Green Function of the Poisson Equation: D=2,3,4

We study the Green function of the Poisson equation in two, three and four dimensions. The solution g of the equation nabla^2 g(x - x') = delta^(D)(x - x'), where x and x are D-dimensional position vectors, is customarily expanded into radial and angular coordinates. For the two-dimensional case (D=2), we find a subtle interplay of the necessarily introduced scale L with the radial component of zero magnetic quantum number. For D=3, the well-known expressions are briefly recalled; this is done in order to highlight the analogy with the four-dimensional case, where we uncover analogies of the four-dimensional spherical harmonics with the familiar three-dimensional case. Remarks on the SO(4) symmetry of the hydrogen atom complete the investigations.


I. INTRODUCTION
Solutions of the equation enter a myriad of physical problems, from the elementary Coulomb problem in electrostatics (D = 3), to the attraction among vortices in two-dimensional systems (D = 2), and on to the four-dimensional formulation of the hydrogen Green function (D = 4, see Ref. [1]). Here, we shall attempt to provide a unified treatment of the radial and angular decompositions of the two-, three-and four-dimensional Green functions, which are solutions to Eq. (1). In D = 2, a scale has to be introduced, which corresponds to a physically irrelevant overall constant term, while in D = 3, the formulas are very familiar (see Refs. [2,3]). In D = 4, we attempt to reveal a structure of (re-)defined associated ultraspherical polynomials (Gegenbauer polynomials), which highlights analogies to the associated Legendre functions that enter the case D = 3.

II. TWO-DIMENSIONAL CASE
Spherical coordinates in two dimensions have a cylindrical symmetry; hence, for definiteness, we denote the twodimensional position vectors as ρ and ρ ′ , and their moduli as ρ = | ρ| and ρ ′ = | ρ ′ |. The Green function solution of the Poisson equation, introduces a scale L, which ensures that the argument of the natural logarithm is dimensionless. In terms of the Green function, the scale L adds nothing but an overall constant term, In order to show that g fulfills the Poisson equation, one specializes the divergence theorem to an infinitesimal area A. For example, A might be chosen as the inner area of a circle of infinitesimal radius ǫ, about the center ρ ′ . With ∂A denoting the boundary of A, i.e, the circle of radius ǫ about ρ ′ , one must have Using the formula (2) and the radial component of the gradient operator in two-dimensional coordinates, one verifies that indeed, Let us now turn to the angular-momentum decomposition of Eq. (2). The spherical representation of the twodimensional Dirac-δ is An appropriate ansatz for the Green function is The two-dimensional representation of the Laplacian is It acts on the Green function as follows, Now, one multiplies both sides with the factor and integrates over dϕ, resulting in the equation Setting and renaming m ′ → m after this operation, one obtains the radial equation Inspired by textbook treatments [2,3] of the three-dimensional Green function, one uses the following ansatz for nonzero m, where ρ < = min(ρ, ρ ′ ) and ρ > = max(ρ, ρ ′ ), and integrates Eq.
This results in the relation and amounts to the condition with the result .
The case m = 0 requires special treatment. One sets because this term matches the asymptotic limit of Eq. (2) for ρ → ∞, ρ ′ → 0. In this case, Eq. (15) translates into the condition with the result D = 1/(2π). Adding the terms for m = 0 and m = 0, one has A numerical check of this relations is successful. For ρ = 0.2ê x + 0.1ê y and ρ ′ = 1.1ê x + 1.5ê y , and L = 10.7 , the expression in Eq. (2) evaluates to while the m = 0 term from Eq. (20) is Adding the sum over the nonzero m, one obtains We have checked the equality T 1 = T 2 + T 3 for a number of example cases. It is interesting to note that Eq. (20) does not seem to have appeared in the literature before.

III. THREE-DIMENSIONAL CASE
Let r and r ′ denote coordinate vectors in three-dimensional space. It is well known that fulfills the Poisson equation The well-known expansion into (three-dimensional) spherical harmonics reads as follows, where r < = min(r, r ′ ), r > = max(r, r ′ ). The Laplacian in three dimensions reads as where L = −i r × ∇. The radial part of the Green function (26) is assembled from homogeneous solutions of the radial equation, in much the same way as in the derivation extending from Eq. (13) to Eq. (17). The transformation from Cartesian to spherical coordinates is, with r = 3 i=1 x iêi , The infinitesimal solid angle element is The well-known spherical harmonics are given as with the orthonormality and completeness properties The summation limits are ℓ = 0, . . . , ∞ and m = −ℓ, . . . , ℓ. The generating function for the Legendre polynomials [4] is while the associated Legendre polynomials are given by They have the property These formulas are recalled with the notion of clarifying the analogies with the four-dimensional case, as will be done in the following.
The summation limits are n = 0, . . . , ∞, ℓ = 0, . . . , n, and m = −ℓ, . . . , ℓ. The four-dimensional Laplacian is and the radial part of the decomposition (37) is assembled from homogeneous solutions of the radial component of the four-dimensional Laplacian. The transformation from Cartesian to spherical coordinates is, with x 2 = r sin ϕ sin θ sin χ, (39b) The infinitesimal solid angle element is The four-dimensional spherical harmonics can be given in terms of the analogue of Eq. (30) as with the orthonormality and completeness properties The generating function for the Gegenbauer-type Q polynomials is an analogue of Eq. (32), The associated Gegenbauer-type polynomials can be defined in complete analogy with Eq. (33), They have a property analogous to Eq. (34), Steps toward a unified treatment of the four-dimensional spherical harmonics were made in Ref. [5], but it appears that the normalization prefactor in Eq. (41) was not given in explicit form. The connection to the usual associated Gegenbauer polynomials C ℓ n (x) (in the canonical form, see Ref. [4]) is found as Finally, we should mention the addition theorem Connections of these formulas to the hydrogen wave functions are discussed in the Appendix.

V. CONCLUSIONS
The most important formulas of this brief paper can be found in Eqs. (20), (26) and (37): We derive [and in the case of Eq. (26), just recall] the decomposition of the two-, three-and four-dimensional Green functions of the Poisson equation into radial and angular parts. For D = 2, only one "quantum number" is introduced, namely, the "magnetic" (azimuthal) quantum number m; for D = 3, one has the orbital angular momentum ℓ and its magnetic projection m, while in D = 4, a third additional quantum number has to be introduced which can be associated with a "principal" quantum number n; it is associated with the additional angular coordinate χ in four dimensions [see Eq. (39)]. The latter interpretation is ramified by the fact that indeed, the momentum-space wave functions of the nonrelativistic hydrogen atom (for nuclear charge number Z = 1) can be written as [cf. p. 39 of Ref. [6]] where and θ and ϕ are the polar and azimuth angles of the unit vector in the momentum direction, i.e., in the direction of the unit vectorp = p/| p|. The Bohr radius in a 0 = /(αm e c), where α is the fine-structure constant, m e is the electron mass, and c is the speed of light. These wave functions are normalized as (2π) −3 d 3 p |ψ nℓm ( p)| 2 = 1. In our treatment of the four-dimensional Green function, we find it useful [see Eqs. (43) and (44)] to define polynomials Q n (x), and associated function Q ℓ n (x), which are related to, but not equal to, the Gegenbauer, and associated Gegenbauer, polynomials [4]. Hence, we refer to them as "Gegenbauer-type" functions. Analogies to the threedimensional case (Legendre and associated Legendre functions) are highlighted. The most intriguing problem in the calculation of the two-dimensional Green function lies in the matching of the m = 0 term from Eq. (18) with the m = 0 term from Eq. (13); the consideration of the asymptotic limit ρ > → ∞ helps in finding the matching coefficients [see Eq. (23)].
The angular-momentum decomposition (20) for D = 2 reveals that the dominant logarithmic term in the interaction of vortices in the two-dimensional sine-Gordon model is exclusively due to S-wave interactions. The result might become useful as one tries to augment previous studies on high-T c Josephson-coupled, and magnetically coupled superconductors [7,8] by the inclusion of higher-order derivative terms [9].
with an "asymmetric" distribution of the factors 2π, are almost universally adopted in the physical literature. With = c = ǫ 0 = 1, the Coulomb potential, in momentum space, is V ( p − p ′ ) = − 4πZα ( p− p ′ ) 2 . The defining equation for the Green function thus becomes, in momentum space, The factor (2π) 3 is not present in the first (unnumbered) equation of Ref. [1]. The fifth (unnumbered) equation of Ref. [1] contains two nontrivial identities. It is useful to derive the equation |ξ0| for the area element on the three-dimensional unit sphere, embedded in four-dimensional space. Here, one should remember that the three-dimensional components of the four-dimensional vector (ξ 0 , ξ) may have varying magnitude, but one considers them, according to Ref. [1], on the four-dimensional unit sphere ξ 2 0 + ξ 2 = 1. One needs to remember that the appropriate generalization to the three-dimensional "surface" of a manifold embedded into four-dimensional space is, with x = x(t 1 , t 2 , t 3 ), y = y(t 1 , t 2 , t 3 ), z = z(t 1 , t 2 , t 3 ), a = a(t 1 , t 2 , t 3 ) being the fourth coordinate, where a is the fourth coordinate, One calculates first the four-dimensional vector described by the determinant, and then calculates its vector modulus. The three-dimensional unit sphere, embedded in four-dimensional space, can be interpreted as a three-dimensional manifold, parameterized by the coordinates x = ξ x = t 1 , y = ξ y = t 2 , and z = ξ z = t 3 , while a = ξ a = 1 − ξ 2 x − ξ 2 y − ξ 2 z . One finds that This finally shows the first identity in the fifth equation of Ref. [1]. In order to show the second identity, one has to calculate a further Jacobian, transforming d 3 ξ into d 3 p (in the conventions of Ref. [1], keeping the zeroth (or fourth) component X = p 0 of the four-dimensional (Euclidean) momentum constant. There is a second nontrivial point which we found to be not very well explained in Ref. [1], and it concerns the fourth unnumbered equation (from the bottom) on the second page of Ref. [1]. The "version of the expansion" referred to in Ref. [1] necessitates the use of the following trick, which is to enter the angular-momentum expansion formula given for D(ξ − ξ ′ ) = −g(ξ − ξ ′ ) with the following values for ξ = r 1 , and ξ ′ = r 2 , as follows, with 0 < ρ < 1, so that r 1 = ρ = r < , and r 2 = 1 = r > . The identity then follows, leading to which is the desired identity used in Ref. [1]. We note that a representation of the Y nℓm (χ, θ, ϕ) in terms of elementary functions is not given in Ref. [1].