What is the solution to this equation in d dimensions: (−∇2d+m2)G(x,x′)=Aδ(x−x′), with the boundary condition that lim|x−x′|→∞G(x,x′)=0?
Answer
The first step is to recognize that equation is invariant under d-dimensional rotations around \mathbf{x} - \mathbf{x}' = \mathbf{0} and simultaneous identical translations of \mathbf{x} and \mathbf{x}', so we can make the following step: \begin{align}(-\nabla_d^2 + m^2)G(\mathbf{x}, \mathbf{x}') &= A \delta(\mathbf{x} - \mathbf{x}') \\ & \mathrm{let: }\ r \equiv |\mathbf{x} - \mathbf{x}'| \Rightarrow \\ \frac{A}{\Omega_d r^{d-1}} \delta(r) &= - \frac{1}{r^{d-1}} \frac{\partial}{\partial r} \left[r^{d-1} \frac{\partial G(r)}{\partial r}\right] + m^2 G(r),\end{align} where the factor \Omega_d r^{d-1} comes from the volume element \operatorname{d} V = \Omega_d r^{d-1} \operatorname{d}r and the derivatives on the right hand side (rhs) are the radial term of \nabla_d^2 in d-dimensional spherical coordinates (wiki link).
The next step is to integrate both sides of the equation over a spherical volume centered at the origin with radius r, then take the limit as r \rightarrow 0. This yields the normalization condition for G: \lim_{r\rightarrow 0} \left[r^{d-1} \frac{\partial G}{\partial r}\right] = -\frac{A}{\Omega_d}, and handles the part of the equation where the delta function is non-zero.
The region where the delta function is zero, the homogeneous region, becomes: 0 = \frac{\partial^2 G}{\partial r^2} + \frac{d-1}{r} \frac{\partial G}{\partial r} - m^2 G. The equation in the homogeneous region can be brought into a more familiar form by the function substitution G(r) = f(r) r^{-(d/2 - 1)} giving: 0 = r^2 \frac{\partial^2 f}{\partial r^2} + r \frac{\partial f}{\partial r} - \left(\frac{d}{2} - 1\right)^2 f - m^2 r^2 f. The familiar form to this equation is the modified Bessel's equation. The most general solution to this equation is: f(r) = C K_{d/2-1}(mr) + D I_{d/2-1}(mr), with I_{d/2-1} and K_{d/2-1} modified Bessel functions of the first and second kind, respectively, and C and D constants fixed by the boundary conditions.
The boundary condition at r\rightarrow \infty requires D=0, giving the following form for G: G(r) = \frac{C}{r^{d/2-1}} K_{d/2-1}(mr). Plugging our solution for G into the left hand side (lhs) of the r \rightarrow 0 boundary condition derived above gives: \lim_{r\rightarrow0} \left[ r^{d-1} \frac{\partial G}{\partial r}\right] = -\Gamma\left(\frac{d}{2}\right) 2^{d/2-1} m^{1-d/2} C, after application of the small argument limit form of K_\nu. This implies that: \begin{align}C &= \frac{A m}{2^{d/2-1}\Gamma(d/2) \Omega_d} \\ & = \frac{A m^{d/2-1}}{2^{d/2} \pi^{d/2}},\end{align} where the explicit form of \Omega_d = S_{d-1} has been inserted.
Finally, replacing C gives: G(r) = \frac{A}{(2\pi)^{d/2}} \left(\frac{m}{r}\right)^{d/2-1} K_{d/2-1}(mr).
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