Bi-Local Fields in AdS${}_5$ Spacetime

Recently, the bi-local fields attract the interest in studying the duality between $O(N)$ vector model and a higher-spin gauge theory in AdS spacetime. In those theories, the bi-local fields are realized as collective one's of the $O(N)$ vector fields, which are the source of higher-spin bulk fields. Historically, the bi-local fields are introduced as a candidate of non-local fields by Yukawa. Today, Yukawa's bi-local fields are understood from a viewpoint of relativistic two-particle bound systems, the bi-local systems. We study the relation between the bi-local collective fields out of higher-spin bulk fields and the fields out of the bi-local systems embedded in AdS${}_5$ spacetime with warped metric. It is shown that the effective spring constant of the bi-local system depends on the brane, on which the bi-local system is located. In particular, a bi-local system with vanishing spring constant, which is similar to the bi-local collective fields, can be realized on a low-energy IR brane.


Introduction
In the development of theories of massless higher-spin fields (HS), it is recognized that a cosmological constant of background spacetime is necessary to construct consistent theories of those fields [1][2][3]. In particular, higher-spin field theories under AdS backgrounds are expected as an important route studying AdS/CFT correspondence [4,5]. According to this line of approach, there are many attempts to study AdS dual of conformal vector fields, which are sources of HS in AdS spacetime. It is also expected that those HS are realized in tensionless limit of string theory. In particular, recently, there arises an interesting point of view such that the aggregate of those HS is dual to a bi-local collective field out of conformal vector fields in the large N limit. In terms of O(N ) vector fields φ a (x), (a = 1, 2, · · · , N ), the bi-local collective field, there, is given by (1.1) 2 Embedding of a bi-local system in AdS 5 spacetime The AdS 5 spacetime with anti de Sitter radius l is realized as the hyper-surface described by coordinates (X µ , X 4 , X 5 ) satisfying η AB X A X B = η µν X µ X ν + (X 4 ) 2 − (X 5 ) 2 = −l 2 , where (A) = (µ, 4,5) and diag(η µν ) = (− + ++). The transformation X µ = e −ky x µ , X 4 = l sinh(−ky) − 1 2l x 2 e −ky , and X 5 = l cosh(−ky) + 1 2l x 2 e −ky with k = l −1 can define another coordinate system of independent variables (xμ) = (x µ , y). In this coordinate system, the spacetime can be characterized by the warped metric η AB dX A dX B = gμνdxμdxν = e −2ky η µν dx µ dx ν + dy 2 , (2.1) which is used in Randall-Sundrum model to address the Higgs Hierarchy Problem [15,16].
In what follows, we consider the AdS 5 spacetime described by the coordinate system (x µ , y), (0 ≤ y ≤ L) with this warped metric 2 . Now, we set the action of a bi-local system in this curved spacetime so that 3 1 . Some of those results were presented at "CST & MISC Joint Symposium on Particle Physics, 2015". 2 We set that the Planck energy scale brane and the low-energy brane are located respectively at y = 0 and y = L; and, we regard kL ≃ 32 ∼ 50 so that e −kL ≃ 10 −14 ∼ 10 −22 . 3 In this paper, we use the unit = c = 1. Figure 1. The figure shows the geodesic interval σ 2,1 connecting two particles in a curved spacetime so that x (1) (τ ) = x(ξ 0 ) and x (2) (τ ) = x(ξ 1 ). The real lines are world lines of respective particles; and, the dotted line designates the geodesic having x (i) , (i = 1, 2) as its ends.
where τ and e (i) (i = 1, 2) are respectively a time ordering parameter of dynamical variables and einbeins in τ space. The θ is an auxiliary variable, which transforms in the same way asẏ under the transformation of τ . We have introduced the θ term to restrict the relative motion of y [17], although the covariance of this formalism is spoiled unless θ → 0. The U (x (2) , x (1) ), U 2,1 simply, is a bi-scalar function representing the interaction of two particles at x (i) , (i = 1, 2) with the same numerical value of τ . According to a previous paper [18] we define this interaction term in such a way that where κ and ω are positive constants with dimension of mass square; and, σ(x (2) , x (1) ) is the geodesic interval defined by [19] σ( The geodesic equation is equivalent to the Euler-Lagrange equation reading L as a Lagrangian. Substituting the solution with conservative quantities along the geodesic for (2.4), the geodesic interval is obtained as function of both ends of the geodesic (Fig.1); that is, σ 2,1 becomes a function of xμ (i) (τ ), (i = 1, 2) only. The σ(x (2) , x (1) ), σ 2,1 simply, is equal to one half the square of the distance along the geodesic between x (1) and x (2) , which tends to 1 2 (1) )ν according as gμν → ημν. Thus, in such a flat spacetime limit, the S in (2.2) represents the action of a two-particle system bounded by a relativistic harmonic oscillator potential with a spring constant κ 2 .
It is obvious that y ′ = 0 at the turning point L = 1 2k log 2K V 2 ; and so, we have K = V 2 2 e 2kL at y = L. Now, in a single-valued region of y(ξ) such as 0 ≤ y(ξ) ≤ L bounded by y(ξ 0 ) = 0 and y(ξ L ) = L, the geodesic equations for x µ (ξ) and y(ξ) two kinds of constants along the geodesic γ such that where x ′μ = dxμ dξ . Equation (2.6) says that if we read y as the coordinate of a particle with unite mass under the potential 1 2 e 2ky V 2 , then K becomes a total energy of the particle, which is related to the turning point L byK = V 2 2 e 2kL (Fig.2). Since K is nothing but L, one can obtain the expression in a single-valued region of y. When we write down the right-hand side of this equation by K and V µ , we firstly be careful about the distinction of two kinds of geodesics xμ + (ξ) with y ′ (ξ) > 0, (ξ 0 ≤ ξ ≤ ξ L ) and xμ − (ξ) with y ′ (ξ) < 0, (ξ L ≤ ξ ≤ ξ 0 , ). Then, using the abbreviation (x µ i,j ) ± = x µ ± (ξ i ) − x µ ± (ξ j ) and |x i,j | ± = (x i,j ) 2 ± , we can get the following formula (Appendix A): on condition that ke −kL |x i,j | ± < 1. The (2.8) is the same for ±; and, as a result, we do not have to worry about that signs. In what follows, we deal with the geodesic starting with y = L along x − line without notice. The potential U (x (2) , x (1) ) defined by (2.8) is, then, not a function of the translational invariant variable x 2,1 due to the curvature in the AdS 5 spacetime. Furthermore, it is not applicable for a long-distance interval ke −kL |x j,i | ≫ 1. However, if we confine our attention to a case such that the two particles are located near y = L brane, the low-energy IR brane, then the situation will be changed. In this case, the geodesic interval can be expressed as follows: where the tilde denotes the scaled variablesL = kL,x = kx, and so on. The result implies that the geodesic interval becomes ∼ 1 2L 2 near e −L |x 2,1 | ∼ 1. Since, however, we are interested in the bi-local system with an extension such as |x 2,1 | > eLl ≫ l, we should apply (2.9-b) to define the potential (2.3) for the present practical application. Thus, the two-body potential under those considerations is where ω is a free parameter effecting on the ground state mass of the bi-local system. We also stress that we may regard the |x 2,1 | in the right-hand side is independent of y due to e −L |x 2,1 | ≫ 1; and, that the resultant two-body potential is fortunately invariant under the translation of four-dimensional variables x µ (i) , (i = 1, 2).

The wave equation of bi-local system in AdS 5 spacetime
The wave equation of the bi-local field in AdS 5 spacetime is q-number representation of the constraints derived from the action (2.2). The Lagrangian out of this action defines the canonical momenta (p (i) , π (i) ) conjugate to (x (i) , y (i) ) in the following form: Under the definition of those canonical momenta, the variations of the Lagrangian with respect to e (i) , (i = 1, 2) and θ give rise to the constraints The constraints (3.4) and (3.5) are not compatible each other; then, we eliminateπ with its conjugate variableȳ = y (1) − y (2) strongly by means of the Dirac bracket for the second class constraintsπ =ȳ = 0. After that, we do not need to worry about the degrees of freedom (ȳ,π). Then using the combinations 1

the constraints (3.4) can be written as
where P = p (1) + p (2) ,p = 1 2 p (1) − p (2) , and π = π (1) + π (2) are the momenta conju- (2) , and y = 1 2 y (1) + y (2) , respectively. The canonical quantization is carried out by replacing the Dirac bracket by the commutator. Then the q-number counterparts of (3.6) and (3.7) define respectively a master wave equation and its subsidiary condition. In the case of flat (k = 0) spacetime, those equations are reduced to bi-local field equations in five-dimensional Minkowski spacetime. In such a reduced system, the condition (3.7) is understood in the sense of expectation value by a physical state Ψ|T |Ψ = 0 or by T (+) |Ψ = 0, where T (+) is a part of T written by the annihilation operators defined out of (p,x). Then the equations H|Ψ = 0 and T (+) |Ψ = 0 come to be compatible each other; and so, there arise no ghost states due to time-like oscillations of the bi-local system.
In the curved spacetime with k = 0, we can not apply this method directly to equations (3.6) and (3.7). First, we have to make clear the operator ordering of e −2ky π 2 in q-number theory. In what follows, we simply take the Weyl ordering Thus the wave equation and its subsidiary condition in q-number theory become where the definition of (P ·p) (+) is not given in this stage. The operator W has the eigenstates φ λ (z) = z ) (Appendix B) associated with the boundary condition d dz φ λ (z) y=L = 0, whose roots r 1 , r 2 , · · · determine the eigenvalues so that λ i = e −L kr i 2 , (i = 1, 2, · · · ). Then the Ψ satisfying the boundary condition ∂ y Ψ| y=L = 0 can be expanded by a Fourier-Bessel series such as where the coefficient a n decreases according as n increases, since φ λn rapidly oscillates for a large n. Until now, the spring constant κ 2 andL are free parameters; in what follows, we put restriction on those parameters by the conditions in UV and IR branes. First, in UV brane with y = 0, we require by taking |x| > eLL into account that the order ofx-potential term becomes e −2L (κL) 2 |x| 2 > ( κ kL 2 ) 2 ≫ λ n ∼ e −2L k 2 ; then, we obtain the first condition κ ≫ e −L k 2 /L 2 . In this case, the order of the eigenvalues λ n 's are negligible small compared with that of U 2,1 in UV brane even for a large r n , since the a n in (3.11) itself is decaying according as r n → ∞. Thus, we discard the W term in (3.9) at UV brane.
Subsequently, we move the bi-local system from UV brane to the brane with y > 0 (Fig.3); then, (3.9) and (3.10) take the following simple forms: As a matter of course, hereafter, the y in those equations should be treated as a parameter instead of a dynamical variable, otherwise the bi-local system allows contiguous spectrum. The next task is to determine the (P ·p) (+) ; for this purpose, we introduce y-dependent κ y = e −(L+ỹ) κL. Then, we can say that κ 0 = e −L κL and κ L = e −2L κL are the square roots of spring constants in UV and IR branes, respectively. With this y-dependent κ y , we define the y-dependent oscillator variables such that to which one can verify [x µ , In terms of those oscillator variables, (3.12) can be written as from which one can say that α ′ y = (8κ y ) −1 is the Regge slope parameter in a y-fixed brane. Then, as the second condition on (κ,L), we require κ L 10 −20 k 2 so as to obtain almost infinite slope parameter α ′ L at IR brane. Both conditions at UV and IR branes give rise to a possible choice such as (κ,L) ∼ (k 2 , 50).
As the final step in this section, we set (P ·p) (+) = −i κy 2 P · a, then (3.13) becomes a subsidiary condition compatible with (3.12). Therefore, in what follows, we read (3.13) as P · aΨ(X,x, y) = 0. (3.16) 4 The bi-local fields in a brane near IR one Let us consider the solutions of (3.15) and (3.16) in each y-fixed brane. First, the ground state of the oscillator variables (a µ , a † µ ) defined by a µ |0 = 0 can be solved as 4 to which (3.15) yields the mass-square eigenvalue In this stage, we adjust ω so as to be M L (0) 2 = 0; that is, we put ω = −4κL.
To construct the excited states of relative oscillation, one can use the physical oscillator variablesâ † µ = Λ µν a †ν , (Λ µν ≡ η µν − P µ P ν /P 2 ), which tend to (â †µ ) = (0, a †i ) in the rest frame (P µ ) = (P 0 , 0) of the bi-local system. In terms of those physical oscillator variables, one can write a complete basis so thatâ † µ 1â † µ 2 · · ·â † µ J |0 , (J = 0, 1, · · · ). Since those states belong to reducible representations of rotation group in the rest frame of the bi-local system, it is convenient to use those states under the following combination: whereâ † (µâ † ν · · ·â † ρ) is the totally symmetric and traceless combination ofâ † µâ † ν · · ·â † ρ ; further, the state with J = m = 0 is read as the ground state given by (4.1). One can verify that the state |Φ In particular, since the spin operator S of the bi-local system in the rest frame satisfies S 2 = N (N + 1) − Q, the state |Φ (0) µ 1 ,··· ,µ J belongs to an irreducible spin representation with the highest spin J. Then (4.6) implies that the particles represented by |Φ (0) µ 1 ,··· ,µ J exist on 4 The accurate representation of ground state should be |0y , although we have used a simple notation |0 . The normalization of x|0 in the indefinite metric formalism is given by a leading Regge trajectory with a slope parameter α ′ y ≡ (8κ y ) −1 (Fig.4). Thus, the general solution of (3.15) and (3.16) with a fixed J has the form where is the eigenstate such asx µ |x ′ =x ′µ |x ′ with the normalization The factor π κy in the right-hand side of (4.7) is introduced for the normalization of Φ y;µ 1 ···µ J (X,x) in the limiting case of κ L ∼ 0; strictly speaking, κ L ∼ 0 means that the order of κ L comes to be 0 compared with the energy scale in IR brane. It should be noticed that the states (4.7) and (4.8) contain the parameter y through κ y . Now, from (4.8), it is not difficult to evaluate π κ y x|Φ (m) The φ (m) P (x (i) ), (i = 1, 2) are scalar fields associated with respective particles with the mass 1 2 M L (0, m) ∼ 0 because of κ L ∼ 0; that is, the masses of those particles are almost degenerate. If we truncate the summation with respect to m in (4.11) to some number, the resultant bi-local field becomes the one, which should be compared with the bi-local collective field (1.1).  Figure 4. The black and gray circles designate respectively leading Regge and their daughter particles. The particles on horizontal dashed line have a common spin, whose mass will be degenerate to the ground-state one in the limit α ′ y → ∞ (y → L).

Summary and discussion
In this paper, we have discussed the relation between a bi-local system embedded in AdS 5 spacetime with warped metric and the higher-spin bulk fields emerging as bi-local collective fields. We tried to formulate the bi-local system in AdS 5 curved spacetime in such a way that the system is reduced to two-particle bound system with a covariant harmonic oscillator potential in flat Minkowski spacetime. As a counterpart of such a harmonic oscillator potential in curved spacetime, we used the geodesic interval connecting two particles. We also modify the kinetic term of the bi-local system so that the internal relative motion is suppressed with the aid of an auxiliary gauge variable overlooking the full covariance of this formalism. The resultant bi-local system is characterized by three kinds of constraints. Two of them are associated with the invariance of the action under the reparametrization of time ordering parameters of respective particles; and, the other is the one due to the auxiliary gauge variable θ. In canonical formalism, the first two constraints are corresponding to onmass-shell condition of the system and physical condition eliminating some relative motions of the bi-local system respectively.
In q-number theory, those two constraints become the wave equation of the bi-local system and its subsidiary condition, which extracts the physical states of the bi-local system, the one-particle wave function of the bi-local field. As for the constraint suppressing internal relative motion, we eliminated it beforehand as a second class constraint in the stage of classical theory. However, the remaining constraints are still not compatible each other. Then, first, we discarded the W term, the kinetic term of internal center of mass variable y, in the UV brane; then, y becomes simply a parameter designating each brane, on which the bi-local system is placed. We further treated the T condition suppressing timelike oscillations of the bi-local system in a form of expectation value; and then, the wave equation and it subsidiary condition, T condition, become compatible each other.
The on-mass-shell solutions of resultant wave equation represent the particles overlying on Regge trajectories with the slope parameter α ′ y = (8κ y ) −1 . Here, the κ 2 y is a spring constant in a y-fixed brane, which tend to 0 according as y comes close to L, the place of IR brane. Strictly speaking √ κ L is almost 0 compared with the energy scale of IR brane. To realize these setup on the bi-local system, we have chosen the parameters in our model so that (κ,L) ∼ (k 2 , 50), which derive the reasonable order of √ κ y such as ( √ κ 0 , √ κ L ) ∼ k(10 −10 , 10 −21 ) for k ∼ Planck energy scale.
Hence, in the IR brane, all particles degenerate in almost massless one's; furthermore, non-zero spin components of the bi-local field can be shown to vanish naturally on that brane. Therefore, the bi-local field on the IR brane behaves as the bi-local collective field (1.1) out of higher-spin bulk fields as we wanted to show. It should be, however, noticed that the respective particles described by the bi-local field (4.11) hold a common center of mass momentum as their hysteresis of a bound system in bulk.
Further, from the bi-local field (4.11), we cannot say anything about 1) the bound system of particles laid on different branes and 2) the bi-local system with very small interval such as |x 2,1 | e −L l. In relation with 2), we should also notice that the practical interaction between two particles on the IR brane can take place only under discreet distances with the unit of e −L l. This property of the interaction evokes the behavior of elementary domains proposed by Yukawa [20,21]. Those are important and interesting future problems to make clear the relation between the bi-local system and the bi-local collective field.
Regarding L as the Lagrangian of generalized coordinates (x µ (ξ), y(ξ)) in AdS 5 described by (2.1), the equations (2.5) and (2.6) are respectively direct results of equations of motion with respect to x µ (ξ) and y(ξ). As can be seen from Fig.5, the y(ξ) is a multi-valued function of ξ. Changing the variable ζ(ξ) = e −ky(ξ) , (2.5) can be rewritten as which can be integrated easily in a single valued region of y so that where the ± in (A.2) designate the sign of y ′ ; the c * is a constant related to one end of γ such as y * = y(ξ * ). In what follows, we choose simply y * = L (ξ * = ξ L ) one turning point of γ, which leads to c * = 0 because of e −kL = V 2 2K as pointed out in Fig.2. In this case, x µ + (ξ) and x − (ξ) become functions defined respectively in the region ξ 0 ≤ ξ ≤ ξ L and ξ L ≤ ξ ≤ ξ 0 , where ξ 0 and ξ L are points such as by y(ξ 0 ) = 0 and y(ξ L ) = L. Then, substituting (A.2) in this case for (2.5), we can integrate x ′µ ± (ξ) as is a constant unit vector for the direction of x ′µ . Hereafter, we write It is also convenient to use the symbolÃ = kA, which is the A measured by k −1 ∼ l. Then (A.3) can be written as from which the following follows providing e −L |x i,L | ± < 1, (i = a, b). The result means that we do not need to worry about the ± when we representξ b,a in terms ofx i,L . Further, (A.2) yields another expression tõ If we apply (A.5) formally to |ξ 0,L |, then σ 0,L ≃ L 2 /2 requires e −L |x 0,L | = tanh(L) ∼ 1. Since, however, e −L |x 0,L | ∼ 1 is near the applicable limit of (A.5), we must be careful to evaluate it. The right value of |x 0,L | can be obtained from To extent this relation to multi-valued regions of y(ξ), we have to take the successive Figure 5. The symbolic figure of turning geodesic. The superscript (n) in ξ (n) , y (n) designate that they characterize the geodesic in R (n) region in the figure. turnings of geodesic at y = 0 and y = L branes in (Fig.5) into account. Writing, here, ξ we are able to get the following expression: where [n] is the largest integer being not greater than n. In this paper, we are interested in the bi-local bound system such as n ≫L; then, one can evaluate 2 n+1 2 ≃ n ≃ e −L |x b (n) ,a (0) |. Therefore, under those approximations, the geodesic interval extended to n turned regions becomes 14) in which the discrete indices (n) and (0) are no longer important to attach. Here, if we try to apply (A.14) to b (n) = 0 (1) and a (0) = L (0) , then we can get σ 0,L = 1 2 L 2 , the result of (A.7). This implies that the|x b,a | in (A.14) is applicable from |x b,a | eLl to infinity. On the other side, (A.9) is holds in the n = 0 single-valued region with |x i,L | ≪ 1, (i = a, b).

B Eigenvalue problem of W
The eigenvalue problem of W can be solved easily by using the variable z(y) = e ky . Then, by taking d dy = kz d dz into account, the eigenvalue equation of W can be written as ; that is, 3) The solutions of this equation are reduced to Bessel functions multiplied by a power of z; indeed, it is known that the equation has z α Z ν (βz) as its solution [25], where Z ν (x) is one of J ν (x), Y ν (x), H ν (x), and H where the "prime"denotes the derivative with respect to z. This means that under the boundary conditions: with z 0 = z(0) = 1 and z L = z(L) = eL, we can put the normalization

C Spin eigenstates
In the rest frame of the bi-local system with P = (M, 0, 0, 0), the hatted oscillator variables are reduced to (â µ ) = (0, a 1 , a 2 , a 3 ) and (â † µ ) = (0, a † 1 , a † 2 , a † 3 ). In terms of those reduced oscillator variables, the spin operator of the bi-local system is defined by S i = −iǫ ijk a † j a k , (i, j, k = 1, 2, 3), to which by taking ǫ ijk ǫ ilm = (δ jl δ km − δ kl δ jm ) into account, one can verify where N = a † ·a and Q = a †2 a 2 . Since N and Q are commute each other, there are common eigenstates of those operators. In particular the spin eigenstates with zero eigenvalue of Q have the following form: where (a,b) stands for the summation over two indices (a, b) taken at a time out of J different objects (1, 2, · · · , J); and, One can see that the T j i ···j J i 1 ···i J is symmetric and traceless with respect to both of the subscripted indices (i 1 , · · · , i J ) and the superscripted indices (j 1 , · · · , j J ); then, it is not difficult to verify that Qa † (i 1 a † i 2 · · · a † i J ) |0 = 0, (C.4) S 2 a † (i 1 a † i 2 · · · a † i J ) |0 = J(J + 1)a † (i 1 a † i 2 · · · a † i J ) |0 .