Induced vacuum currents in anti-de Sitter space with toral dimensions

We investigate the Hadamard function and the vacuum expectation value of the current density for a charged massive scalar field on a slice of anti-de Sitter (AdS) space described in Poincar\'{e} coordinates with toroidally compact dimensions. Along compact dimensions periodicity conditions are imposed on the field with general phases. Moreover, the presence of a constant gauge field is assumed. The latter gives rise to Aharonov-Bohm-like effects on the vacuum currents. The current density along compact dimensions is a periodic function of the gauge field flux with the period equal to the flux quantum. It vanishes on the AdS boundary and, near the horizon, to the leading order, it is conformally related to the corresponding quantity in Minkowski bulk for a massless field. For large values of the length of the compact dimension compared with the AdS curvature radius, the vacuum current decays as power-law for both massless and massive fields. This behavior is essentially different from from the corresponding one in Minkowski background, where the currents for a massive field are suppressed exponentially.


Introduction
In quantum field theory on curved backgrounds, the anti-de Sitter (AdS) spacetime is interesting from several points of view. The corresponding metric tensor is a maximally symmetric solution of the Einstein equations in the presence of a negative cosmological constant and, because of the high degree of symmetry, numerous physical problems are exactly solvable on its background. These investigations may help to gain deeper understanding of the influence of gravity on quantum matter. The early interest to the dynamics of quantum fields in AdS spacetime was motivated by principal questions of the quantization on curved backgrounds. The presence of both regular and irregular modes and the possibility of getting an interesting causal structure, lead to a number of interesting features that have no analogs in Minkowski spacetime field theory. Being a constant negative curvature manifold, AdS space provides a convenient infrared regulator in interacting quantum field theories [1], cutting infrared divergences without reducing the symmetries. The importance of AdS spacetime as a gravitational background increased after the discovery that it generically arises as a ground state in extended supergravity and in string theories, and also as the near horizon geometry of the extremal black holes and branes.
More recent interest in this subject was generated in connection with two types of models where AdS geometry plays a crucial role. The first one, called AdS/CFT correspondence [2] (for a review see [3]), represents a realization of the holographic principle and provides a mapping between string theories or supergravity in the AdS bulk and a conformal field theory living on its boundary. In this mapping the solutions in AdS space play the role of classical sources for the correlators in the boundary field theory. The AdS/CFT correspondence has many interesting consequences and provides an opportunity to investigate non-perturbative field-theoretical effects at strong couplings. Among the recent developments of such a holography is the application to strong-coupling problems in condensed matter physics (holographic superconductors, quantum phase transitions, topological insulators). The second type of model with AdS spacetime as a background geometry is a realization of a braneworld scenario with large extra dimensions, and provides a solution to the hierarchy problem between the gravitational and electroweak mass scales (for reviews see [4]). Braneworlds naturally appear in string/M-theory context and give a novel setting for discussing phenomenological and cosmological issues related to extra dimensions, including the problem of the cosmological constant.
The presence of extra compact dimensions is an inherent feature of all these models. In quantum field theory, the boundary conditions imposed on the field operator along compactified dimensions lead to a number of interesting physical effects that include topological mass generation, instabilities in interacting field theories and symmetry breaking. These boundary conditions modify the spectrum of the zero-point fluctuations, as a result the vacuum energy density and the stresses are changed. This is the well-known topological Casimir effect. This effect has been investigated for large number of geometries and has important implications on all scales, from mesoscopic physics to cosmology (for reviews see [5]). The vacuum energy depends on the size of extra dimensions and this provides a stabilization mechanism for moduli fields in Kaluza-Klein-type models and in braneworld scenario. In particular, motivated by the problem of radion stabilization in Randall-Sundrum-type braneworlds, the investigations of the Casimir energy on AdS bulk have attracted a great deal of attention 1 . The Casimir effect in AdS spacetime with compact internal spaces has been considered in [7]. The vacuum energy generated by the compactification of extra dimensions can also serve as a model of dark energy needed for the explanation of the present accelerated expansion of the universe.
An important characteristic associated with charged fields is the vacuum expectation value (VEV) of the current density. Although the corresponding operator is local, because of the global nature of the quantum vacuum, this expectation value carries information about both the geometry and topology of the background space. Moreover, this VEV acts as the source in the semiclassical Maxwell equations and therefore plays an important role in modeling a self-consistent dynamics involving the electromagnetic field. The VEV of the current density for a fermionic field in flat spaces with toral dimensions has been investigated in [8]. Applications were given to the electrons of a graphene sheet rolled into cylindrical and toroidal shapes and described in terms of an effective Dirac theory in a two-dimensional space. Combined effects of the compactification and boundaries are discussed in [9]. The finite temperature effects on the current densities for scalar and fermionic fields in topologically nontrivial spaces have been studied in [10]. The VEV of the current density for charged scalar and Dirac spinor fields in de Sitter spacetime with toroidally compact spatial dimensions are considered in [11]. The effects of nontrivial topology induced by the compactification of a cosmic string along its axis have been discussed in [12].
In the present paper we investigate the VEV of the current density for a charged scalar field in a 1 See references in [6].
slice of AdS spacetime covered by Poincaré coordinates assuming that a part of spatial coordinates are compactified to a torus. In addition to the background gravitational field, we also assume the presence of a constant gauge field interacting with the field. Though the corresponding field strength vanishes, the nontrivial spatial topology gives rise Aharonov-Bohm-like effect on the current density along compact dimensions. This current is a source of magnetic fields in the uncompactified subspace, in particular, on the branes in braneworld scenario. The problem under consideration is also of separate interest as an example with gravitational and topological polarizations of the vacuum for charged fields, where one-loop calculations can be performed in closed form.. The outline of the paper is as follows. In the next section we describe the geometry of the problem and evaluate the Hadamard function for a charged massive scalar field obeying general quasiperiodicity conditions along compact dimensions. By using this function, in section 3, the VEV of the current density is investigated. The behavior of this VEV in various asymptotic regions of the parameters is discussed. The main results of the paper are summarized in section 4. The main steps for the transformation of the Hadamard function to its final expression are described in Appendix.

Hadamard function
As a background geometry we consider (D + 1)-dimensional AdS spacetime. In Poincaré coordinates the corresponding line element is expressed as where η ik = diag(1, −1, . . . , −1), i, k = 0, 1, . . . , D−1, is the metric tensor for D-dimensional Minkowski spacetime and −∞ < y < +∞. The parameter a is the AdS curvature radius and is related with the cosmological constant. Note that the Poincaré coordinates cover a part of the AdS manifold and there is a horizon corresponding to the hypersurface y = +∞. In what follows we assume that the coordinates x l , with l = p + 1, . . . , D − 1, are compactified to circles with the lengths L l , so 0 x l L l . For the coordinates x l , with l = 1, 2, . . . , p, one has −∞ < x l < +∞. Hence, the subspace perpendicular to the y-axis has a topology R p × (S 1 ) q , where q + p = D − 1. The coordinates in uncompactified and compactified subspaces will be denoted by x p = (x 1 , . . . , x p ) and x q = (x p+1 , . . . , x D−1 ), respectively. Introducing a new coordinate z = ae y/a , 0 z < ∞, the line element is presented in a conformally-flat form The hypersurfaces identified by z = 0 and z = ∞ correspond to the AdS boundary and horizon, respectively. In figure 1 we have displayed the spatial geometry corresponding to (1) for D = 2 embedded into the 3-dimensional Euclidean space. The compact dimension corresponds to the circles. We have also displayed the flux of the gauge field strength which threads the compact dimension (see below). Note that for a given z, the proper length of the lth compact dimension is given by L (p)l = aL l /z and it decreases with increasing z.
In this paper we are interested in the evaluation of the VEV of the current density associated with a massive charged scalar field, ϕ(x), in the presence of an external classical gauge field, A µ . In (4), D µ = ∇ µ + ieA µ , with ∇ µ being the standard covariant derivative. The corresponding equation of motion is given by where ξ is the curvature coupling parameter. The most important special cases correspond to minimally and conformally coupled fields with the parameters ξ = 0 and ξ = (D −1)/(4D), respectively. In the geometry under consideration for the scalar curvature one has R = −D(D + 1)/a 2 . In models with nontrivial topology one need also to specify the periodicity conditions obeyed by the field operator along compact dimensions. Here we consider quasiperiodicity conditions with constant phases α l and with e l being the unit vector along the lth compact dimension, l = p + 1, . . . , D − 1. The special cases of untwisted and twisted fields correspond to α l = 0 and α l = π, respectively.
In what follows we assume that the gauge field is constant, A µ = const. Though the corresponding field strength vanishes, the nontrivial topology of the background space gives rise to Aharaonov-Bohm-like effects on the VEVs of physical observables. By a gauge transformation, the problem with a constant gauge field may be reduced to the problem in the absence of the gauge field with the shifted phases in the periodicity conditions (6). Indeed, let us consider two sets of the fields (ϕ, A µ ) and (ϕ ′ , A ′ µ ) related by the gauge transformation ϕ = ϕ ′ e −ieχ , A µ = A ′ µ + ∂ µ χ. Choosing χ = A µ x µ , we see that in the new gauge the vector potential vanishes, whereas the field operator obeys the periodicity condition with the new phases In this expression, A l L l = dx l A l is the flux of the field strength which threads the lth compact dimension. Note that in models with compact extra dimensions the gauge field fluxes generate a potential for the moduli fields providing a stabilization mechanism for them (for a review see [13]). In the evaluation of the VEVs we can use both the sets (ϕ, A µ ) and (ϕ ′ , A ′ µ ). In the following, the evaluation procedure for the VEV of the current density is based on the set (ϕ ′ , A ′ µ ). The field obeys the equation (5) and the current density is given by (4) with D µ = ∇ µ in both the formulas. For simplicity of the notations we shall omit the prime in the discussion below.
The VEV of the current density can be expressed in terms of the Hadamard function being |0 the vacuum state. Here we consider the vacuum corresponding to the Poincaré coordinate system, the so called Poincaré vacuum. For the VEV of the current density one gets Let us denote by {ϕ σ (x)} the complete set of normalized positive-and negative-energy solutions to the field equation obeying the periodicity condition (7). The set of quantum numbers σ that label the mode functions will be specified below. Expanding the field operator in terms of the complete set we obtain the following expression for the Hadamard function For the problem under consideration, we can choose the mode function in the factorized form where k p = (k 1 , . . . , k p ) and k q = (k p+1 , . . . , k D−1 ). For the components of the wave vector along uncompactified dimensions one has −∞ < k l < +∞, l = 1, . . . , p, and for the components along compact dimensions from the condition (7) we find with n l = 0, ±1, ±2, . . .. Substituting (12) into the field equation we obtain the equation for the function f (z). The solution of the latter is presented in the form where Z ν (x) is a cylinder function of the order and Now, the set of quantum numbers is specified as σ = (k p , n q , λ), with n q = (n p+1 , . . . , n D−1 ). For imaginary values of ν AdS spacetime is unstable [14] and in what follows we assume that the parameter ν is real. The factor C in (14) is determined from the normalization condition where δ σσ ′ is understood as Kronecker delta for discrete components of σ and as the Dirac delta function for the continuous ones. For ν 1, from the normalizability of the mode functions it follows that one should take Z ν (λz) = J ν (λz) being J ν (x) the Bessel function of the first kind. For the solution with the Neumann function Y ν (x), the normalization integral in (17) diverges on the AdS boundary z = 0. Note that in AdS/CFT correspondence normalizable and non-normalizable modes are dual to states and sources, respectively. For 0 ν < 1, in order to uniquely define the mode functions for the quantization procedure, it is necessary to specify boundary conditions at the AdS boundary [14,15]. The Dirichlet and Neumann boundary conditions are the most frequently used ones. The general class of allowed boundary conditions of the Robin type, has been specified in [16] on the base of general analysis in [17]. In what follows, for the modes with 0 ν < 1, we shall choose Z ν (λz) = J ν (λz) that corresponds to Dirichlet condition on the AdS boundary. In this case, from (17) for the coefficient in (14) one gets where V q = L p+1 · · · L D−1 is the volume of the compact subspace. Substituting the mode functions into (11), for the Hadamard function we obtain where By using the transformations described in appendix, this function is presented in the final form where we have defined with ∆z = z − z ′ . In (20), we have introduced the function being Q µ α (x) the associated Legendre function of the second kind and F (a, b; c; u) is the hypergeometric function. By using the recurrence relations for the associated Legendre functions we can obtain the following relation Note that in (20) the term with n q = 0 corresponds to the Hadamard function in AdS spacetime with trivial topology of the subspace perpendicular to the y-axis. The remaining terms are induced by the compactification. The quantity v nq with n q = 0 is related to the invariant distance d( For a conformally coupled massless field one has ν = 1/2 and the hypergeometric function in (20) is expressed as For the Hadamard function this gives where G with the notation (∆x It can be shown that the function (26) coincides with the Hadamard function for a massless scalar field in (D + 1)-dimensional Minkowski spacitme with uncompactified dimensions (x 1 , . . . , x p , z) and compactified dimensions (x p+1 , . . . , x D−1 ), in the presence of a planar boundary at z = 0 on which the field obeys Dirichlet boundary condition. The formula (25) is the standard conformal relation between the conformally connected problems [18]. For a conformally coupled massless field the problem at hand is conformally related to the corresponding problem in Minkowski spacetime with a planar boundary. The presence of the latter is a consequence of the boundary condition we have imposed on the AdS boundary. Note that the part in the function G M (x, x ′ ) with the first term in the square brackets of (26), is the Hadamard function in boundary-free Minkowski spacetime with toroidally compactified dimensions. The remaining part is induced by the boundary.

Vacuum current density
With the Hadamard function from (20), the VEV of the current density is evaluated by making use of formula (10). The charge density and the components along uncompactified dimensions vanish. For the component along the lth compact dimension, by using the relation (23), one gets where n q−1 = (n p+1 , . . . , n l−1 , n l+1 , . . . , n D−1 ), and the summation goes over all values of −∞ < n i < +∞, i = l. Note that in (28) instead of cos(α q−1 · n q−1 ) we can also write D−1 i=1, =l cos(α i n i ). As is seen, the current density along the lth compact dimension is an odd periodic function of the phaseα l and an even periodic function of the phasesα i , i = l. In both cases the period is equal to 2π. In particular, the current density is a periodic function of the magnetic fluxes with the period equal to the flux quantum 2π/|e|. In the absence of the gauge field, the current density along the lth compact dimension vanishes for untwisted and twisted fields along that direction. Of course, the latter is a direct consequence of the problem symmetry under the reflection x l → −x l in these special cases.
The charge flux through the (D − 1)-dimensional spatial hypersurface x l = const is determined by the quantity n l j l , where n l = a/z is the corresponding normal. From (28) we see that this quantity depends on the coordinate lengths of the compact dimensions L i in the form of the ratio L i /z. For a given z, the latter is the proper length of the compact dimension, L (p)i = aL i /z, measured in units of the AdS curvature radius a. This property is a consequence of the maximal symmetry of AdS spacetime. Because j z = 0 and the VEV (28) depends on the coordinate z only, the covariant conservation equation ∇ l j l = 0 is satisfied trivially.
The function q µ α (x) in (28) is defined by (22). By using the relation (23), it can also be expressed in the form for D = 2n, n = 1, 2, . . ., and in the form for D = 2n − 1. In what follows we shall need the asymptotic expressions in different limiting cases. For x ≫ 1 one has If x is close to 1, x − 1 ≪ 1, by using the asymptotic formula for the hypergeometric function, we get Finally, for large values of the order, ν ≫ 1, one has An alternative expression for the VEV of the current density is obtained by using the integral representation (57) given in the appendix for the Hadamard function: This representation is useful in the discussion of limiting cases. For a conformally coupled massless field one has ν = 1/2 and for the current density we get where n l sin(α l n l ) n q−1 cos(α q−1 · n q−1 ) is the current density for a massless scalar field in Minkowski spacetime with toroidally compactified dimensions, in the presence of Dirichlet boundary at z = 0. The part with the first term in figure braces corresponds to the VEV in the boundary-free Minkowski spacetime. It is obtained from the general result of [10] in the zero mass limit. Now we turn to the investigation of the current density in various asymptotic regions of the parameters. First let us consider the Minkowskian limit corresponding to a → ∞ for a fixed value of the coordinate y. In this limit ν ≈ ma ≫ 1 and for the coordinate z one has z ≈ a + y. Introducing in (35) a new integration variable x = νu we use the uniform asymptotic expansion for the function I ν (νu) for large values of the order [19]. The dominant contribution to the integral comes from large values of u and, after the integration, to the leading order we get n l sin(α l n l ) being K (D+1)/2 (x) the MacDonald function. The expression in the right-hand side coincides with the corresponding result for a scalar field in Minkowski spacetime with toroidally compactified dimensions obtained in [10].
If the proper length of the one of the compact dimensions, say x i , i = l, is large compared with the AdS curvature radius, L i /z ≫ 1, the dominant contribution into (28) comes from the n i = 0 term and the contribution of the remaining terms is suppressed by the factor (z/L i ) D+2ν+2 . To the leading order, we obtain the current density for the topology R p+1 ×(S 1 ) q−1 with the uncompactified direction x i . In the opposite limit corresponding to small proper length of the dimension x i , L i /z ≪ 1, the dominant contribution comes from large values of n i and for the corresponding series in (35) we use the relation The behavior of the current density depends crucially on whether the phaseα i is zero or not. Forα i = 0, to the leading order, for the combination (aL i /z) j l we obtain the expression which coincides with the formula for j l in D-dimensional AdS spacetime, obtained from the geometry under consideration by excluding the dimension x i , with the replacement ν(D − 1) → ν(D), where the function ν(D) is defined by (15). Forα i = 0, after the substitution (39), the dominant contribution in the integral of (35) comes from the values x z/L i . Replacing the modified Bessel function by its asymptotic expression for large values of the argument, we can see that the dominant contribution comes from the term n l = 1, n r = 0, r = i, l, and to the leading order one gets In this case the current density is exponentially small. Now let us consider the behavior of the current j l for large and small values of L l . For large values of the proper length compared with the AdS curvature radius, L l /z ≫ 1, the argument of the function q (D+1)/2 ν−1/2 (x) in (28) is large and by using (32) in the leading order we find n l sin(α l n l ) The dominant contribution comes from large values of n i , i = l, and we replace the summation n q−1 by the integration. Forα i = 0, i = p + 1, . . . , D − 1, i = l, to the leading order one finds with the power-law decay as a function of L l for both massless and massive fields. In this sense, the situation for the AdS bulk is essentially different from that in the corresponding problem for Minkowski background. For the latter, in the massless case and for large values of L l the current density decays as 1/L p l , whereas for a massive field the current is suppressed exponentially, by the factor e −mL l . This shows that the influence of the background gravitational field on the VEV is crucial. If L l /z ≫ 1 and at least one of the phasesα i , i = l, is not equal to zero, the dominant contribution comes from the n l = 1 term and one gets where β q−1 = ( D−1 i=p+1, =lα 2 i /L 2 i ) 1/2 . In this case the current density, as a function of L l , decays exponentially.
For small values of L l , L l /z ≪ 1, the dominant contribution to the current density comes from the term with n q−1 = 0 and to the leading order we obtain In this limit, the contribution of the terms with n q−1 = 0 is suppressed by the factor e −σ l |L·n q−1 |/L l , where σ l = min(α l , 2π −α l ), 0 <α l < 2π. The expression in the right-hand side of (44), multiplied by (a/z) D+1 , coincides with the VEV of the current density for a massless scalar field in (D + 1)dimensional Minkowski spacetime compactified along the direction x l to the circle with the length L l . Now let us investigate the behavior of the current density near the AdS boundary and near the horizon for fixed values of the lengths L i . Near the boundary, z → 0, the argument of the function q (D+1)/2 ν−1/2 (x) in (28) is large and to the leading order we find Hence, the current density vanishes on the boundary. Near the horizon one has z → ∞, the argument of the function q (D+1)/2 ν−1/2 (x) is close to 1 and we use the asymptotic expression (33). This gives, where n l sin(α l n l ) is the corresponding current density in Minkowski spacetime for a massless scalar field. And finally, for large values of the mass, ma ≫ 1, by using the asymptotic expression (34) we see that the dominant contribution in (28) comes from the term with n i = 0, i = l, and n l = 1. To the leading order we get with u = L l /z. As we could expect, for large values of the mass we have an exponential suppression. The suppression is stronger near the AdS boundary.
In the special case of a single compact dimension (dimension x D−1 ) with the length L D−1 = L from the general formula one finds bulk is given by the right-hand side of (38), specified to the special case under consideration. The graphs are plotted forα = π/2 and the numbers near the curves are the corresponding values of the parameter ma (mass measured in units of the AdS energy scale). As before the full and dashed curves correspond to minimally and conformally coupled fields. Note that in the case of the AdS bulk one has L (p) = aL/z, with L being the coordinate length. In the Minkowski spacetime with compact dimension the proper and coordinate lengths coincide. From figure 3 we see the feature already described before on the base of the asymptotic analysis: for a massive field and for large values of the proper length the decay of the current density in the Minkowski bulk is stronger than that for AdS background.

Conclusion
In the present paper we have investigated combined effects of geometry and topology on the VEV of the current density for a charged scalar field. In order to have an exactly solvable problem we have taken highly symmetric background geometry corresponding to a slice of AdS spacetime with a toroidally compactified subspace. Currently, the AdS spacetime is among the most popular backgrounds in gravitational physics and appears in a number of contexts. The information on the vacuum fluctuations is encoded in two-point functions and, as the first step, we have evaluated the Hadamard function for general values of the phases in the periodicity conditions obeyed by the field operator along compact dimensions. Additionally, the presence of a constant gauge field is assumed. The latter may be excluded by a gauge transformation that leads to the shift in the phases of periodicity conditions. The shift is expressed in terms of the ratio of the magnetic flux enclosed by compact dimension to the flux quantum. The Hadamard function is expressed in the form (20), where the part corresponding to AdS geometry without compactification is explicitly separated and is presented by the term with n q = 0. Because the toroidal compactification does not change the local geometry, in this way the renormalization of the VEVs for physical quantities bilinear in the field is reduced to the one for the uncompactified AdS spacetime.
With a given Hadamard function, the VEV of the current density is evaluated by using the formula (10). The vacuum charge density and the components of the current along uncompactified dimensions vanish and the current density along lth compact dimension is given by the expression (28). The latter is an odd periodic function of the phaseα l and an even periodic function of the phasesα i , i = l. In particular, the current density is a periodic function of the magnetic fluxes with the period equal to the flux quantum. For a conformally coupled massless field the current density is conformally related to the corresponding quantity in (D + 1)-dimensional Minkowski spacitme with toral dimensions, in the presence of a planar boundary on which the field obeys Dirichlet boundary condition. The appearance of the latter is a consequence of the boundary condition we have imposed on the AdS boundary. As an additional check, we have shown that in the Minkowskian limit, corresponding to a → ∞ for a fixed value of the coordinate y, the result obtained in [12] is recovered.
In order to clarify the dependence of the current density on the values of the parameters characterizing the geometry and the topology, we have considered various limiting cases. Near the AdS boundary, the VEV of the current density behaves as z D+2ν+2 and, hence, it vanishes on the AdS boundary. For fixed values of the coordinate lengths of compact dimensions, this region corresponds to large values of the proper lengths. Near the horizon, corresponding to z = ∞, the VEV of the current density is related to the current density in Minkowski spacetime with toral dimensions by the relation (46) and behaves as z D+1 . It is also of interest to consider the behavior for large and small proper lengths of compact dimensions for a fixed value of z. If the proper length of the ith compact dimension is small, then the behavior of the current density along the lth dimension depends crucially on whether the phaseα i is zero or not. For the zero value of this phase, in the leading order, the expression for the quantity L (p)i j l coincides with the formula for j l in D-dimensional AdS spacetime, obtained from the geometry at hand by excluding the dimension x i and replacing ν(D − 1) → ν(D), where the function ν(D) is defined by (15). Forα i = 0 the current density j l is suppressed by the factor e −L l σ i /L i , where σ i is defined in the paragraph after formula (39). For large values of the proper length of the lth dimension, compared with the AdS curvature radius, and for zero values of the phases along the remaining directions, the decay of the VEV j l , as a function of L l is power-law, as 1/L p+2ν+2 l for both massless and massive fields. For massive fields this behavior is crucially different from that for the Minkowski bulk where the decay is exponential. If at least one of the phasesα i , i = l, is not equal to zero, the suppression of the current density is exponential, given by (43). For small values of the proper length of the lth dimension, in the leading order, the VEV j l does not depend on the phases along other directions and the current density behaves as 1/L D+1 l . Figure 4: The contour of the integration in the representation (51).
For the further transformation we note that in (54) the series corresponding to the lth compact dimension has the form +∞ n l =−∞ e ik l ∆x l −s 2 k 2 l . From the Poisson resummation formula it directly follows that wheref (y) = +∞ −∞ dx e −iyx f (x). From this formula one gets +∞ n l =−∞ e ik l ∆x l −sk 2 l = L l 2 √ πs +∞ n l =−∞ e in lαl e −(∆x l −L l n l ) 2 /(4s) .
With this transformation, the Hadamard function is expressed as whereα = (α p+1 , . . . ,α D−1 ), and v nq is defined by (21). The integral in (57) is expressed in terms of the hypergeometric function and we obtain the representation (20).