Classical Limit of Large N Gauge Theories with Conformal Symmetry

In this paper we study classical limit of conformal field theories realized by large N gauge theories using the generalized coherent states. For generic large N gauge theories with conformal symmetry, we show that the classical limit of them is described by the classical Einstein gravity. This may be regarded as a kind of derivation of the AdS/CFT correspondence.


Introduction and summary
According to the AdS/CF T correspondence [1], a certain class of d-dimensional conformal field theories (CF T d ) correspond to d+1-dimensional quantum gravity theory on an asymptotically AdS d+1 spacetime. This conjecture has been investigated intensively and there are many evidences for this conjecture, although there is no proof. The explicit relation of the AdS/CF T correspondence is the GKPW relation [2,3] where a CF T partition function with the source terms is identified with the partition function of a quantum gravity on AdS with appropriate boundary conditions corresponding to the source terms. The extrapolation formula [4], which state that the boundary value of the bulk field is the CF T primary field, can be used as an explicit relation between the two theories. These are the "dictionaries" of the AdS/CFT correspondence and the basic assumptions of the most of the studies.
Alternatively, we can say that the AdS/CF T correspondence is the equivalence of the two theories as a quantum theory. More explicitly, the correspondence means that the Hilbert spaces and the Hamiltonians of the two theories in the operator formalism are equivalent. 2 In this formulation, a proof of the AdS/CFT correspondence means showing the spectrum of CF T d is equivalent to the spectrum of a quantum gravity on asymptotic AdS d+1 .
In [5], instead of assuming such dictionaries or existence of a bulk dual, we studied the (low energy) spectrum of generic large N gauge theories with conformal symmetry, in the leading order in the large N limit, and found that it is identical to the spectrum of the Einstein gravity theory in global AdS d+1 in the free theory limit, following some earlier works [6,4,7]. Here, "generic" means that the theories satisfy the following two properties. The first one is that the low energy spectrum is determined only by the conserved symmetry currents whose conformal dimension is protected against the quantum corrections. 3 In this paper, we further assume that the symmetry of the CF T d is the conformal symmetry only, for simplicity. The second one is that the spectrum generated from the primary states by acting the conformal symmetry generators is completely independent except the relations imposed by the symmetry. 4 These properties are highly expected for the large N strongly coupled gauge theories. 5 With this explicit identification of the spectrum of CF T to the spectrum of the gravity on AdS space, we derived the GKPW relation.
In this paper, we include the 1/N corrections to the study of [5], which are expected to correspond to the interactions in the gravity side, although we do not assume the existence of the gravity dual as in [5]. What we assume is the above two properties and the large N factorization which is satisfied for the large N gauge theories. 6 In the gravity dual, what we consider in this paper is the classical limit, → 0, and because G N is an overall factor of the action, this classical limit is G N → 0. 7 More explicitly, we would like to understand what is the classical limit of the generic large N gauge theories with conformal symmetry. In order to answer this, an important object is the algebra of the energy momentum tensor. This is an analogue of the Virasoro algebra for d > 2. As conserved charges of the theory, an analogue of the Virasoro algebra is the algebra of the conformal symmetry generators. However, if we regard the generators of the Virasoro algebra as modes of the energy momentum tensor, we can consider the commutator algebra of the modes of the energy momentum tensor for d > 2. Because of the assumptions, the generators of this algebra spans all operators on the low energy theories. Thus, the classical limit of the theory is given by the classical limit of this algebra. It is important to note that this algebra is almost unique, like Virasoro algebra, and the Brown-York tensor also forms the algebra by the Poisson bracket. We will show that the classical limit of the generic large N gauge theory with conformal symmetry is the classical dynamics of the Einstein gravity on asymptotic AdS d+1 . This may be regarded as a kind of derivation of the AdS/CFT correspondence from CFT.
Conversely, we can say that the classical dynamics of gravity on AdS is a classical dynamics of energy momentum tensor of CFT. Thus, the quantization of gravity can be considered as finding a quantum system which has an "energy momentm tensor" which reduced to the energy momentum tensor of CFT with a large central charge C T . (We also 3 This is similar to the hydrodynamics. 4 Conversely, this complete independence is needed for the CF T d to have a gravity dual, in the limit. 5 Here, the theories with weak t'Hooft coupling is not regarded as a generic theories. 6 More explicitly, we will assume that only the energy momentum tensor is the low energy primary field which has large C T in (2.26) and the states generated by this are completely independent for the energy below some C T -dependent large energy scale. This C T plays the role of N 2 . 7 Note that G N will be identified as N 2 , thus, what we will consider is a large N limit which is different from the one in [5].
require that it has the generic spectrum in the large C T limit.) Of course, the natural choice for such theory is the CFT and the tensor as the energy momentum tensor itself. In general, the (classical) dynamics of an appropriate system in low energy limit will be described by the hydrodynamics in which only the energy momentum tensor appears. Because only the energy momentum tensor of the CFT appears in the classical gravity, this might mean that the classical gravity is a kind of thermal physics.
There are many things we do not understand in this paper. In particular, the black holes in the brick wall picture [8,9] in the classical dynamics obtained in this paper will be interesting to be investigated. We hope to report this in near future.
This paper is organized as follows. In the next section, we study the algebra of the energy momentum tensor by expanding it on the cylinder. In section three, we consider the classical limit of the CF T d using the generalized coherent states. It is shown that the classical limit of the CF T d is the classical dynamics of the Einstein gravity on asymptotic AdS d+1 in the final section. In the appendix, relation between the classical limit of AdS/CFT and large N limit is explained.

Algebra for the energy momentum tensor
In this section, we will expand the energy momentum tensor T µν of the CF T d on the cylinder by the "spherical harmonics" of S d−1 and the energy, which is the eigen value of the dilatation D, to the infinitely many operators. These operators are analogues of the generators of the Virasoro algebra of the CF T 2 . We will divide these operators to three classes (positive, negative and isotropy), which we will explain. Then, the commutator algebra of them will be studied from the OPE. Because the low energy states are spanned by these operators acting on the vacuum, the algebra is considered as the operator algebra of the low energy theory of the generic large N gauge theory with conformal symmetry.
For a review of the CF T d , see, for example, [10].

Scalar case
First, as a a warming up example, let us consider the (normalized) scalar primary field O ∆ (x), instead of T µν , on R d . Here, we regard the operators are defined by the radial quantization, thus, it is expanded by the spherical harmonics and the radial direction where O ∆ωlm is the operator acting on the state of the theory on the S d−1 , whose coordinates are denoted by Ω, and ω corresponds to the energy. Note that should be half integer because operators which violate this integer condition gives states which can not be in the states spanned by a descendant state of the primary operator. 8 m max (l) is the number of the independent spherical harmonics, which depends on d and l. Here, this scalar field is assume to be Hermite. The scalar field on the cylinder, ds 2 = dτ 2 + dΩ 2 where τ = log r, is given by then, the reflection positivity on the cylinder (or the usual Hermite conjugate on the Lorentzian cylinder) requires where we take Y lm real.

OPE and the commutator algebra
In this subsection, we will consider the relation between the OPE and the commutator algebra. It is well known that the commutator algebra of the conserved charges are given by OPE using the deformation of the integration contour, like in CF T 2 . However, this deformation technique can not be used for the operators which are not conserved charges, for example, the generic modes of the energy momentum tensor.
Here, we will explain how to derive the algebra defined by the commutators of the operators O ∆ωlm from the OPE of the corresponding primary field O ∆ (x), which is assumed to be given as where · · · includes the 1/N suppressed terms.
In the radial quantization, the commutator of the two fields is given as and y are decomposed to the radial direction |y| and Ω ′ for S d−1 , we find where P ± [f (x, y)] is defined as follows: Let us consider a function f (x, y) which has an expansion of the form as has such expansions depending on the sign of |x| − |y|. We will denote r > and r < as the larger and smaller ones of |x| and |y|, respectively.
We will first concentrate on the large N leading term in the OPE (2.11). Then, we have the expansion of the correlation function as where η = x µ yµ |x||y| and is the Gegenbauer polynomial [11] which reduces to the Legendre polynomial for ∆ = 1/2. By the rotational invariance, we can write where Ω and Ω ′ are the angular variables of x and y, respectively, and (d ∆ ) l q = 0 for l > q or q − l is odd. [11]. This implies the relations between Y * Y and (2η) q−2p : Using these relations, we find .

(2.19)
With these expansion formulas, we can evaluate (2.14) as which can be non-zero for ∆ ≤ ω ′ ≤ ∆ + l, and which can be non-zero for ∆ ≤ ω ≤ ∆ + l. Thus, the commutators is given by which are consistent with (2.9), where · · · means terms suppressed in the large N limit. Note that the two different expansions, which originate from the different limits of the integration contours, of the function give the different results. Note also that the finite number of the modes of the expansions give non-zero results. For other terms in the OPE (assuming the parity invariance), we can write it as is the (not necessary primary) fields with spin l and conformal weight ∆ ′ . Then, the 1 (x−y) 2∆+l−∆ ′ factor can be expanded by (2.15) and O ∆ ′ µ 1 µ 2 ,···µ l (y) also can be expanded by the spherical functions. In order to compute the commutator, as for the leading order, we just need to decompose the products of irreducible representations of SO(d). For d = 3 the Clebsh-Gordon coefficients are well known and for other d we can compute it order by order at least. Thus, in principle, we can compute the commutator and it contains only a finite number of terms as for the leading order, It should be noted that the terms (2.23) with 2∆ + l − ∆ ′ ≤ 0, which is non-singular at x − y = 0, do not contribute to the commutator because the two expansions are same and P + = P − for such terms. Therefore, the commutators of the operators can be determined by the singular terms of the OPE.

Algebra for the energy momentum tensor
As for the scalar, we will consider the commutator algebra for the energy momentum tensor. For the energy momentum tensor, two point function is given by and C T is a normalization constant. For the energy momentum tensor, we use the usual normalization in which the conformal generators are given by the mode expansions of the energy momentum tensor, like the Virasoro algebra. This means C T = O(N 2 ) for the large N gauge theories. The three point function is also fixed in [12] with the three coefficients A, B, C with which C T is written as Note that the OPE between the energy momentum tensors is fixed, if the two and the three point functions are given, at least in principle, by the general argument of the CFT [?, ?, 10]. Thus, the singular part of the OPE is also given with only three unknown coefficients A, B, C as where s µ 1 ν 1 µ 2 ν 2 µ 3 ν 3 (x, ∂) consists of terms proportional to A/C T , B/C T and C/C T . Because of the large N factorization, we find A = O(N 2 ), B = O(N 2 ) and C = O(N 2 ). 9 We will expand the energy momentum tensor with the traceless and conserved properties. First, we will use the coordinates r = |x| and z i as the coordinates of S d−1 , where (i = 1, . . . , d − 1), with the flat metric ds 2 = dr 2 + r 2 g S d−1 ij dz i dz j . We decompose it as 28) 9 We assumed that there are no low energy fields other than the energy momentum tensor. and e µ i = ∂x µ ∂z i e µ r = ∂x µ ∂r = x µ r . Then, for the S d−1 directions, we will uniquely decompose them and Here, the rank r symmetric (traceless) tensor harmonics on unit radius where z i (i = 1, 2, · · · , d − 1) is the coordinate, D i is the covariant derivative on S d−1 , and g ij S d−1 is the inverse metric of unit radius S d−1 . Here, l = r, r + 1, r + 2, · · · and m runs from 1 to the number of the independent harmonics which depends on l and r. This harmonics Y (r)lm i 1 ,i 2 ,··· ,ir is the unitary representation of SO(d) which corresponds to the Young diagram labeled by [l, r, 0, . . . , 0]. More details for the symmetric tensor harmonics, see [13,14,15]. The energy ω should be an integer because there are no states with non-integer energy for the primary field with the integer conformal weight. There are more restrictions on the ω as we will see below.
We will show that the rank r symmetric (traceless) tensor harmonics can be represented by where l ≥ r and s rlm µ 1 µ 2 ···µr ;ν 1 ν 2 ···ν l is a traceless constant tensor which is given by the antisymmetrization of the r pairs of the following indices: (µ 1 , ν 1 ), (µ 2 , ν 2 ), · · · , (µ r , ν r ), and then the symmetrization for the µ a and ν a . This (anti)symmetrization procedure corresponds to the Young diagram labeled by [l, r, 0, . . . , 0]. Using where ∇ is the covariant derivative on R d , we can see that g ij S d−1 Y lm i,j,i 3 ··· ,ir = 0 because s rlm µ 1 µ 2 ···µr;ν 1 ν 2 ···ν l−r is traceless and anti-symmetric for the interchanging a pair of µ and ν. Next, note that the Christoffel symbols in the coordinates {z i , r} of R d are given by is the Christoffel symbols of the S d−1 , and others vanish. Then, we can show that Using these, we can see that D i Y lm i,i 2 ,··· ,ir = 0 and D i D i Y lm j 1 ,j 2 ,··· ,jr = (−l(l + d − 2) + r)Y lm j 1 ,j 2 ,··· ,jr hold because of the traceless and (anti)symmetric properties of s nlm .
In the expansion (2.29), the traceless condition of the energy momentum tensor is just t trace ωlm = −s S ωlm . For the conservation condition ∂ µ T µν = 0, we will use following formula: With these, we find, for l ≥ 2, where we have used (D i D j − D j D i )V k = V l R l ikj and R ijkl = g il g jk − g ik g jl for unit radius sphere. Thus, for l ≥ 2, only the s S ωlm , v V ωlm , t T ωlm are the independent operators for the energy momentum tensor. We find, for l = 1, with t S ωlm = t V ωlm = t T ωlm = 0, and, for l = 0, Thus, the non-trivial operators for l = 0, 1 are v V 0 1 m , s S ±1 1 m = v S ±1 1 m and s S 0 0 0 . All of these correspond to the generators of the conformal symmetry. Indeed, inserting the expansion of the T µν (x) into the definition of the generators, where ǫ ν is the conformal Killing vector and integration is on S d−1 at a fixed r, we find that D ∼ s S 0 0 0 , M µν ∼ v V 0 1 m , P µ ∼ s S 1 1 m , K µ ∼ s S −1 1 m for any choice of r as required from the conservation law. Note that all the generators of the conformal symmetry have l = 0, 1 in our notation.
We will denote for the non-trivial s S ωlm , v V ωlm , t T ωlm as where the index A takes S, V, T . Let us consider the spectrum of the low energy theory. The energy momentum tensor is Hermite on the cylinder, then, where we have taken Y lm i 1 ,i 2 ,··· ,ir real. Here, the energy momentum tensor on the cylinder (r = e τ ) is given by where extra 1/r factors are from the normalizing dx µ /dz i . 10 This energy momentum tensor indeed satisfies the conservation law.
As for the scalar case, we will require the regularity of T µν (x)|0 at r = 0. For L T ωlm , this means L T ωlm |0 = 0 for ω < d+l because ∂z i ∂x µ ∂z j ∂x ν Y lm ij (z i ) = s rlm µ 1 µ 2 ;ν 1 ν 2 ···ν l δ µ 1 µ δ µ 2 ν 1 r 2 x ν 1 r x ν 2 r · · · x ν l r . Similarly, we can see that where r[A] = 0, 1, 2 for A = S, V, T , respectively. This can be checked by considering T µν (x) = c µν , which is constant and traceless. This is the lowest regular term and in the polar coordinates, T ij (x) = e µ i e ν j c µν , T rj (x) = x µ r e ν j c µν , T rr (x) = x µ r xν r c µν . This corresponds to s S ωlm with ω = d, l = 2, which is the boundary of the condition (2.48). For L V ωlm , we can check (f2.48) by considering T µν (x) = c µνρ x ρ .
The Hermite condition implies that 0|L Aωlm = 0 for ω > −(d + l + r[A] − 2). Thus the operators L Aωlm with |ω| < d + l + s[A] − 2 satisfy 0 = L Aωlm |0 = 0|L Aωlm and we will denote these operators as L iso , which includes the generators of the conformal group. We will also denote the operators L Aωlm with ω ≥ d + l + s[A] − 2 and ω ≤ −(d + l + s[A] − 2) as L + and L − , respectively. For L ± ωlm , ω should be restricted to satisfy ω − (d + l + r[A]) ∈ 2Z. This restriction comes from the fact that the spectrum are constructed by acting P µ on the primary states.
It is important to note that L + ωlm correspond to the creation operators of the free gravity theory in the asymptotic AdS d+1 [16] in the large N limit in [5].
For a generic strong coupling large N gauge theories with conformal symmetry, the low energy primary field is energy momentum tensor only and the spectrum generated by the it are expected to be independent as assumed in [5]. Thus, the (low energy) states are spanned by A,ω,l,m where N Aωlm is a non-negative integer and

H(
A,ω,l,m These coincide the states of the Fock space in the weak coupling limit of the gravity on AdS space as shown in [5]. These states are independent by the assumption in the limit taken in [5], however, will not be independent if we consider high-energy states for a large, but, finite N case.
As for the scalar case, the commutation relations between the L Aωlm are fixed by the singular part of the OPE of the energy momentum tensor although we will not calculate them because of the technical difficulties. Thus, the commutators of the modes of the energy momentum tensor is given as where g ij and f k ij are N-independent constants and we represented L Aωlm as L i . These constants are, in principle, fixed by the OPE (2.26) which is fixed by the two and three point functions given in [12] which has only two parameters other then N. This algebra (2.51) forms an infinite dimensional Lie algebra including the identity operator 1 like the Virasoro algebra. Note that the operators in L iso forms a sub algebra, which we will call isotropy algebra, because 0 = [L iso i , L iso j ]|0 = 0|[L iso i , L iso j ]. For the space of states (2.49), which are valid in the low energy approximation, the operators in L iso (and any opeartor) can be represented as a (formal) sum of the polynomials of the operators in L + and L − . Indeed, for a state |N Aωlm = ( A,ω,l,m (L + Aωlm ) N Aωlm )|0 with the energy E = A,ω,l,m N Aωlm ω, we can show ( A,ω,l,m (L + Aωlm ω > E because the vacuum is the lowest energy state. Furthermore, the states |N Aωlm which have a same energy are independent by the assumption. Thus, we can construct an operator L which has the matrix element N Aωlm |L|N Aωlm from the polynomials of L + and L − order by order according to the energy of the ket.

Classical limit of the CF T d
In this section, we consider what is the classical limit of the large N CF T d . For this, we will first introduce two more different normalizations of L i . First, we define This normalization corresponds to the usual normalization of the primary field other than conserved currents, up to an O(N 0 ) factor, while L i include the conformal generators with an O(N 0 ) factor. This normalization is suitable especially for the free limit where we neglect the last term in (3.1), then they becomes the creation and annihilation operators (and L iso ) on the Fock space. The other normalization is defined by Note that the r.h.s. of this is N-independent. If we regard N 2 as 1/ , it is expected, for where {L cl i , L cl j } P is a Poisson bracket of the corresponding classical theory where L cl i is identified as Ψ|L cl i |Ψ for a "classical" state |Ψ . We will see this below. First, let us explain how to obtain the classical limit of the quantum mechanics, for example, for a particle. Note that a quantum mechanical system need not to have a classical limit if the theory is not obtained from a quantization of a classical system. There exist purely quantum mechanical systems. On the other hand, let us consider a quantum mechanical system where we can choose operatorsp,x with the commuta- where ǫ is a small parameter, and the Hamiltonian H = 1 ǫĤ ′ (p,x) + O(ǫ 0 ). Here, the time evolution of a operatorÔ is assumed to be given by dÔ dt = i[Ĥ,Ô]. 11 Then, regarding ǫ as , this quantum system may be derived by the quantization from the classical system with the Poisson bracket {x i , p j } P = δ i j and the Hamiltonian H cl =Ĥ ′ which satisfies dO dt = −{H cl , O} P . For this identification with the classical system, 12 we need to consider a state with x i = O(ǫ 0 ) and p j = O(ǫ 0 ), which implies H cl = O(ǫ 0 ), and then this state will have large "quantum numbers" because ǫ is small.
This classical limit of the quantum system is an approximation, where states which have same x i , p j up to O(ǫ) differences are identified. For the operators, we need to consider the "classical operators" O(x i , p j ) which are also defined up to O(ǫ) differences. For the CFT case, L cl i indeed correspond tox,p above and give the classical limit with ǫ ∼ 1/N 2 . 13 Note 11 If we regard ǫ as , the usual definition of the HamiltonianĤ ′ (= ǫĤ) satisfies ǫ dÔ dt = i[Ĥ ′ ,Ô]. Thus, our definition of the Hamitonian has a different normalization from the usual one. This is because we would like to consider the CFT where there is no notion of generically. 12 Note that if we start from the quantum mechanics, instead from the classical mechanics, the classical limit and the parameter emerge if the theory satisfies some requirements. In particular we need a small parameter, which for our case 1/N 2 ∼ 1/C T . 13 More precisely, L iso are not independent, thus we should exclude these.
that if we use z a (x,p) as basis of operators instead ofx,p, i ǫ [x i ,p j ] = −δ i j + O(ǫ) will be replaced by i ǫ [z a , z b ] = −f ab (z) + O(ǫ) and these should satisfy the Jacobi identities. The commutation relations (3.2) are in this generalized form.
We have seen that the classical limit of the CF T d may exist and is given from (3.2). The corresponding classical states for our case are known as generalized coherent states based on Lie groups [17] [18]. We basically follows [19] to consider the generalized coherent states and their properties. 14 First, we define the coherent group G which consist of the following unitary operators:Û where L i are taken to be Hermite and c i , c are N-independent real constants. We also define the isotropy subgroup H of the coherent group whose elementV satisfiesV |0 = |0 up to a phase factor, i.e.V = e i(c i L iso i +cN 2 1) . Then, the generalized coherent states are defined byÛ|0 with parameters c i , c in (3.4). 15 Using the Baker-Campbell-Hausdorff formula, we can rewrite it aŝ where α i are some complex functions of c i , c which are N-independent and C = e iN 2 θ |e α where θ is a N-independent real complex function of c i , c . Thus, generalized coherent states are parametrized by α i and the states with same α i should be identified. 16 We can also see that the classical operators defined in [19] are the operators constructed from L cl i . In order to see this, let us consider two coherent statesÛ|0 = Ce α i L + i |0 and is also a coherent state and the overwrap of the two coherent states is given by 0|Û †Û ′ |0 = e iN 2 θc 0|e α i By the Baker-Campbell-Hausdorff formula, we expect the following rewriting: eᾱ i where β i , γ i , φ are some N-independent constants determined by α i . Using this expression, we have Re(ln 0|Û †Û ′ |0 ) = −N 2 φ/2 where Re means the real part. Then, if α i c is non zero for some i, φ > 0 because | 0|Û †Û ′ |0 | < 1. Similarly, we can also show that Re(ln 0|Â clÛ † =Â cl whereÂ cl also is an operator constructed from L cl i without N-dependent coefficients. 14 There could be some differences between the large N limit taken in [19] and this paper. The large N limit in [19] seems to correspond to the free limit because the factorization of the correlation functions were discussed. Here, (3.2) include the 1/N corrections which violate the factorization properties. 15 Because the Lie algebra is infinite dimensional, we need to require some properties for c i such that the coherent state is well-defined, in particular, the state should have a finite energy. A simple requirement for this is that only a finite number of c i do not vanish. This will be too strong condition and it is desirable to find an appropriate condition although we just assume the state is well-defined in this paper. 16 The coherent states are parametrized by the the coadjoint orbit as we will see later.
The classical operators defined in [19] are the operators, say,Â such that 0|U †Â U|0 / 0|Û †Û ′ |0 is finite in the N → ∞ limit. Therefore, we find that the classical operators are indeed the operators constructed from L cl i . We can also easily show that two coherent states,Û |0 = Ce (α i L + i ) |0 andÛ ′ |0 = C ′ e (α ′i L + i ) |0 , are classically equivalent [19], which means 0|Û †Â clÛ |0 = 0|Û ′ †Â clÛ ′ |0 for anyÂ cl , if α i = α ′i . Even if α i = α ′i , two states can be classically equivalent. Including such identification, the coherent states are parametrized by the coadjoint orbit. We will shortly explain this below. First, denoting g as the Lie algebra of the coherent group G, we can define the dual space g * whose elements are linear functionals acting on g. Then, the expectation values of L cl i for a coherent state, 0|Û † L cl iÛ |0 , can be regarded as an element ζÛ in g * if we regard L cl i as basis of g. In particular, we will denote ζ 1 as for the element corresponding to |0 for which the components are given by ζ 1 i = 0|L cl i |0 . The coadjoint orbit Γ is the set of ζÛ in g * generating byÛ . Note that the expectation values of the operators constructed from L cl i are fixed by the ζÛ because of the factorization of the expectation values [19]. Thus, classical equivalence class of the coherent states are identified as the coadjoint orbit 17 and then, the classical phase space is identified as the coadjoint orbit.
There are some requirements [19] such that the classical limit considered here is indeed behaves as the classical dynamics. The one is the irreducibility of the representation of G and this is satisfied because we assumed (2.49) are all independent and our algebra reduced to the free harmonic oscillators, i.e. Heisenberg algebras, for the small c i , c. It also required that if 0|U †Â U|0 = 0 for any U, U ′ ∈ G, thenÂ = 0. This is also satisfied. The requirement about the overwrap between the coherent states was already shown to be satisfied above.
The last requirement is about the Hamiltonian. The classical Hamiltonian h cl is given by h cl = 1 N 2Ĥ , whereĤ = D is the Hamiltonian in our theory, and h cl is indeed the classical operator. Here, in the classical limit h cl is regarded as a function on the coadjoint orbit. Thus, all the requirements are satisfied and we conclude that the (classical) equations of motion for a function f on the coadjoint orbit, which are parametrized by L cl i , is with the Poisson bracket (3.3) for the classical limit of the CFT. Until now, we have only used the properties of generic large N gauge theories with the conformal symmetry to derive the classical limit of the CFT. It is expected that this classical system is identified as the classical (Einstein) gravity on asymptotic AdS d+1 space because the classical system certainly reduces to the linearized gravity in the large N limit taken in [5]. We will show this identification is indeed correct in the next section. Finally, we note that the this classical description is, of course, an approximation. In particular, the Hilbert space we consider is the low energy approximation and the classical approximation will be violated if the energy of the states are sufficiently large.

Classical gravity on asymptotic AdS d+1 and the CF T d
In this section, we will consider the classical gravity on asymptotic AdS d+1 in the Hamiltonian formalism, in particular using the Brown-York tensor. We will see that the classical dynamics of generic large N gauge theories with the conformal symmetry is equivalent to the Einstein gravity on asymptotic AdS d+1 . (We will explain the difference between the large N expansion from the free theory and the classical limit in the Appendix A.) The Einstein-Hilbert action of the gravitational theory, with appropriate boundary terms, is 2l 2

AdS
and we set the AdS scale l AdS = 1 in this section. The Gibbons-Hawking term S GH is needed to allow the Dirichlet boundary condition as a consistent boundary condition and S ct is needed for making the action finite [20,21], although this term does not play any role in the equations of motion of the classical dynamics which we concentrate on this paper. The metric of the vacuum solution is the AdS d+1 metric: where 0 ≤ r < ∞, −∞ < t < ∞ and dΩ 2 d−1 is the metric for the d − 1-dimensional round unit sphere S d−1 . Let us parametrize the metric as g µν = g AdS µν + h µν and consider h µν as the varying fields.
First, we consider the free limit of the gravity, i.e. the linearized gravity. The e.o.m. of this limit was explicitly solved in [16] using the gauge invariant combinations. We can easily see that in the (classical) Hamiltonian formalism, the results can be expressed as and where (A, ω, l, m) are the labels for the L + Aωlm [5]. Here, we required that the only the normalizable modes of h µν are dynamical and the non-normalizable modes of h µν are set to be zero, which corresponds to fix the boundary conditions. This classical system is equivalent to the system with the Poisson bracket (3.3) and the Hamiltonian as the dilatation if we neglect the terms proportional to f k ij , which are 1/N corrections to the free limit.
the variables a Aωlm and a † Aωlm where we fix the boundary metric g (0) µν as the cylinder. We can expand T µν bndy by the symmetric tensor harmonics and obtain the corresponding modes L bndy Aωlm as in the previous section. 20 Thus, the modes L bndy Aωlm are functions of a Aωlm and a † Aωlm . Conversely, {a Aωlm , a † Aωlm } can be regarded as functions of {L +bndy Aωlm , L −bndy Aωlm } (at least if the theory is close to the free theory) because the number of the independent variables are same. We can regard the map between these two as a field redefinition although in order to obtain such a map explicitly we need to solve the equations of motion. Note that the modes defined at the boundary can be equivalent to the whole bulk modes because the diffeomorphism gives the constraints and the system in the AdS space is like the system in a box [25,22]. Therefore, the Poisson bracket algebra, which satisfies the Jacobi identities, of the boundary stress tensor is same as the algebra of the energy momentum tensor with the three parameters. However, if we require the unitarity and the causality, 21 it have been shown [27,28] that there remains only one parameter C T in (2.26) which is O(N 2 ) for the gauge theory. For the T µν bndy , we can see C T ∼ 1/G N . This is because T µν bndy ∼ 1/G N by definition and {h ,ḣ } P ∼ G N where h is the some component of h µν , schematically. We can fix the coefficient of this relation by computing above precisely for the free theory limit and the result should agree with the result assuming the AdS/CFT correspondence as [26]. The Hamiltonian of this gravitational system can be identified as the dilatation in the conformal symmetry. Thus, we conclude that the classical limit of the generic large N gauge theory with conformal symmetry is the classical Einstein gravity on asymptotic AdS d+1 because the Hamiltonian and the Poisson bracket are same. main parts of this paper.
Let us consider the following metric around the AdS space: where l AdS is the AdS (length) scale and h µν is the fluctuation around the AdS space. The action of the gravitational theory in the derivative expansion is, schematically, where (l p ) d−1 = 8πG N , α i are dimensionless constants and x ν and h µν are also dimensionless. We assume that the AdS space is the solution of the e.o.m. of this action, as for the Gauss-Bonnet gravity or g AdS µν is modified from the AdS metric to be the solution with the higher derivative terms.
Let us define the dimensionless parameter N as N 2 = l AdS lp d−1 , then, the action for the fluctuation is, schematically, given by where we abbreviated the various contractions of the indices and f i (h) is a function such that f (h = 0) is finite. There are several choices for large N limits. One is the perturbation around the AdS geometry or the liner approximation. For this, we will normalize h µν such that the kinetic term will be the canonical one. Thus, we need to takeh µν ∼ Nh µν small, but finite. If the higher derivative terms vanish, i.e. α i → 0, in the large N limit, we have the free theory ofh in the leading order in this large N limit, with the spectrum given in [16] corresponding to the energy momentum tensor of the CFT [5]. Thus,h is directly related to the creation/annihilation operators of the free theory. The sub-leading terms are interactions which include the D 2 R terms. In general, this 1/N expansion withh include the quantum gravity effects.
Another choice of the large N limit is the classical limit where h µν is finite and N 2 is regarded as 1/ . This means that the v.e.v. of the creation/annihilation operators, i.e. h, should have O(N) values. Note that this large N limit contains the whole interactions of the Einstein gravity as a leading term, which are non-leading term in the previous 1/N expansion withh. In this paper, we will consider this large N limit.

A.1 Coherent states for the linearized gravity
We can consider classical states which is very close to the vacuum, i.e. h µν = O(N 0 ), but h µν ∼ ǫ ≪ 1. This is the liner approximation of the Einstein gravity (at least if α i → 0 in the large N limit). Here, we describe the coherent states for this approximation, which have been considered in [29,30].
Let us remember the coherent states for the free field (harmonic osscilator). The coherent state for it is defined as |α = e αa † −ᾱa |0 , where [a, a † ] = 1 and a|0 = 0, which satisfies a|α = α|α and C d 2 α|α α| = π. The overlap between the coherent state and the normalized energy eigen state |n = 1 n! (a † ) n |0 is given by n|α = e − 1 2 |α| 2 α n √ n! = e − 1 2 |α| 2 +f (n)+O(ln n) , (A. 4) where f (n) = − 1 2 n(ln n − 2 ln α − 1). Thus, particle numbers where dominant contributions comes from for the coherent state is n ∼ α 2 because ∂f (n) ∂n = 0 at n = α 2 . Therefore, the classical state which is very close to the vacuum in the CFT is where β i = O(N 0 ) are complex constants, ǫ is a small parameter. This state is expected to be the coherent state a i |β i ∼ Nǫβ i |β i , where we assumed [a i , a † j ] = δ ij + O(N −1 ) and the O(N −1 ) terms are neglected in the small ǫ limit. The time evolution (in the Schrödinger picture) is given by i ∂ ∂t |β i =Ĥ|β i = D|β i where D is the dilatation operator and [D, a i ] = ω i a i . Thus, the solution is |β i (t) = e N ǫ i (β i (t)a † i −β i (t)a i ) |0 , where ∂β i (t) ∂t = ω i β i (t). 22 Of course, if ǫ is not small, then the state (A.5) is not regarded as a classical state because it is not regarded as a coherent state.