Noncommutative Spacetime Realized in $AdS_{n+1}$ Space

In $\kappa$-Minkowski spacetime, the coordinates are Lie algebraic elements such that time and space coordinates do not commute, whereas space coordinates commute each other. The non-commutativity is proportional to a Planck-length-scale constant $\kappa^{-1}$, which is a universal constant other than the light velocity under the $\kappa$-Poincare transformation. In this sense, the spacetime has a structure called as"Doubly Special Relativity". Such a noncommutative structure is known to be realized by SO(1,4) generators in 4-dimensional de Sitter space. In this paper, we try to construct a nonommutative spacetime having commutative n-dimensional Minkowski spacetime based on $AdS_{n+1}$ space with SO(2,n) symmetry. We also study an invariant wave equation corresponding to the first Casimir invariant of this symmetry as a non-local field equation expected to yield finite loop amplitudes.


Introduction
The κ-Minkowski spacetime is a noncommutative spacetime characterized by an algebraic structure with a constant κ other than the light velocity; in this sense, the framework of κ-Minkowski spacetime is called as doubly special relativity (DSR) [1]. The κ is a Planck energy scale constant, which is usually said to be a trace of quantum gravity through the combination κ ∼ G/Λ (or /G) in some limit of G, Λ → ∞ [2]. Here, G and Λ are respectively the gravitational constant and the cosmological constant.
Associated with this dimensional constant, the coordinates of the κ-Minkowski spacetime form the Lie algebra characterized by where i runs over (1,2,3). The characteristic relations (1) and (2) spoil the symmetry under the Lorentz boost in the Minkowski spacetime, although they are symmetric under the space rotation. Nevertheless, this framework is symmetric under the κ-Poincare transformation, which reduces to the Lorentz transformation according as κ → ∞.
Historically, another Lie algebra type of noncommutative spacetime was first discussed in 1947 by H. S. Snyder [3] in the context of non-local field theory extended in spacetime with the fundamental length l ∼ κ −1 . Although the algebraic structure of Snyder's spacetime is slightly different from that of the κ Minkowski spacetime, the symmetry under the Lorentz boost is broken in this spacetime too. In both types of noncommutative space times, the dispersion relation of particles embedded in this spacetime becomes highly non-linear due to κ = 0. As a result of those non-linear dispersion relations, the wave equations of particles possess non-local structure by regarding those equations as effective ones in the usual commutative spacetime.
On the other hand, it is well known that H. Yukawa proposed an attempt of non-local field theory [4], the bi-local field theory, in the same period as Snyder's noncommutative spacetime theory appeared. Yukawa's attempt is motivated by a unified description of elementary particles and to get divergence free field theories by introducing a fundamental length in spacetime. After long history according to this line of thought, Yukawa arrived at an idea of field theory such as elementary domains [5], which obeys a difference equation instead of a differential equation. The equation of domain keeps the Lorentz invariance, but it is not consistent with the causality, since the field equation allows timelike extension of fields.
Although the field theories based on κ-Minkowski spacetime and domain like non-local field theories are standing on different bases of thought, they look like to have close connection each other. The purpose of this paper is, thus, to study the relationship between a domain type of field theory and a non-local field theory based on a κ-Minkowski like spacetime, which is modified so as to be symmetric under the Lorentz boost.
In the next section, we formulate the κ-Minkowski spacetime from the viewpoint regarding noncommutative coordinates as SO (1,4) generators in dS 4 space, the 4-dimensional de Sitter space. As an extension of this formulation, in section 3, we discuss a modified non commutative spacetime realized in AdS n+1 space so that the framework is symmetric under the Lorentz transformation. I section 4, we discuss a wave equation of a non-local field characterized by the first Casimir invariant in this space; and, a detailed analysis of on-mass-shell particles is given. In section 5, discussions on φ 3 type of interaction of such a field are made. Therein, we show that some loop diagrams become finite due to a non-unitary structure of such a field in the energy scale of κ . Section 6 is the discussion and summary. In appendices A and B, the mathematical background of §2 and §3 is summarized.
2 κ-Minkowski spacetime based on dS 4 A simple way to construct the Lie algebraic coordinates (1) and (2) is to start from dS 4 with coordinates y = (y A ) = (y 0 , y i , y 4 ), (i = 1, 2, 3) characterized by In terms of these coordinates, the generators of SO (1,4) isometric group can be written as to which one can verify that It is obvious that there are ten independent components of these generators; that is, the generators of space rotation {M ij }, the Lorentz boost {M i0 }, and remaining four generators {M µ4 }, ((µ) = (0, i)). The noncommutative coordinatesx 0 andx i are constructed out of those remaining generators byx Here, the inverse sign g 00 = −g 44 in the metric is essential to realize the commutation relations (1) and (2) 1 . For the latter purpose, it is convenient to introduce light-cone variables between (y 0 , y 4 ); that is, we put (y A ) = (y i , y + , y − ), (A = i, ±), where y ± = y 0 ±y 4 . Then, we can write the invariant length in dS 4 as y 2 =ḡ AB y A y B = −(y i ) 2 + y + y − with the metric In this basis, the noncommutative coordinates can be expressed aŝ In order to find the invariant wave equation in the κ-Minkowski spacetime, let us consider the unitary transformation U (ω) = e i 2 ω AB MAB , which causes a finite SO(1, 4) transformation in dS 4 in such a way that y A = U (ω)y A U † (ω) = (e κ −1 ω ) A B y B with ω = (ω A B ). In particular, for the contracted transformation defined by ω BC = a B b C − b B a C with (b i , b − , b + ) = (0, 2, 0), the exponent of U (ω) reduces to 1 2 ω BC M BC = a −x0 − a ixi ; namely, U (ω) becomes an exponential function in the κ-Minkowski spacetime in this contracted case. Furthermore, a i and a − are related to four momenta conjugate to {x µ } in some sense, which will be discussed soon. Now, SO (1,4) transformation of a c-number vector u = (u A ), (u 2 = −κ 2 ) in dS 4 is defined by U (ω)(u · y)U (ω) † = {(e −κ −1 ω ) A B u A }y B = u(ω) · y, so that u(ω) 2 = −κ 2 holds. If we choose, for sake of simplicity, u = (u 0 , u i , u 4 ) = (0, 0, 0, 0, κ), then u(ω) A = κ(e −κ −1 ω ) A 4 becomes a non-linear realization of a vector in dS 4 in terms of (a i , a − ). Here, a little calculation leads to the explicit form of e −κ −1 ω such that (Appendix A) Thus, (x µ ) tends to the p-representation of coordinates in flat Minkowski spacetime in the limit κ → ∞. In other words, {y µ } is the momentum space in flat Minkowski spacetime.
By taking (ω 04 , ω i4 , ω 44 ) = (−a − , −a i , 0) and (ω 2 04 , ω 2 i4 , ω 2 44 ) = (−a 2 , −a − a i , −(a − ) 2 + a 2 ) into account, we thus arrive at the expressioñ where we have writtenũ A (ω) = κ −1 u(ω) A ; henceforth, we use the same notationf = κ −1 f for any f . Further, we note that the vector (ũ A ) is usually introduced in relation to the bicovariant differentials of U (ω), sinceũ(ω) A = (e −κω ) A 4 satisfies (Appendix A) The next task is to identify U (ω) to an ordered exponential function e −ik 0x0 +ik ixi by some way: the typical cases arê where For example, if we put U (ω) =ê R (k), the right ordering of exponential function , we get the well known expression of the vector in dS 4 such that 2 2 Similarly, the substitutions (a − , a i ) → −k L/S give yields the other expressions of vectors in dS 4 : Therefore, if we put P A (k) = −κũ A (k) R , then eq.(15) can be read as where Finally, we discuss the SO(1, 3) transformation realized in {ũ(k) µ } through the transformation of {k µ }; then, their resultant form should bê so that the constraintũ Aũ B = −1 holds. TheL µν 's are actions on k µ causing non-linear transformations in general. The space rotation is simple, sinceũ(k) 0 andũ(k) i transform respectively as a scalar and a vector under the rotation of {k i }; then, the action ofL ij oñ k µ becomes as usualL The Lorentz boost is somewhat difficult, and it is given by (Appendix B) The closed algebra (28)-(30) consisting of (L ij ,L 0i , k µ ) is known as the κ-Poincare algebra [1], in which by definition, C 1 = P 4 (k) and C 2 = P (k) µ P (k) µ form respectively the first and the second Casimir invariants. The invariant wave equation under the κ-Poincare transformations is, thus, P 4 (k)Ψ = 0 or P (k) µ P (k) µ Ψ = 0. We may read those equations as non-local field equations in the Minkowski spacetime by substitution k µ → i ∂ ∂x µ . Then, those equations describe non-local fields with timelike extension, which spoils the symmetry under the Lorentz boost. It is, however, likely that if we start with a higher-dimensional spacetime with a timelike extra-dimension, then we can realize the non-commutativity between the usual spacetime and the extra-dimension, so that the Lorentz covariance is maintained.
3 Space non-commutatively realized in an AdS n+1 spacetime We are interested in the (n+1)-dimensional noncommutative spacetime with the coordinates (xμ,x n ) characterized by whereμ = (µ, i) runs over (µ) = (0, 1, 2, 3) and (i) = (4, 5, · · · , n − 1). The metric, here, is assumed to be gμν = diag(+, −, −, · · · , −−). One can realize the closed algebra (31) and (32) by the combination of the generators of isometry group of AdS n+1 with coordinates (y A ) = (yμ, y n , y n+1 ) defined by In terms of those coordinates, the generators of isometry group, the SO(2, n), can be written as , (A, B =μ, n, n + 1), to which the same type of algebra as (5) holds. The light-cone variables in this case is defined by y ± = y n+1 ± y n , by which the invariant length (33) for y = (yμ, y + , y − ) can be written asḡ AB y A y B = gμνyμyν + y + y − with the metric Then, it is easy to verify that the combination satisfy Eqs. (31) and (32). As in the previous section, we can again construct the vector in AdS n+1 space using the contracted SO (2, n) Then, each component of the vectorũ to whichũ Aũ A = 1 is satisfied obviously. It is also straightforward to rewrite those components in terms of the wave numbers, associated with an ordered exponential function. In what follows, for a reason of symmetry, we consider the case of symmetric ordering such aŝ which leads to k S = (kμ S , k n S ) = kn ek n /2 − e −k n /2 kμ, k n .
The Lorentz boost which causes the mixing between a new time componentũ n+1 and the spacetime componentsũμ can be again represented as a nonlinear transformation among {kμ}; and, the resultant form is Lμ ,n+1kn = e −k n 2kμ .
As in the case of previous section, P A = κũ A is a momentum vector in AdS n+1 space; and, under the transformations from (48) to (51), C 1 =ũ n (k) and C 2 =ũμ(k)ũμ(k) + (P n+1 (k)) 2 are the first and the second Casimir invariants, respectively.

Non-local field in the background of noncommutative spacetime
Let us, now, consider the wave equation for a scalar field, which is invariant under the SO(1, n − 1) transformation in {ũ A } space. It is obvious that the linear combinations of the first and the second Casimir invariants are those candidates, which tend to the Klein-Gordon equation in the limit κ → ∞. In what follows, we consider a wave equation with the first Casimir invariant only because of its simple structure; that is, we put as the free field equation. Here, m = κm is a κ-dependent mass-dimension parameter that is introduced in the meantime to adjust the lowest mass for this field. In this stage, the dimensional parameter in the theory other than the additional m 0 is κ only, which characterizes the spacetime in the Planck scale physics. We are, now, intended to modify the above field equation by introducing a new energy scale µ(< κ) according to the following two steps: In the first, we note that the k n is nothing but the a − in ω AB = a [A b B] , which define the vectorũ A = (e −ω ) A,n+1 ∈ AdS n+1 . Since b is a fixed vector in AdS n+1 with b + component only, a A may be a vector in AdS n+1 with a free a + component. Then, by shifting a + → a + + κ 2 /a − , we can put a A at a projective boundary of AdS n+1 such as a A a A = 0, on which SO(1, n + 1) acts as conformal transformation. Secondly, we break this conformal symmetry by introducing a scale parameter µ lower than κ in such a way that Since, this equation gives rise tok n = − 1 µκ kμkμ = − κ µkμkμ , then the field equation (52) is modified so that [6] kμkμ + 2 sinh(κ µkμkμ ) −m 2 Φ = 0, where κ µ = κ µ . Here, the scale parameter µ is introduced by hand without any principle; however, it may be natural to read κ µ ≃ 10 4 ∼ 10 5 , the order of unification.
Substituting i∂μ for kμ in K(k), the free field equation (55) becomes nothing but a nonlocal one in {xμ} space, which is longer a noncommutative space. In a practical model, further, the space of extra dimensions (x i ) = (x 4 , x 5 , · · · , x n−1 ) must be compact. For example, if we require the U (1) cyclicity x i ≡ x i + 2πr 0 , then k i takes the spectrum k i = li r0 , (l i = 0, ±1, · · · ), which we assume, henceforth, for sake of simplicity in addition to r 0 ∼ κ −1 .   Fig.2 The solutions associated with the intersection at x > 0 and x < 0 in Fig.1 are corresponding to even n and odd n poles, respectively.
The solutions of K(k) = 0, then, appear at the intersections of y = −2 sinh(x) and y = κ −1 µ x−m. From Fig.1, it is obvious that the intersection gives rise to a real k 2 > 0; i.e., a realtimelike-five momentum k. Other than such a time-like solution, there are complex solutions of k 2 ; and, the whole solutions are approximately expressed ask 2 ≃ κ −1 µ (−1) n 2m 2 + iπn , (n = 0, ±1, ±2, · · · ; |n| 2κ µ ) 3 . Therefore, the mass square M 2 = k µ k µ =k 2 + k i k i of particles in 4-dimensional spacetime takes the spectra where l = (l 4 , · · · , l n−1 ). In the right-hand side of equation (57), the first term is the order of κ 2 except the ground state l = 0. The third term add an imaginary component to M 2 n,l , which may spoil the unitarity in the energy of the order of √ µκ. In other words, the present effective theory will beyond the limits of validity in larger energy scale than √ µκ, where the spacetime gets back to noncumulative one. We finally note that the K −1 (k), the propagator of free field, has simple poles at as a function of z = κ µk 2 . Then, one can verify that are residues of K −1 (k) at z = z n characterized by K −1 z≃zn ≃ R n (z − z n ) −1 . Those poles are expected to play an effective role in internal lines of loop diagrams, though those poles are negligible in low energy physics.

An attempt of interacting fields
In the usual κ-Minkowski spacetime, it is not easy to formulate the interaction of fields because of its noncommutative structure amongx 0 andx i , (i = 1, 2, 3) [7]. In our approach discussed in the previous section, the resultant spacetime is commutative one, although the fields on it obey a non-local field equation. Nevertheless, local interactions of such fields are not excluded in principle, we here study, in attempt, a φ 3 type of interaction of the field, which is characterized by the free equation (55) and the following action: 3 In terms of z = ξ + iη, the equation κ −1 µ z + 2 sinh(z) −m 2 = 0 is decomposed into simultaneous equations κ −1 µ ξ + 2 cos(η) sinh(ξ) −m = 0 and κ −1 µ η + 2 sin(η) cosh(ξ) = 0. The latter leads to sin(η) = −(2κµ cosh(ξ)) −1 η ≃ 0; and so, we obtain y ≃ πn, (n = 0, ±1, · · · ; |n| 2κµ). Substituting these values for the former, the equation for ξ becomes κ −1 µ ξ + 2(−1) n sinh(ξ) −m = 0. The Fig.1 show that the solutions for ξ exist near ξ = 0 only; and so approximating sinh(ξ) ≃ ξ, we obtain ξ ≃ For simplicity, we confine our attention to the case of n = 5 with the compact fifth dimension such as x 4 ≡ x 4 + 2πr 0 ; and so, the wave number vector in (55) has the form (kμ) = (k µ , r −1 0 l) = κ{k µ , (κr 0 ) −1 l}, (l = 0, ±1, · · · ). Now, the sinh term in the free propagator K −1 (k) plays a roll of ultraviolet convergent for both regions of timelike and spacelike kμkμ in Feynman diagrams. To see this situation in detail, let us study the propagator up to the order of one-loop corrections consisting of connected diagrams described by Fig.3, to which we have the expression where K −1 * K −1 is the convolution of K −1 . By this convolution, the self-energy term Σ(p), the Fourier transform of x| − ig 2 2 K −1 * K −1 (i∂)|y , can be expressed as Further, I = x|K −1 (i∂)|x is the tadoploe term of Fig.4, to which we have the expression We first evaluate the tadpole term (63) in detail, since its structure is rather simple. For this purpose, it is not available to apply a simple Wick rotation with respect to k 0 , since K −1 (k) has poles on complex k 0 plane. However, remembering that K(k) is a function of κ µk 2 = (µκ) −1 (k 2 − r −2 0 l 2 ), we can write (63) in the following form: where K[z] = K(k)| z=κµk 2 . We have also approximated the summation with respect to l to the leading (l = 0) term only 4 , since l = 0 makes to damp (63) by the factor e −λκµl 2 . The next task is to evaluate the z integration in (64). This can be down by deforming the integration contour so as to surround poles of K −1 [z] in (58) by taking their residues (59) into account. Then, replacingm 2 →m 2 − iǫ as usual, we obtain where The parameter N (∼ 2κ µ ) plays a role to exclude the region λ N −1 . Indeed, if we approximate simplym = 0 in (66), we can verify that where θ(x) is the step function defined so that θ(x) = 0 or 1 according as x < 0 or x > 0. Unfortunately, however, since the integrand of λ integration in (64) has the form λ −2 Θ(λ), a logarithmic divergence still remains in I; that is, that the tadpole term I can be evaluated as The second term in the right-hand-side of the above equation is logarithmic divergent one; and so, we need a renormalization with a cut off to handle this term. Next, let us study the self-energy term defined in (62) according to the same line of approach to I. By the same reason as in (63), we again discuss the case l = 0 in both of external and internal lines in the above integral; that is, we put p = (p, 0) and k = (k, 0). Then, we obtain the expression Here, since Θ(λ) = 0 for λ > 0,we can insert ∞ 0 dτ δ(τ +λ 1 +λ 2 ) = 1 into the above integral. Then carry out the integration with respect to λ 2 after the scaling λ i = τλ i , we arrive at the expression where x = √ τ κ µ (λ 1 + 1 2 ) and One can find that D τ (x) equals 1 for almost region of |x| < √ τ κµ 2 and vanishes for |x| > √ τ κµ 2 ; that is, the interval of the integration with respect to Strictly speaking, near both limits of integration, we have to modify the edges of D(x) τ so as to approach continuously 0 reflecting the behavior of Θ(λ) near λ = 0. The condition that the both ends of the interval of x integration close to 0 should be up to the order of x 2 . On the other side, we may extend the interval of 2 ) to all over x axis for a finite τ (≫ κ −1 µ ). Therefore, we can roughly evaluate the x integration so that As for the former, in section 2, we could show that the noncommutative coordinates (x 0 ,x i ) in four-dimensional κ-Minkowski spacetime are nothing but generators of transformations between light-cone coordinate y + and others (y − , y i ) in dS 4 . The plane wave in the κ-Minkowski spacetime, then, has the meaning of a finite SO (1,4) transformation. From this definition of the plane wave, the five-momentum P A in dS 4 associated with the bi-covariant differential of the plane wave is naturally understood as a resultant vector obtained by a finite transformation of e 4 = (0, 0, 0, 0, 1).
The invariant wave equations in the κ-Minkowski spacetime are defined in terms of the first or the second Casimir invariants in the background SO (1,4) symmetry. Our attention is that such a wave equation defines a non-local field theory having a similarity to Yukawa's domain theory, though the wave equation spoils four-dimensional Lorentz invariance. To secure the Lorentz invariance, in section 3, we studied a noncommutative spacetime associated with AdS n+1 type of background spacetime. In such a spacetime, there appears another time-like coordinate y n+1 in addition to y 0 , from which one can construct a κ-Minkowski like spacetime characterized by the non-commutativity [x n ,xμ] = iκ −1xμ and [xμ,xμ] = 0. In section 4, we put the wave equation in this spacetime by using the first SO(2, n + 1) Casimir invariant. Then, the wave equation is not invariant under the transformations betweenxμ andx n but is invariant under the Lorentz transformations among {xμ}. Further, by introducing a new scale parameter µ at the projective boundary of the AdS n+1 , the wave equation is reduced to a non-local field equation in commutative {xμ} spacetime, which is invariant under the Lorentz transformation.
In a resultant spacetime, we need not worry about the non-commutativity of spacetime variables; then, in section 5, we have discussed a local interaction of fields, which obeys non-local field equations characterized by a free field equation including a infinite higher derivative term such as sinh{(κµ) −1 ∂ 2 }. There, we tried to evaluate one-loop diagrams by assuming a φ 3 type of local interaction for those fields. At first, it is expected to get finite results for those diagrams, since the sinh term in the propagator plays a roll of strong dumping factor in the both spacelike and timelike regions of momentum square k 2 =kμkμ. However, the situation is not so simple, because the propagator contains complex poles of k 2 , which may spoil the unitarity of the interactions in Planck energy scale. The contribution of those poles, fortunately, again produces dumping factor to internal lines of loop diagrams: the more the number of internal lines increase, the more the dumping effect grows. Those effects are not trivial, and one can expect to get convergent results by the same mechanism in higher loop diagrams too.
We also note that the second scale parameter µ characterizing the resultant spacetime is introduced by hand without enough guiding principles. The wave equation, the first Casimir invariant, then, becomes a non-local field equation that resembles Yukawa's domain one to some points. One of the purposes of domain theory is to improve the divergent problem in local field theories. Therefore, the investigation of the meaning of µ in more detail will be an interesting future problem.
The result is a non-linear realization [8] of dS 4 vectorũ out of ω(a).
Substituting this result for (86), we arrive at The above results are nothing but (50) and (51).