµq i /2 E i, Q i L E i, Q i R. (2.17) i

Hawking radiation 24 r = 0 H + I + H + I + I r = 0 I Figure 2.4: Penrose diagram for a black hole originated by collapse. H represents the event horizon and I + and I future and past null infinity. 2.3 Hawking radiation So far, we have ignored quantum effects. When taking them into consideration on a black hole background, the situation becomes more complicated. Quantum effects on a curved spacetime are responsible for a black hole radiation, known as Hawking radiation [1]. This eventually leads to the information paradox. We will consider a black hole formed by collapse in flat space, as originally studied by Hawking. This case can be generalized to that of the BTZ black hole. In the case of a black hole formed by collapse, the Penrose diagram reduces to the one shown in figure 2.4, since the usual Schwarzschild metric is valid only in the region outside the collapsing matter. The Schwarzschild metric in d dimensions is ds 2 = ( 1 2M r ) dt 2 + dr2 1 2M r + r 2 dω 2 d 2, (2.18) where dω d is the metric on a d dimensional sphere. The regions I and I + are the past and future null infinity. When we consider the limit r, the metric reduces to the flat space one, so these regions are asymptotically flat. Let us now consider a real scalar field φ, which we take to be massless for simplicity. The field satisfies the Klein Gordon equation, which reduces to the usual flat space Klein Gordon equation φ = 0 at infinity. The mode expansion for the scalar field on region I is φ = ω f ω a ω + f ωa ω (2.19) where the coefficients f ω solve the Klein Gordon equation. We consider the f coefficients to contain only positive frequencies so that a ω and a ω are annihilation and creation operators on the region considered. A massless field in the region outside the event horizon will be fully described by the choice of the coefficients f ω. It is possible to do the same considering future infinity instead of the past one. In this case,

25 The ER=EPR proposal we need to consider not only the region I +, but also the event horizon. We can determine the field as φ = ω p ω b ω + p ωb ω + q ω c ω + q ωc ω. (2.20) The p modes represent the outgoing modes on I + and the q modes the ingoing on the event horizon. Again we restrict to positive frequencies only on I +, so that b ω and b ω are annihilation and creation operator. The same cannot be done on the event horizon, since the splitting between positive and negative frequencies is possible when a timelike killing vector exists. On the horizon, the killing vector becomes null. This, however, is not relevant for our computation. Since φ can be determined by information on I or on I + and the event horizon, the coefficients f, p and q or the operators a, b and c are not independent of each other. They are related by some Bogoliubov transformation p ω = ω α ωω f ω + β ωω f ω b ω = ω α ωω a ω β ωω a ω (2.21) and so on. The Bogoliubov coefficients satisfy the condition ω α ωω 2 β ωω 2 = 1 (2.22) which gives consistent commutation relations for the a and b operators. Let us now consider the vacuum for the a operator. This is given by 0 in with the condition a ω 0 in = 0 ω. (2.23) The outgoing vacuum 0 out will be given by the same condition, this time for the operator b. However, in general, 0 out will not be the same as 0 in, which will not be annihilated by a b operator. Let us consider the b number operator N b = ω b ωb ω, and compute its expectation value on the state 0 in. This will be given, using (2.21), by and will not vanish in general. N b in = ωω β ωω 2 (2.24) The value of the β ωω coefficients can be computed (as in [1]) by considering the radial part of the mode decomposition at past and future infinity. The modes on I + can be of two kind. It could be a mode of energy ω on I which is reflected by the Schwarzschild static potential and ends up on I + with the same frequency ω. It could otherwise be a mode that scattered inside the collapsing matter, and ends up on I + with a different frequency ω. The interesting effects are given by the second type of modes, for which the result is β ωω = e πω/κ α ωω (2.25)

The information paradox and the firewall proposal 26 where κ represents the surface gravity of the black hole, in this case 1/4M. Combining this with the condition (2.22), we obtain that ω β ωω 2 = 1 e 2πω/κ 1. (2.26) This is the expectation value of b ωb ω, the expected number of particles in every ω mode. It is a Bose-Einstein distribution for a temperature of T = κ 2π = 1 8πM. (2.27) Hawking radiation can also be pictured in a simple manner. It consists of two virtual particles being created just outside the black hole horizon. There is the possibility that, before they annihilate each other, one of the two falls inside the black hole. The other particle will move away from the black hole. The sum of many similar processes gives rise to Hawking radiation. Since the time translation Killing vector becomes spacelike inside the horizon, the energy of the infalling quanta is considered to be negative. The black hole loses mass and eventually evaporates. In the case of the BTZ black hole, the situation is slightly different. This black hole did not originate by collapse, and its Penrose diagram is the one shown in figure 2.2. Because of the Unruh effect [21], a distant static observer sees a non zero temperature for the black hole. Since the space is asymptotically AdS, the black hole is thermodynamically stable [22], and is eternal. AdS acts as a confining box, so all the particles emitted by Hawking radiation will eventually come back to the black hole. It is still possible to study the evaporation of a BTZ black hole by coupling it to sources. 2.4 The information paradox and the firewall proposal The presence of the Hawking radiation has important consequences on the physics of black holes. Consider for example a black hole that originated from the collapse of a star, which is in an initial pure state. Particles emitted through Hawking radiation are entangled with particles inside of the black hole. After the black hole evaporates, we are left with a thermal density matrix, and it looks like we evolved from a pure to a mixed state 1. This is not allowed by quantum mechanics, since it would violate unitarity. The loss of information is also not consistent with the AdS/CFT correspondence, according to which the black hole is dual to a thermal theory which follows the rules of quantum mechanics. Unitarity in this latter theory implies that the black hole evolves unitarily as well. Let us now introduce the concept of typical state, which will play an important role later on in the discussion. A typical state is a state for which the expectation value of simple observable 1 It was argued by [23] that even though there could be small corrections to Hawking radiation, these will not be enough to make this process unitary. However, in [24, 25], the authors argue that the presence of small corrections becomes important once Page time is reached and eventually unitarizes Hawking radiation. This last construction presents a small violation of locality, which however cannot be detected within the limits of effective field theory.

27 The ER=EPR proposal quantities is close to the thermal expectation value. For example, one expects that almost all of the one point functions will be close to their thermal expectation value. If we consider an evaporating system in a typical state, information will significantly start leaking out after half of the degrees of freedom of the system have already evaporated [26]. In the case of black holes, this time is called Page time. This happens because subsystems smaller than half of the system contain a negligible amount of information. The leaking out of information is possible thanks to the entanglement between the late and early radiation. This, however, presents a different problem. Let us call C a quantum of the early Hawking radiation, and B a quantum emitted in the late Hawking radiation, i.e. after the Page time. B has also a partner just inside the horizon, A. Hawking radiation is a process which creates pairs of particles in a highly entangled pure state. Therefore, we can approximate the quantum B to be maximally entangled with A; due to the leaking out of information, B must be maximally entangled with C. We are facing a serious paradox, since monogamy of entanglement, the principle for which one quantum system cannot be maximally entangled with two independent systems, is violated. This explanation of the paradox is intuitively clear, but it is not very rigorous, since it does not take into account energy conservation; for example, the quanta A and B are in a state which is not maximally entangled, but rather thermally entangled. A more rigorous version of the paradox can be presented using the strong subadditivity of entropy [3]. Let us consider three quantum systems A, B and C and the total Hilbert space, H = H A H B H C. The density matrix on H is ρ ABC, while ρ AB is the reduced density matrix obtained by tracing out H C, and so on. S i is the entropy associated with a density matrix, i.e. S i = Tr(ρ i log ρ i ). Then the strong subadditivity of entropy states that S AB + S BC S ABC + S B. (2.28) Now let us consider again the evaporating black hole. If we assume that nothing unusual happens when crossing the horizon (this assumption is often referred to as the no drama hypothesis), then B and A, which were created in a pure state, are still in a pure state. This implies S AB = 0 and S ABC = S C. Besides, if the black hole is old, its entropy is decreasing, because once the black hole is evaporated we will obtain a pure state. Therefore, S BC < S C. Finally, the strong subadditivity of entropy reduces to S C > S B + S C, which is of course wrong, since B has a thermal density matrix. In the AMPS paper [3], it was argued that the following hypothesis cannot be valid at the same time: 1. the process of evaporation of a black hole obeys unitarity; 2. outside of the horizon, physics is approximately semiclassical; 3. the crossing of the horizon does not present any special feature for a freely falling observer (no drama hypothesis).