Feynman and Wheeler Scalar Field Propagators in the ADS/CFT Correspondence

Transcription

1 arxiv: v [hep-th] 7 Nov 08 Feynman and Wheeler Scalar Field Propagators in the ADS/CFT Correspondence A. Plastino,3,4, M.C.Rocca,,3 Departamento de Física, Universidad Nacional de La Plata, Departamento de Matemática, Universidad Nacional de La Plata, 3 Consejo Nacional de Investigaciones Científicas y Tecnológicas (IFLP-CCT-CONICET)-C. C. 77, 900 La Plata - Argentina 4 SThAR - EPFL, Lausanne, Switzerland November 9, 08 Abstract We discuss in rigorous fashion, without appealing to any conjecture, the boundary-bulk propagators for the scalar field, both in the non-massive and massive cases. Our calculations are carried out in two instances: (i) when the boundary is a Euclidean space and (ii) when it is of Minkowskian nature. In this last case we compute also three propagators: Feynman s, Anti-Feynman s, and Wheeler s (half advanced plus half retarded). For an operator corresponding to scalar field we obtain the two points correlations functions in the three instances above mentioned PACS.5.-w, Cf, 0.30.Sa, Db. KEYWORDS ADS/CFT correspondence; Boundary-bulk propagators; Feynman s propagators, Wheeler s propagators.

2 Contents Introduction 3 Euclidean Case 4. Massless Scalar Field Propagator Massive Field Propagator Approximate Massive Field Propagator Minkowskian Case 0 3. Massless Field Propagator Massive Field Propagator Approximation Two points correlation functions in Euclidean space 5 4. Massless case Massive case Two points correlation functions in Minkowskian space 7 5. Massless case Massive case Discussion 8

3 Introduction Propagators and correlators are one of the essential tools to work, for example, in Quantum Field Theory (QFT) and String Theory (ST). In particular, in formulating the correspondence ADS/CFT (Anti-de Sitter/ Conformal Field Theory)/ This correspondence was established by Maldacena [] in 998 and is universally regarded as a very useful model for many purposes. The bibliography on this subject, for scalar fields, is quite extensive. We give here just a small part of it in [, 3, 4, 5, 6, 7, 8, 9]. For a more complete bibliography the reader is directed to the report [0] One of the ADS/CFT correspondence s prescriptions (see []) will allow us to evaluate the correlators on the boundary of ADS space The first boundary-bulk propagator was calculated by Witten a few months after the appearance of [], entitled Anti de Sitter space and holography. In this case the boundary is an Euclidean space [, 3]. In this work, instead, we evaluate the boundary-bulk propagators for the case in which the boundary is a Minkowskian space. In such regard, remark that some attempts have been made before in [,, 3]. The only previous (and almost ) attempt to try to calculate boundarybulk propagators in the Minkowskian boundary for the Anti-de Sitter space [in the ADS/CFT correspondence] was made by Son and Starinets (SS) in 00 [4]. However, SS needed to formulate a conjecture that we show here to become unnecessary if one uses the full distributions-theory of type S (of Schwartz). This important work was entitled Minkowski-space correlators in AdS/CFT correspondence: recipe and applications. We must also mention the work of Freedman et al. [], in which the authors deal with the case of a Euclidean boundary. Freedman, however, did not treat the case of a Minkowskian boundary, at least in the way that Son and Starinets did. Let us remark, as this is an important point for us, that in this paper we do not evaluate renormalized correlation functions. We will do that in a forthcoming paper using the method given in []. Thus, in the present effort we evaluate, without any conjecture, the boundarybulk propagators corresponding to the following three cases i) Feynman, ii) Anti-Feynman, and iii) Wheeler. We do this both, for a massless scenario and for the massive ones, (a scalar field involved). Later we calculate the two points correlators (TPC) for operators corresponding to this scalar field in the three instances previously mentioned. We clarify that in this paper we do not evaluate the renormalized TPC. We will do that in a next paper 3

4 using the method given in []. In the present effort we demonstrate that the Feynman propagator must be a function of ρ + i0 (see below for the notation) in momentum space, and thereforeafunctionofx i0inconfigurationspace. Weshowthatsomething similar happens with the Anti-Feynman propagator. For the first time ever, we calculate the Wheeler s propagator as well. Note that, until the 90 s, the only field propagators that had been calculated were Anti-de Sitter (spatial) ones. The paper is organized as follows: Section deals with with the Euclidean case. In it, the three different propagators referred to above can not be distinguished (neither in the massive nor in the massless instances). In section 3 we tackle similar scenarios as those of section, but now in Minkowski s space, where the three propagators can be distinguished. In section 4 we compute in Euclidean space the TPC for a scalar operator corresponding to a scalar field via Witten s prescription In section 5 we generalize the calculations of section 4 to Minkowski s space. We obtain in this fashion the two-points correlations functions corresponding to the three different propagators of our list above. Finally, some conclusions are drawn in section 6. Euclidean Case. Massless Scalar Field Propagator The Klein-Gordon equation in ADS + for the scalar field φ(z, x) reads, in Poincare coordinates, z z φ(z, x)+( )z zφ(z, x)+z φ(z, x) ( )φ(z, x) = 0, (.) where ( ) 0 plays the role of m. We exclude tachyons from of this treatment. Here is the conformal dimension, the boundary s dimension, and x their coordinates. The Fourier transform in the variables x of the field φ(z, x) is ˆφ(z, k) = φ(z, x)e i k x d x. (.) Using (.), (.) takes the form z z ˆφ(z, k)+( )z zˆφ(z, k) [z k + ( )]ˆφ(z, k) = 0. (.3) 4

5 We analyze now the massless case given by = 0,. For it we have the motion equation or equivalently (for z 0) z z ˆφ(z, k)+( )z zˆφ(z, k) z k ˆφ(z, k) = 0, (.4) zˆφ(z, k)+ ( ) zˆφ(z, k) k ˆφ(z, k) = 0. (.5) z In the variable z, this equation is of the Bessel type (see [8]) F (z)+ ( α) ] F (z) [k + µ α F(z) = 0 (.6) z z The pertinent solution (that does not diverge when the argument tends to infinity) is F(z) = z α K µ (kz). (.7) Thus, the solution of (.5) becomes ˆφ(z,k) = z K (kz). (.8) One easily verifies that, for infinitesimal z [8], Γ ( ) K(kz) = +O ( (kz) +) (kz), (.9) and therefore lim z Γ ( ) K (kz) =. (.0) z 0 Inotherwords, thesolutionisregularattheoriginandvanishesatinfinity(in the variable z). Accordingly, we have, for the field in the bulk, the solution φ(z, x) = z (π) k a( k)k (kz)e i k x d k. (.) This solution must reduce itself to the field φ 0 (x) on the boundary, so that φ(0, x) = φ 0 ( x) = Γ ( ) a( (π) k)k e i k x d k = 5

6 (π) ˆφ 0 ( k)e i k x d k. (.) From this last equation we can obtain a(k) as a function of ˆφ 0 and then write φ(z, x) = z (π) Γ ( ) k K (kz)ˆφ 0 ( k)e i k x d k, (.3) or, equivalently, φ(z, x) = z (π) Γ ( ) k K (kz)φ 0( x )e i k ( x x ) d kd x. (.4) From (.4) we then obtain an expression of the boundary-bulk propagator K(z, x x ) = z (π) Γ ( ) k K (kz)e i k ( x x ) d k. (.5) To carry out the integration in the variable k we appeal to the expressions for the Fourier transform and its inverse obtained by Bochner [7]. For the Fourier transform we have ˆf(k) = and for its inverse f(r) = (π) f( x)e i k x d x = (π) k ˆf( k)e i k x d k = Using these relations we have now 0 (π) r r J (kr)f(r)dr, (.6) 0 k J (kr)ˆf(k)dk. (.7) K(z, x x ) = z (π) Γ ( (π) ) x x 0 k K (kz)j (k x x )dk. (.8) So as to evaluate the last integral we appeal to a result from [8] 0 x µ++ K (bx)j µ (ax)dx = µ+ a µ Γ(µ+ +) b (a +b ) 6 µ++, (.9)

7 Our deduction follows a different, simpler and complete path than that of []. Our approach also has a didactic utility. K(z, x x ) = Γ() [ ] π Γ ( z ), (.0) z +( x x ) which leads to φ(z, x) = an expression that, in turn, leads to. Massive Field Propagator K(z, x x )φ 0 ( x )d x, (.) lim ) = δ( x x ). (.) z 0 K(z, x x We now consider the massive case 0,. The equation of motion for this case reads z z ˆφ(z, k)+( )z zˆφ(z, k) [z k + ( )]ˆφ(z, k) = 0, (.3) or equivalently, z ˆφ(z, k)+ ( ) [ zˆφ(z, k) k + ( ) ] z z The solution for this last equation is ˆφ(z, k) = 0. (.4) ˆφ(z,k) = z Kµ (kz), (.5) with µ = ± + ( ). (.6) 4 Since K µ (z) = K µ (z), we select for µ in (.6) the plus sign. We have then φ(z, x) = z a( (π) k)k µ (kz)e i k x d k. (.7) For 0, this solution is not regular at the origin. To overcome this problem we select φ(ǫ, x) = φ ǫ ( x) = ǫ a( (π) k)k µ (kǫ)e i k x d k = 7

8 (π) where ǫ is infinitesimal. From (.8) we have then Replacing the result of (.9) into (.7) we obtain or similarly, φ(z, x) = ( z ) (π) ǫ φ(z, x) = ( z ) (π) ǫ ˆφ ǫ ( k)e i k x d k, (.8) a( k) = ˆφ ǫ ( k) ǫ K µ (kǫ). (.9) Kµ (kz) K µ (kǫ) ˆφ ǫ ( k)e i k x d k, (.30) Kµ (kz) K µ (kǫ) φ ǫ( x )e i k ( x x ) d kd x. (.3) From this last equation we see that the propagator is K m (z, x x ) = ( z ) (π) ǫ As a consequence we can write φ(z, x) = From (.33) we immediately gather that Kµ (kz) K µ (kǫ) e i k ( x x ) d k. (.3) K m (z, x x )φ ǫ ( x )d x. (.33) K m (ǫ, x x ) = δ( x x ). (.34).3 Approximate Massive Field Propagator Wearenowgoingtodiscuss anon-validapproachforthefunctionk(kǫ). The issue here is that, although ǫ is infinitesimal, it can not adopt the 0 value. As k is an unbounded variable, when k, we have kǫ. Notice first that K µ (kǫ) = µ Γ(µ) (kǫ) µ +O((kǫ) µ ). (.35) We now make the approximation K µ (kǫ) = µ Γ(µ) (kǫ) µ. (.36) 8

9 From (.3) we obtain an approximation for the propagator K that we shall call M. We have then M m (z, x x ) = ( z ) ǫ µ k µ K (π) ǫ µ µ (kz)e i k ( x x ) d k. (.37) Γ(µ) Using again the Bochner formula we arrive at k µ K µ (kz)e i k ( x x ) d k = (π) x x By recourse to (.9) we then have M m (z, x x ) = ǫµ Γ ( ) µ+ [ Γ(µ) We now define π 0 k µ+ Kµ (kz)j (k x x )dk. z z +( x x ) (.38) ] µ+. (.39) γ = +µ = + + ( ), (.40) 4 so that we can write [ ] γ M m (z, x x ) = ǫγ Γ(γ) π Γ ( z ). (.4) γ z +( x x ) We now realize that, by construction, and define M m (ǫ, x x ) δ( x x ), (.4) N m (z, x x ) = M m (z, x x )ǫ γ, (.43) which allows us to write for N m the expression N m (z, x x ) = [ ] γ Γ(γ) π Γ ( z ). (.44) γ z +( x x ) Therefore, we have constructively proved that lim N m(z, x x ) δ( x x ). (.45) z 0 Note that (.44) is indeed the well known expression for the boundary-bulk propagator for a scalar field in configuration space. However, this expression can only be used as an approximation to the propagator K when µ =. 9

10 3 Minkowskian Case 3. Massless Field Propagator Let is now deal with the case in which the boundary of the ADS + is the -dimensional Minkowskian space. In the massless case the field-equation is z z ˆφ(z,k)+( )z zˆφ(z,k)+z k ˆφ(z,k) = 0, (3.) where k = k 0 k = ρ. Thus, we can write z z ˆφ(z,ρ)+( )z zˆφ(z,ρ)+z ρˆφ(z,ρ) = 0, (3.) or, rewriting this last equation, z z ˆφ(z,ρ)+( )z zˆφ(z,ρ) z [ i(ρ±i0) ] ˆφ(z,ρ) = 0. (3.3) The distribution (ρ±i0) λ is defined as (see reference [6]) (ρ±i0) λ = ρ λ + +e ±iπλ ρ λ, (3.4) and can be cast in terms of H(x), the Heaviside step function [6]. We recast now (3.3) in the form of a Bessel equation z ( ) ˆφ(z,ρ)+ zˆφ(z,ρ) [ i(ρ±i0) ] ˆφ(z,ρ) = 0. (3.5) z The solution of this equation that is i) regular at the origin and ) vanishes for ρ, becomes ˆφ(z,k) = z K [ i(ρ±i0) z]. (3.6) One must take into account that lim k e ikx = 0 (see below in this section and [9]). Γ ( ) ) K [ i(ρ±i0) z] = +O ([ i(ρ±i0) [ i(ρ±i0) z] z] +. (3.7) We have then φ (z,x) = z (π) a( k)k [ i(ρ±i0) z]e ik x d k = 0

11 ˆφ(z,k)e ik x d k. (3.8) From this last equation we deduce that φ (z,x) = z (π) Γ ( ) or, equivalently, φ (z,x) = z (π) Γ ( ) The ensuing propagator becomes then K (z,x x ) = z (π) Γ ( ) Thus, the corresponding Feynman s propagator is K F (z,x x ) = z (π) Γ ( ) [ i(ρ±i0) ] K [ i(ρ±i0) z]ˆφ0 (k)e ik x d k, (3.9) [ i(ρ±i0) ] K [ i(ρ±i0) z] φ 0 (x )e ik (x x ) d kd x. (3.0) [ i(ρ±i0) ] K [ i(ρ±i0) z]e i k (x x ) d k. (3.) [ i(ρ+i0) ] K [ i(ρ+i0) z]e i k (x x ) d k. (3.) Note that the Feynman propagator is a function of ρ+i0, as it should. For the anti-feynman propagator we have instead K AF (z,x x ) = z (π) Γ ( ) The expression for the Wheeler s propagator is: Using the relations [i(ρ i0) ] K [i(ρ i0) z]e i k (x x ) d k. (3.3) W(z,x x ) = [K F(z,x x )+K AF (z,x x )]. (3.4) K F (z,ρ) = z Γ ( ) [ i(ρ+i0) ] K [ i(ρ+i0) z], (3.5)

12 and K AF (z,ρ) = z Γ ( ) [i(ρ i0) ] K [i(ρ i0) z], (3.6) we can define, as usual, the retarded propagator K R (z,ρ) = H(k 0 )K F (z,ρ)+h( k 0 )K AF (z,ρ), (3.7) and the advanced propagator K A (z,ρ) = H(k 0 )K AF (z,ρ)+h( k 0 )K F (z,ρ). (3.8) Wearegoingtoshow nowthat lim e ikx = 0(see [9]) Let ˆφbeatest function k belonging to a sub-space S of Schwartz s one [5, 6]. Its Fourier transform is φ(k) = where φ belongs to S. Then one can verify that 0 = lim φ(k) = lim k k As a consequence, we obtain ˆ φ(x)e ikx dx = ˆ φ(x)e ikx dx, (3.9) ˆ φ(x) lim k e ikx dx. (3.0) lim k eikx = 0 (3.) The Feynman propagator is, according to (3.,) K F (z,x) = z (π) Γ ( ) [ i(ρ+i0) ] K [ i(ρ+i0) z]e ik x d k. (3.) Since K is exponentially decreasing or oscillating, we can evaluate the integral that defines K F by means of a Wick rotation over k 0. Therefore we have the change of variables k 0 = ik 0E, x 0 = ix 0E, ke = k 0E + k, and x E = x 0E + x. Casting the integral that defines the propagator in terms of these new variables, we obtain K F (z, x E ) = iz (π) Γ ( ) k E K (k Ez)e i k E x E d k E. (3.3)

13 Using Bochner s formula together with (.9) we have K F (z,x E ) = iγ() [ ] π Γ ( z ). (3.4) z +x E Now, making the change to Minkowskian variables and taking into account that the Fourier transform of a distribution that depends on ρ i0 is a distribution that depends on x +i0, we obtain K F (z,x) = iγ() [ π Γ ( ) z z x i0 ], (3.5) which is the expression of the Feynman propagator in terms of the variables of the configuration space. For the anti-feynman propagator we analogously find K AF (z,x) = iγ() [ ] π Γ ( z ). (3.6) z x +i0 3. Massive Field Propagator For the massive case, the field-motion equation is { z ( ) [ ] } ˆφ(z,ρ)+ zˆφ(z,ρ) i(ρ±i0) ( ) + ˆφ(z,ρ) = 0, z z (3.7) with, again, µ = + ( ). (3.8) 4 The pertinent solution is now ˆφ (z,ρ) = z Kµ [ i(ρ±i0) z]. (3.9) The field-expression in configuration space is then φ (z,x) = z a( (π) k)k µ [ i(ρ±i0) z]e ik x d k. (3.30) Once again we choose φ(ǫ,x) = φ ǫ (x) = ǫ (π) a( k)k µ [ i(ρ±i0) ǫ]e ik x d k = 3

14 and from (3.3) we obtain (π) ˆφ ǫ (k)e ik x d k, (3.3) a(k) = We have then the following relation for the solution φ (z,x) = ( z ) (π) ǫ so that the propagator is now K m (z,x x ) = ( z ) (π) ǫ ˆφ ǫ (k) ǫ K µ [ i(ρ±i0) ǫ]. (3.3) Kµ [ i(ρ±i0) z] K µ [ i(ρ±i0) ǫ] φ ǫ(x )e ik (x x ) d kd x, The corresponding Feynman s propagator becomes K mf (z,x x ) = ( z ) (π) ǫ (3.33) Kµ [ i(ρ±i0) z] k (x x ) K µ [ i(ρ±i0) ǫ] e i d k. (3.34) Kµ [ i(ρ+i0) z] k (x x ) K µ [ i(ρ+i0) ǫ] e i d k, (3.35) and for the anti-feynman propagator we obtain the expression K maf (z,x x ) = ( z ) (π) ǫ Kµ [i(ρ i0) z] k (x x ) K µ [i(ρ i0) ǫ] e i d k. (3.36) Finally, the definition of Wheeler propagators, retarded and advanced, is similar to that of the preceding subsection. 3.3 Approximation We now evaluate in approximate fashion the propagator K µ [ i(ρ+i0) ǫ] entailing M mf (z,x) = z ǫ µ (π) µ Γ(µ) K µ [ i(ρ+i0) µ Γ(µ) ǫ] = ( i) µ (ρ+i0) µ ǫ µ, (3.37) K µ [ i(ρ+i0) z][ i(ρ+i0) ] µ e ik x d k. 4 (3.38)

15 Effecting again the above Wick s rotation we obtain M mf (z, x E ) = iz ǫ µ k µ (π) µ E Γ(µ) K µ(k E z)e i k E x E d k E. (3.39) This integral is evaluated as in the previous cases. One has M mf (z, x E ) = iǫµ Γ ( ) µ ( ) µ+ z. (3.40) π Γ(µ) z +x E Changing variables as above we arrive at ( ) γ M mf (z,x) = iǫγ Γ(γ) π Γ ( z ), (3.4) γ z x i0 where γ = + + ( ). (3.4) 4 Now we return to the inequality The following relation is valid for N F M mf (ǫ,x) δ(x) (3.43) N mf (ǫ,x) = ǫ γ M mf (ǫ,x). (3.44) Proceeding in analogous fashion with the Anti-Feynman propagator we obtain the approximation ( ) γ M maf (z,x) = iǫγ Γ(γ) π Γ ( z ) (3.45) γ z x +i0 4 Two points correlation functions in Euclidean space 4. Massless case To evaluate the two-points correlation function of a scalar operator we use the result obtained in [0]. This is < O( x )O( x ) >= g µ K(y 0, y x ) µ K(y 0, y x )d + y, (4.) 5

16 where 0 y 0 = z <, y µ = x µ, µ 0, and then < O( x )O( x ) >= lim z 0 [z K(z, x x ) z K(z, x x )]d x. (4.) Boundary As lim z 0 K(0, x x ) = δ( x x ), we obtain < O( x )O( x ) >= lim z 0 [z z K(z, x x )]. (4.3) Using now the expression for K given in Eq.(.0) we have < O( x )O( x ) >= Γ( +) π Γ ( ) ( x x ). (4.4) Accordingly, we have here arrived to the usual, well-known result. 4. Massive case For the massive case we obtain similarly < O( x )O( x ) > m = [z K m (z, x x ) z K(z, x x )]d x. (4.5) Boundary As K m (ǫ, x x ) = δ( x x ) we can write < O( x )O( x ) > m = δ( x x )[z z K(z, x x )] z=ǫ d x. (4.6) Thus we arrive at < O( x )O( x ) > m = [z z K m (z, x x )] z=ǫ. (4.7) Now, we use the expression for K m given in (.3) and write [ (z < O( x )O( x ) > m = ǫ ) Kµ (kz) (π) z ǫ K µ (kǫ) e i k ( x x ) d k or, equivalently, < O( x )O( x ) > m = ǫ 3 (π) [ z Kµ (kz) K µ (kǫ) e i k ( x x ) d k + 6 ], (4.8) z=ǫ

17 z k K µ (kz) ] K µ (kǫ) e i k ( x x ) d k Using now the following result, given in [8], we obtain < O( x )O( x ) >= ǫ (µ )δ( x x ) ǫ (π) z=ǫ. (4.9) K µ(z) = µ z K µ +K µ, (4.0) k K µ (kǫ) K µ (kǫ) e i k ( x x ) d k. (4.) Note that we have not renormalized the correlation functions. We will do that using the results of [] in a forthcoming paper. 5 Two points correlation functions in Minkowskian space 5. Massless case Similarly to the Euclidean case we obtain for the Minkowskian one the result < O(x )O(x ) > F = ilim z 0 [z z K F (z,x x )]. (5.) Thus, we obtain for the Feynman s propagator < O( x )O( x ) > F = Γ( +) π Γ ( ) [( x x ) (x 0 x 0 ) +i0], (5.) and for Anti-Feynman one < O( x )O( x ) > AF = Γ( +) π Γ ( ) [( x x ) (x 0 x 0 ) i0]. (5.3) 5. Massive case Again, following the developments of the Euclidean case, we have, for the Minkowskian instance, the two points Feynman s correlator < O( x )O( x ) > mf = i[z z K mf (z,x x )] z=ǫ. (5.4) 7

18 Thus, we have < O(x )O(x ) > mf = i ǫ (π) z or equivalently, < O(x )O(x ) > mf = i ǫ 3 (π) iz (ρ+i0) [ (z Using again (4.0) we finally obtain ǫ (π) ǫ ] ) Kµ [ i(ρ+i0) z] e ik (x x ) d k K µ [ i(ρ+i0) ǫ] (5.5) [ z Kµ [ i(ρ+i0) z] e ik (x x ) d k K µ [ i(ρ+i0) ǫ] ] K µ z] [ i(ρ+i0) e ik (x x ) d k K µ [ i(ρ i0) ǫ] < O(x )O(x ) > F = ǫ (µ )δ(x x )+ z=ǫ. (5.6) (ρ+i0) K µ [ i(ρ+i0) ǫ] e ik (x x ) d k. (5.7) K µ [ i(ρ+i0) ǫ] For the Anti-Feynman propagator we obtain in analogous fashion < O(x )O(x ) > maf = ǫ (µ )δ(x x ) ǫ (π) (ρ i0) K µ [i(ρ i0) ǫ] e ik (x x ) d k. (5.8) K µ [i(ρ i0) ǫ] Note again, that we have not renormalized the correlation functions. We will do that using the results of [] in a forthcoming paper. 6 Discussion In this work we have first calculated, without using any conjecture, the boundary-bulk Feynman, Anti-Feynman, and Wheeler propagators for both a massless and a massive scalar field, by recourse to the theory of distributions. Note that such a conjecture was made in 00 by Son and Starinets [4], only for the Feynman propagator, and that they did not get a correct result 8 z=ǫ,

19 As further novelties, in the paper we show that, for massive scalar fields, the expression for the boundary-bulk propagator in Euclidean momentum space does not correspond to the expression used in the configuration space, but it is rather a mere approximation. Subsequently, using the previous results, we have evaluated the correlation functions of scalar operators corresponding to massless and massive scalar fields. Unlike the results obtained in [4], with the ones obtained here you can calculate the n-points correlation functions from gravity. This is feasible for a scalar operator when n is an arbitrary natural number. 9

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