Quantization of a Scalar Field in Two Poincar\'e Patches of Anti-de Sitter Space and AdS/CFT

Two sets of modes of a massive free scalar field are quantized in a pair of Poincar\'e patches of Lorentzian anti-de Sitter (AdS) space, AdS$_{d+1}$ ($d \geq 2$). It is shown that in Poincar\'e coordinates $(r,t,\vec{x})$, the two boundaries at $r=\pm \infty$ are connected. When the scalar mass $m$ satisfies a condition $0<\nu=\sqrt{(d^2/4)+(m\ell)^2}<1$, there exist two sets of mode solutions to Klein-Gordon equation, with distinct fall-off behaviors at the boundary. By using the fact that the boundaries at $r=\pm \infty$ are connected, a conserved Klein-Gordon norm can be defined for these two sets of scalar modes, and these modes are canonically quantized. Energy is also conserved. A prescription within the approximation of semi-classical gravity is presented for computing two- and three-point functions of the operators in the boundary CFT, which correspond to the two fall-off behaviours of scalar field solutions.


Introduction
Quantization of scalar fields propagating in anti-de Sitter space was attempted in the past. [1] [2] [3] In [1] the problem of a time-like boundary at space-like infinity, through which data can propagate, is studied in a massless case by conformally mapping the spacetime in the global coordinates into upper hemisphere of Einstein static universe (ESU). By using the fact that AdS space is mapped to a half of ESU, it was shown that there are two sets of mode functions, which are characterized by different boundary conditions and are orthonormal and form a complete set of basis by themselves, separately. It was concluded that only one of the two sets of mode functions can be quantized. In [2] this procedure is more elaborated and extended to massive scalars. In [3] mode functions for scalar fields in AdS space in both Poincaré coordinates and global coordinates are obtained. Group theoretic analysis was performed in [4].
On the other hand, AdS/CFT duality was discovered in [5] and its precise definition has been developed. [6] [7][8] [9] To the string compactification on AdS d+1 ×M n , there corresponds a conformal field theory (CFT) living on a space conformal to the d-dimensional boundary of the AdS. To each field Φ in the bulk there corresponds a local operator in the CFT. By fixing the boundary value of Φ and computing the effective action of the bulk theory, this effective action yields the generating functional of the operators in conformal field theory with the boundary value acting as the source function. In the semiclassical supergravity limit, one can compute the effective action by solving the classical equation of motion and just substituting the solution into the action.
In the case of a free scalar field φ of mass m, it falls off like φ ∼ r −(d−∆+) φ 0 +r −(d−∆−) φ 1 near the spacelike boundary r → ∞. Here ∆ ± = d 2 ±ν and ν = (d 2 /4) + m 2 . When ν > 1, only φ 0 acts as a source for an operator O + with a scaling dimension ∆ + in CFT. When 0 < ν < 1, it is argued that either of the two operators O + and O − with scaling dimensions ∆ + and ∆ − can be considered in CFT. To compute two-point functions of O + one should take φ 0 as a source function and functionally differentiate the effective action with respect to φ 0 . [7] To compute two-point functions of O − , however, one needs to Legendre transform the effective function with respect to φ 0 to obtain a generating functional. [8] This restriction of the holographic correspondence is argued to be related to the above peculiarity of the scalar field quantization in AdS space.
Meanwhile, in the context of AdS/CFT for 3d higher-spin gravity coupled to matter fields, it was found [10] that we can compute semi-classically two-point functions of two sets of single-trace operators in boundary CFT by introducing only one set of matter fields B and C. This motivates us to study whether we can quantize a scalar field in AdS space while keeping both two sets of scalar modes.
One of the purposes of this paper is to show that these two sets of scalar modes in AdS space can be quantized altogether by considering a coordinate system which is obtained by patching together a pair of Poincaré coordinates with radial coordinate r > 0 and r < 0, respectively, along the horizon (r = 0). The AdS space can be divided into two Poincaré patches. The boundary of AdS space is also divided into two. Usually, a scalar field is quantized only in one of the two Poincaré patches. In connection with AdS/CFT correspondence, however, conformal symmetry of boundary CFT has an origin in the isometry of AdS space.
Although the metric in a pair of Poincaré coordinates is invariant under special conformal transformations, points in the two Poincaré patches are exchanged and a single Poincaré patch is not invariant. Hence it is not appropriate to restrict analysis of a field theory in AdS space to just within a single patch. 1,2 In this paper, it is shown that the two patches can be joined together by matching the fluxes of a scalar field across the horizon and two boundaries, and that the united coordinate system admits two sets of scalar mode functions. The fluxes across the horizon vanishes, while those across the boundaries do not. These fluxes across the two boundaries, however, cancel out with each other. It is shown that in Poincaré coordinates for AdS d+1 (d ≥ 2), the two boundaries r = ±∞ are connected. Hence the cancellation of the fluxes occurs on the connected boundaries of the hyperboloid. As a result, Klein-Gordon norm (3.9) is conserved. It is also shown that energy is conserved.
After canonical quantization of the scalar field, Wightman function for a scalar field in AdS space is computed by performing explicit integrations. 3 An allowed form of boundary conditions for a scalar field on the two boundaries is also identified. An interesting issue of AdS/CFT is the prescription for semi-classically computing two-point functions of O + and O − for a scalar field theory with a mass in the range −d 2 /4 < m 2 < 1 − d 2 /4. To present this prescription is the second aim of this paper. It turns out that the (renormalized) action integral (in Euclidean AdS (EAdS) space) is given by a sum of bulk action and boundary 1 In [12] a quotient space AdS d+1 /J, where J is an antipodal map Xµ → −Xµ, is considered. This space is invariant under the isometry of AdS d+1 .
2 EAdS space is one piece of the two disconnected hyperbolic spaces, and this single piece has the full conformal symmetry. This is in sharp contrast to the Lorentzian case. 3 Wightman function for a scalar field in AdS space was computed previously by solving a differential equation with respect to an AdS-invariant distance and matching its singularity with that of flat space. [11] terms.
Here r is the radial coordinate which takes the value in the range −∞ < r < ∞. r = 0 is the horizon and r = ±∞ are the two boundaries. Two Poincaré patches are also introduced in the EAdS space corresponding to the Lorentzian version. The metric is given by ds 2 = dr 2 /r 2 + r 2 d y 2 = g µν dy µ dy ν and γ ij is an induced metric on the boundaries and √ γ = |r| d .
Here ℓ is an AdS radius. Line element in E d,2 induces a one on this hyperboloid.
There are several coordinate systems, and the global coordinates and the Poincaré ones are among them. 4 Global coordinates are defined by where radial coordinate ρ and time τ take values in ranges 0 ≤ ρ < π/2 and −π < τ ≤ π, and ρ = π/2 is a boundary. Spherical coordinates Ω i satisfy −1 ≤ Ω i ≤ 1 and i Ω 2 i = 1. The line element is given by To avoid time-like closed loops, one unwraps τ to have range −∞ < τ < ∞, and works with a universal covering space, CAdS d+1 .
Poincaré coordinates are defined by Here t and x i range between −∞ and ∞, and radial coordinate z ranges over 0 ≤ z < ∞.
The line element is now given by The boundary is at z = 0. There is also a Killing horizon at z = ∞. The time-like Killing vector becomes null at this horizon.
Poincaré coordinates cover only half of AdS d+1 , since X 0 −X d = 1/z > 0. The remaining half is covered by coordinates (2.5) with −∞ < z ≤ 0. Usually, when AdS spacetime is studied in Poincaré patch, only a single patch is considered. However, as is explained below, it is necessary to consider a pair of Poincaré patches.   Figure 2. Here a new radial coordinate r = 1 z is introduced. This ranges over −∞ < r < ∞. The line element (2.6) is rewritten as

Two Poincaré Patches
(2.7) The boundaries are at r = ±∞ and the horizon is at r = 0. The metric (2.7) degenerates at the horizon r = 0, but there is no singularity in the curvature tensor R µνλρ = ℓ −2 (g µλ g νρ − g µρ g λν ). Although the two boundaries in Figure 2 are separated far apart, as will be shown below, points on some hypersurface of the boundaries must be identified. The conformal boundary of AdS d+1 is a two-fold cover of conformally compactified Minkowski spacetime Figure 1. And that of the universal cover is Einstein

Conformal Symmetry of Poincaré Patch
Importance of introducing a pair of Poincaré patches is understood by the following ob- also becomes a symmetry transformation of (2.7), and full conformal symmetry is realized.
The factor multiplying r on the righthand side of (2.10) is not positive definite, and this transformation connects the two patches. The situation is completely different for EAdS. In this case a single Poincaré patch has a full conformal symmetry.

Boundaries at r = +∞ and r = −∞ are connected
Let us study the location of the conformal boundary in the Poincaré coordinates. By the definition of the hyperboloid (2.1) it is defined by d i=1 X 2 i → ∞, and given by ρ = π/2 in the global coordinates. In the Poincaré coordinates (2.5), it is given by Here h is a function h(t, x) ≡ t 2 − x 2 + ℓ 2 . Hence the boundary of the pair of Poincaré patches is composed of the following hypersurfaces.
The union of the above corresponds to the boundary of the global patch. Note that the horizon, and the spacial and even the temporal infinities are also part of the boundary. This last point is puzzling, because the conformal boundary in the global coordinates is time-like.
This problem is not persued in this paper. The structure of the boundary is illustrated in

Solutions to Klein-Gordon Equation in a Pair of Poincaré Coordinates
In this section we consider a scalar field φ(r, t, x) of mass m in AdS spacetime in a pair of Poincaré coordinates r > 0 and r < 0. Action integral is defined by Solution will be constructed in such a way that the fluxes across the horizon vanish and those across the boundaries at r = ±∞ cancel out. The resulting solution will be shown to have the following structure in a pair of Poincaré patches. See (3.27)-(3.28).
In order to solve the equation of motion which is derived from the above action, we separate variables as Then χ(r) satisfies the equation Two linearly independent solutions for non-integral ν is given by where J ν (z) is a Bessel function and We will restrict our attention to the case where ν is real and in the range 0 < ν < 1, because then mode functions with two different falloff behaviour can be obtained. For simplicity, we will set ℓ = 1 in what follows.
For ω 2 − k 2 < 0, solutions (3.5) blow up exponentially at either side of the horizon r = 0 and are non-normalizable. Thus from now on we will require ω 2 − k 2 ≥ 0. In this case solutions oscillate near the horizon. The general solution to the Klein-Gordon equation can be written for r > 0 and r < 0 as Here the mode functions are defined by Because the metric (2.7) is degenerate at the horizon (r = 0), the equation for φ is singular.
So, the coefficientsC ± will be connected to C ± in such a way that the fluxes are matched at the horizon and cancel out between the boundaries.

Klein-Gordon Norm
The Klein-Gordon (KG) norm (φ 1 , φ 2 ) for two modes φ 1,2 is given by 5 Although the KG current is divergenceless, for conservation of the norm (3.9), we need to impose some conditions on the solutions. We will show that this norm is conserved (i.e., independent of t 0 ), if the coefficients satisfy the relations Here α and S are real parameters.
When solution (3.7) is substituted into the norm (3.9) and x integral is performed, the norm is given by Now because ψ in (3.8) solves (3.4), ψ 1 and ψ 2 satisfy and for ω 2 1 − ω 2 2 = 0, the norm (3.12) is expressed in terms of boundary values. 6 14) The contributions from the boundaries r = ±∞ are computed by using J ν (z) ∼ (Γ(ν + if Dirichlet or Neumann boundary condition is imposed. There is, however, another solution. This norm also vanishes, if the following condition is satisfied.
This new solution is possible, because a pair of Poincaré patches is introduced. As will be shown in the next subsection, a flux across one boundary matches that from another.
We now turn to the contributions to the norm from the horizon. These are obtained by using the asymptotic form J ν (z) ∼ 2/πz cos(z −(2ν +1)π/4) for z → ∞. The contribution from the upper side of the horizon is given by To simplify M 1 and M 2 , we need to use some formulae for distributions: sin(Λx)/(πx) → δ(x), cos(Λx)/(πx) → 0 for Λ → +∞. [13] In the limit r → +0, functions M 1,2 can be simplified by using these formulae as (3.21) , those terms which contain M 1,2 all vanish, and we get Contribution to the norm at the other side of the horizon, (φ 1 , φ 2 )| r=−0 , can be similarly computed. Finally, KG norm is independent of t and given by (3.23)

Flux
Since the KG norm is conserved, the fluxes must cancel or vanish at the boundaries and the horizon. Let us check this. One can compute the flux across the horizon from the r > 0 patch.
One can show that this vanishes by using √ −g g rr = r d+1 and φ ∼ r −(d+1)/2 cos(· · · ). A calculation similar to that used in deriving (3.23) must be done. Similarly, the flux at r = −0 also vanishes. The fluxes at r = +∞, however, does not vanish. It is given by This takes forms of interference terms between the two kinds of modes. By using (3.16) one can show that this is canceled by the out-going flux at r = −∞.
The above results might seem useless, because the two boundaries in Figure 2 appear to be infinitely separated. As mentioned in subsec 2.4, however, in AdS d+1 space with d ≥ 2, the two boundaries at r = +∞ and r = −∞ are connected. In this way the total flux computed on the boundaries r = ±∞ cancels out at any time t.
To summarize, normalizable modes in the pair of Poincaré patches are given by for r > 0, and cos( ω 2 − k 2 /r − (±2ν + 1)π/4). However, this very rapid oscillation actually makes the mode functions cancel out and vanish at the horizon. We will show in sec.6 that the solution (6.12) to the boundary-value problem, constructed by smearing these mode functions by source functions which have compact supports, has a milder behaviour φ ∼ |r| ∆± for r → 0, where ∆ ± = d/2 ± ν. By means of the coordinate 7 ρ ≡ log |r| this can be written as φ ∼ e ∆±ρ , and φ asymptotes to zero exponentially near the horizonρ = −∞. In this sense, the mode functions are smoothly connected at the horizon.

Conservation of Energy
In AdS d+1 there is a time-like Killing vector and by contracting this with a stress-energy tensor, a formally conserved energy can be defined. To obtain an exactly conserved energy, one needs to show that the energy-flux vanishes or cancels at the horizon and boundaries. In AdS space the Riemann scalar R is constant, d(d+ 1) (in units ℓ = 1), and a coupling R φ 2 is equivalent to a mass term. Hence we may replace the mass squared m 2 by m 2 −ξ d(d+1)+ξ R in the action. Here ξ is a constant and the conformal coupling corresponds to ξ = ξ c ≡ −(d − 1)/(4d). We will leave ξ as a free parameter and fix its value below. 8 The stress-energy tensor is, after substitution of the solution into the equation of motion, given by can be calculated as in the previous subsection for the particle number flux. T rt is given by and it is easily shown that the fluxes at r = +0 and r = −0 vanish. It turns out, however, that for general ξ, the energy-fluxes at r = ±∞ contain an infinity |r| 2ν associated with the modes ψ − . This infinity can be removed by fine tuning ξ.
Interestingly, at ν = 1 2 , this agrees with the conformal value ξ c presented above. There still remain finite (O(r 0 )) fluxes at the two boundaries. It can, however, be shown that the remaining fluxes at r = +∞ and r = −∞ cancel out completely by using (3.16), exactly as in the particle number flux. Hence the energy associated with both kinds of modes ψ ± is conserved in the pair of Poincaré patches.

Mode Expansion of φ and Canonical Commutation Relations
In this section we will perform canonical quantization of a free scalar field in AdS d+1 in a pair of Poincaré coordinates. We use mode expansions (3.27) and (3.28). By replacing the coefficients C +,− by annihilation and creation operators, and integrating over ω and k, we obtain the following operator.
The integration region is restricted to | k| ≤ ω. This operator is defined for both r > 0 and r < 0. The functionsψ ± are obtained by slightly modifying ψ ± , and given bŷ This operator and its canonical conjugate momentum Π(r, t, x) = |r| d−3 ∂ t Φ must satisfy the canonical commutation relations: Because e iα is multiplied by S or 1/S in (4.2), (4.3), we can set α = 0 by allowing S to take positive or negative values.
The role of parameter S is to specify the relative magnitude of the mode functions (4.2)-(4.3) in the two patches. One can replace ψ − by S −1ψ − and a − by Sã − without changing the form of (4.1). Then, S-dependences of [a + , a † + ] and [ã − ,ã † − ] become the same: 1/(1+S 2 ). If one sets S = 0, then the modeψ − is quantized only in the patch with r < 0, while ψ + is quantized only in the r > 0 patch. At present we do not have an argument to determine S, and in this paper we will leave the value of S undetermined.
Next we turn to [Φ(r, t, x), Π(r ′ , t, x ′ )]. In this case we have For rr ′ > 0, this yields and the identities for distributions used in the previous section, it can be shown that (4.8) This vanishes due to (4.9). The commutators with r < 0 and r ′ > 0 also vanish.

Wightman Function
In this section we will compute Wightman function for a scalar field in AdS d+1 space.
Here |0 is a vacuum which is annihilated by a + and a − .
By using the mode expansion (4.1) and the commutation relations (4.4), Wightman function is given by G ± can be expressed as integrals (A.1) of a flat-space Wightman function integrated over a mass parameter µ. We will display the results for space-like separation of the plane (E d−1,1 ) coordinates.
From the structure of the mode functions (4.2), (4.3), we have By using some mathematical formulae in, for example [16], we can show that Here 2 F 1 (a, b, c; z) is a hypergeometric function, and θ(x) is a step function (θ(x) = 1 for x > 0 and 0 for x < 0). P is defined by and related to the chordal distance σ ≡ X · X ′ + 1 by Note that σ = 0 for X = X ′ , and σ = 2 for X = −X ′ . Hence P −1 vanishes either if the points coincide X = X ′ (rr ′ > 0), or if they are antipodal to each other, X = −X ′ (rr ′ < 0). The hypergeometric functions in (5.6) and (5.7) can be singular at −4P = 0, 1, ∞.
As discussed above, condition P = ∞, which is equivalent to is satisfied for X = ±X ′ (coincident and antipodal points). 9 A singularity at P = −1/4 occurs for 1 |r| These singularities (5.10) and (5.11) are associated with a real charge and its image. [12] If d ≥ 2, P = 0 is not a singularity due to the pre-factors P d 2 ±ν in (5.6) and (5.7). The result (5.6) is derived in Appendix A. When r ′ is sent to infinity, the above functions approach the bulk-boundary propagators: For null and time-like separation (P −1 ≤ 0), G is given by analytically continuing the above result by iǫ Feynman propagator of a scalar field in a Poincaré patch of AdS space with a single type of modes was obtained in [11].

AdS/CFT Correspondence
In the preceding sections we have learnt that a general solution φ to the K-G equation in a pair of Poincaré patches has the structure (3.2). As discussed at the end of sec.3, ϕ ± in a coordinateρ ≡ log |r| effectively asymptote to zero exponentially near the horizonρ → −∞, and φ is smooth at the horizon, even if ϕ ± are multiplied by S and −1/S for r < 0. As we will see, this structure imposes some constraints on the boundary conditions for φ at r = ±∞. The above relations remind us of the connection between Fourier series expansion in an interval (−π, π), and sinusoidal and cosinusoidal Fourier series expansions in a half interval (0, π). [1][2] [3] In that case the sinusoidal one is odd under the reflection and the cosinusoidal one is even. Here this correspondence is modified by the extra factors S and 1/S.

Wick rotation
In what follows we will switch to Euclidean Anti-de Sitter space (EAdS d+1 ) by Wick rotation. In contrast to AdS d+1 , the quadric in E d+1,1 is composed of two hyperbolic spaces H d+1 (disconnected balls B d+1 ). Each piece has r > 0 and r < 0, respectively. One of the two is EAdS d+1 . Hence one usually quantizes a scalar field in a single Poincaré patch. When the entire Lorentzian AdS space is considered, however, one cannot go from a Lorentzian signature to a Euclidean one, and then come back through analytical continuation. Our primary concern is to study a scalar field theory in Lorentzian AdS space, not in EAdS. We perform Wick rotations in order to make integrals which contain products of bulk-boundary propagators well-defined, when the UV divergence is regularized by cutoff |r| = finite. Hence, in what follows, we will consider both pieces of the hyperbolic spaces, and glue together the two half spaces at the horizon, which is also part of the boundary. Then, we assume that the structure of the solution (3.2) is the same after Wick rotation, although the topology of the spacetime has changed by Wick rotation. The coordinates on the boundary will be denoted as y instead of (τ = it, x). The bulk action integral for the scalar field is given by Here note that this action has an asymmetric form. This is different from the action in ordinary form by surface terms. This is arranged so that the on-shell value of I 0 vanishes. [8] Surface action integrals I ± will also be introduced later, and the total action is I = I 0 + The metric tensor is given by The equation of motion for φ in the bulk has solutions of a form φ ∼ r −(d−∆) as |r| → ∞.
There are two values of ∆ BF bound [2] is given by m 2 ≥ − d 2 4 . When ∆ − satisfies the unitarity bound ∆ − ≥ d−2 2 , ν will be in the range 0 < ν < 1. 10 If ν is in this range, there are two scalar operators O + , O − with scaling dimensions ∆ + , ∆ − in the boundary CFT. Discussion in this paper will be restricted to this case. Then ∆ − < d 2 < ∆ + .

Green Functions and Solutions to Boundary-Value Problem
Euclidean Green function G E (r, y; r ′ , y ′ ) is obtained from Feynman propagator iG F (y = (τ, x)) by the relation The bulk-boundary Green functions are given by [7][15] (6.5) Near the boundary (r → ∞), these have the asymptotics.

Then we can write down the general solution to the Klein-Gordon equation in the pair of
Poincaré patches for EAdS d+1 .
According to (3.2), these functions must be related bȳ After substituting the above into (6.8) we obtain Now the boundary conditions on φ are Here f ± ( y) andf ± ( y) are functions which are determined in terms of φ ± ( y). 14) The first terms are source functions and the second terms are 'responses' to the sources. In actual calculations of the asymptotics of a given solution, one cannot distinguish the two.
For the integrals in f − andf − , some regularization for the singularities at y = y ′ will be necessary. Now the O(r −∆− ) and O(r −∆+ ) terms in φ(r, y) are fixed on the boundaries, and in the derivation of the equation of motion, the variation of φ is at most δφ(r, y) = O(r −2−∆− ).
Then the variations of the action on the boundaries vanish: d d y √ g φ r ∂ r δφ → 0, Hence the variational problem is well-posed. To determine f ± andf ± in terms of φ ± , one needs to know K ∆± . 11 In an asymptotically AdS space, such as the one in the presence of black holes, one would need to use a bulk-boundary propagator K ′ ∆± of a scalar field in such a background. Let us now turn to the behaviour of the solution (6.12) near the horizon. We consider integrals d d y ′ [r/(1 + r 2 ( y − y ′ ) 2 ] ∆± φ ± ( y ′ ). As far as the source functions φ ± ( y) have compact supports, these integrals can be approximated as r ∆± d d y ′ φ ± ( y ′ ) as r → 0.
and the solution φ to the boundary-value problem behaves near the boundary as φ ∼ r ∆± , although the mode functions (3.27) and (3.28) are blowing up and oscillating rapidly near the horizon. Then, φ and (r ∂ r ) n φ = ∂ ñ ρ φ vanish on the horizon, and the surface terms on the horizon are not required.

Two-point Functions and Boundary Terms
According to AdS/CFT correspondence, in the semi-classical regime, the on-shell action of the scalar field in the AdS background is supposed to give generating functional of twopoint functions of single-trace operators O + or O − in boundary CFT. In this paper we will try to realize AdS/CFT correspondence for O + and O − altogether at the same time.
Since the bulk action I 0 (6.1) vanishes on shell, we need to introduce boundary terms (and counterterms). The choice of the boundary terms defines definite theories. Because there are two boundaries, we can introduce boundary action I ± on each boundary. They must be local functionals of φ and its derivatives. We will consider the following form.
Here α i (i = 1, 2, 3, 4) are constants, and γ ij is an induced metric on the boundaries. It will turn out that the generating functional is universal up to a multiplicative constant, and we can set α 3 = α 4 = 0. These boundary terms are invariant under reparametrizations which keep the boundary unchanged.
We will substitute the general solution (6.8)-(6.9) into (6.19). It is necessary to evaluate integrals of a form lim r→∞ d d y r d (r∂ r ) n K ∆ ( y, y 1 , r) (r∂ r ) n ′ K ∆ ′ ( y, y 2 , r) with n, n ′ = 0, 1, 2. The method will be explained in Appendix B. Then, the on-shell boundary action I + (6.19) is given by 12 The coefficients A 1 , · · · , A 4 are given by There are still free parameters in addition to an overall constant. Note that this finiteness prescription eliminates the coupling between φ + and φ − .
Finally we get We believe that even if further boundary terms are introduced in ( This problem is actually resolved, when we also use I − as given in (6.19) with an overall negative sign with respect to I + . Due to the relative coefficients in (6.12), we have from (6.29), The sum I = I + + I − yields the following two point functions.
, (6.32) To reinstate unitarity, we need to adjust the parameters such that If the bulk action (6.1) is rewritten into an ordinary symmetric form by partial integration, boundary terms − 1 2 r→+∞ d d y |r| d φ r∂ r φ and − 1 2 r→−∞ d d y |r| d φ r∂ r φ will appear, and to cancel the first term we must set α 2 = 1 2 . In this case the second term is not canceled. Then, α 3 = α 4 = 0 and α 1 = 1 2 ∆ − will be the simplest choice of parameters. Hence, S 2 < 1. The above prescription is different from the previous ones. [5][7] [8] To summarize, after partial integration, the action integral is given by 7 Three-point functions The Euclidean Green function satisfies Let us consider a λ φ 3 interaction with λ being of order of 1/N . [14] I Equation of motion ∆ φ − m 2 φ = λ φ 2 can be solved by using the Green function G E : These equations can be solved by iterations. Up to the first order in λ, the solution is given as follows.
(r < 0) (7.6) Figure 6: Graph contributing to three-point functions The generating functional for three-point functions up to order O(λ 1 ) has two kinds of contributions: the bulk part and the boundary one.
The bulk part is obtained by substituting the solution for the free theory (6.12) into the bulk action in (7.7). The corresponding diagram [7] is presented in Figure 6. The wavy lines are bulk-boundary propagators K ∆σ . There are actually a lot of terms and, a typical form of the terms is given by Here σ i = ± and ∆ i ≡ ∆ σi . Integral of the form (7.8) is evaluated in [15] by using an inversion, as where a is a constant given by ). (7.10) The boundary part of the generating function is obtained by substituting the solution (7.5)-(7.6) into the boundary terms in (7.7). The corresponding diagram is depicted in Figure 7, and a typical form of the integrals is given by Here F is defined by F ( y, y 1 , y 0 , r, r 0 ) ≡ ∆ − K ∆1 ( y, y 1 , r) G E (r, y; r 0 , y 0 ) + 1 2 r∂ r K ∆1 ( y, y 1 , r) G E (r, y; r 0 , y 0 ) + 1 2 K ∆1 ( y, y 1 , r) r∂ r G E (r, y; r 0 , y 0 ). (7.12) The propagator G E = G + + G − with (5.4)-(5.5) and (6.7) is to be substituted into (7.11).
Upon substitution, each boundary term gives divergences. However, the linear combinations in the boundary terms work correctly, and the sum of all turns out finite. Moreover, the integral (7.11) can be explicitly carried out, and the result is proportional to the result of integral (7.9). The three-point functions are obtained by summing the bulk and boundary contributions. The details will be reported elsewhere. Here only the results of a three-point function of O + is presented.

Discussion
We showed a prescription for quantizing two sets of scalar modes in a pair of Poincaré patches of AdS space, and also presented a prescription for semi-classically obtaining twoand three-point functions in the boundary CFT. This is possible since the two boundaries at r = ±∞ are connected, as a result of which the KG norm is conserved. Needless to say, more analysis is necessary. This will be left to future study. There are a few comments.
If we want to quantize only a single set of scalar modes, or if ν > 1 and only modesψ + in (4.2) are allowed, we can still do this in a pair of Poincaré patches. Mode expansion is (4.1) with only a + and a † + retained. Canonical commutation relation is [a + (ω, k), a † Wightman function is proportional to G + in (5.6), and the commutator [Φ(r, t, x), Π(r ′ , t, x ′ )] contains a term iS δ(r+r ′ ) which is harmless since a singularity at r ′ = −r is beyond the horizon. In the study of this paper, a parameter S which parametrize quantization is introduced.
The role of this parameter is to specify the relative magnitude of the mode functions in the two patches. We have not reached a concrete use of this degree of freedom yet. It is also discussed in sec.4 that by setting S = 0 one can quantize only one of the two sets of the scalar modes on each of the two Poincaré patches.
The procedure of this paper can be extended to the black hole geometry. Schwarzschild-AdS d+1 black hole solution in Poincaré coordinates is given by where f (r) = r 2 1 − r + r d .

(8.2)
Event horizon is at r = r + (> 0) and temperature is T = d 4π r + . To quantize two sets of modes of a scalar field with mass m in the range 0 < (mℓ) 2 + d 2 4 < 1 in this background, we consider a pair of Poincaré patches with r > 0 and r < 0. Event horizons are at r = ±r + .
The line element (8.1) is to be used in both patches. In Lorentzian space, the boundary conditions for scalar field at the event horizons must be in-going conditions. The fluxes across the horizons vanish due to f (±r + ) = 0. At the boundaries, the boundary conditions for the scalar field must be such that the fluxes at the boundaries cancel out. These will be (6.10) and (6.11). It is interesting to compute partition functions and entropies for the black hole geometries. Q µ ν (z) = e µπi Γ(ν + µ + 1) Γ( 1 2 ) 2 ν+1 Γ(ν + 3 2 ) (z 2 − 1) P is defined in (5.8). Similar expression for G − can be obtained by replacement ν → −ν in G + and multiplying the result by S 2 . For example, for d = even, G = G + + G − behaves near the singularity P → ∞ as This does not depend on S, ν or ∆. Hence normalization of the singularity of Wightman function cannot be used to fix the value of S. It can be checked that eq (A.11) with S = 0 agrees with eq (7.4) for iG F (x, x ′ ) in [11] after replacements λ ± = ∆ ± , n = d + 1, a = ℓ −1 = 1, 2/u = −4P and substitution y 0 − y ′ 0 = 0.

B Calculation of the integrals necessary for evaluating boundary actions I ±
Here the method for evaluating integrals of a form lim r→∞ d d y r d (r∂ r ) n K ∆ ( y, y 1 , r) (r∂ r ) n ′ K ∆ ′ ( y, y 2 , r) will be explained for the cases n = n ′ = 0. First, let us consider an integral, Here y 12 = y 1 − y 2 . In the r → ∞ limit, regions near t = 0 and t = 1 have dominant contributions. These contributions from 0 ≤ t ≤ ǫ and 1 − ǫ ≤ t ≤ 1 with ǫ = r −1 can be evaluated by setting t = r −2 y −2 12 z or t = 1 − r −2 y −2 12 z and replacing t integral by z integral. These two contributions have the same values and we have Here B(a, b) is Euler's beta function. There is also a region of t which must be taken into account. For the region ǫ < t < 1 − ǫ, we can replace 1 + r 2 t(1 − t) y 2 12 in the denominator of (B.3) by r 2 t(1 − t) y 2 12 . This yields a contribution proportional to For ν > 0 this damps in the r → ∞ limit. So finally, we obtain a finite result.
In the above calculation it is assumed that y 1 = y 2 , and ultra local terms such as δ (d) ( y 1 − y 2 ) are neglected.
We then consider an integral Actually, y integration converges only for d − 4∆ − < 0. If d ≥ 4, this condition is satisfied, because 0 < ν < 1. Otherwise, ν must be in the range 0 < ν < d/4. At the end of the calculation, we will analytically continue the results in variable ν to its remaining region.
Contribution to the integral from regions 0 ≤ t ≤ ǫ and 1 − ǫ ≤ t ≤ 1 is