I am trying to derive retarded potentials or retarded Green's function for d'Alembert eq. in (2+1)-dimensional space-time. In the literature I encounter two distinct approaches, one "mathematical", based on Fourier analysis and one "physical", in which retardation is introduced straightforwardly. The earlier is found in J.D. Jackson's Classical Electrodynamics (Wiley, 1962, Sect 6.6, pp. 183-186) and is dimension-independent, hence, gives one and the same form of the Green's function for all dimensions that is impossible, and the latter is rather speculative and gives product of delta-function which provides the retardation, and logarithm which is purely spatial Green's function.
Have you checked "Classical Dynamics" from Greenwood. I think that you can derive it from them
I believe you confuse the Fourier domain and the space-time domain. If you Fourier transform the wave equation in both space and time, you will indeed obtain, quite correctly, a dimension independent expression for the spacetime Fourier transform of the Green function, namely G(k, omega).
But in order to obtain G(x, t), you need to perform an inverse Fourier transform, which gets the dimension dependence back in: the point is, you have
G(x, t) = int G(k, omega) exp(-i k dot x) exp(i omega t) d^d vector x d omega
and the volume element will provide, when going to polar coordinates, a factor r^(d-1) which gives you all the dimension dependence you need.
Thank you. Now, will you please show me what retarded Green's function in 2+1 dimensions looks like?
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