In 3 dimensions, the wave equation $$\Box\psi=\delta(t)\delta(\vec{x})$$ has the retarded and advanced solutions $$\psi=A_R \frac{\delta(t-x)}{4\pi x} + A_A \frac{\delta(t+x)}{4\pi x}.$$ How does this generalize to higher dimensions?

  • $\begingroup$ Are you asking what the differential equation is in higher dimensions? Or what the solutions are? $\endgroup$ – G. Smith Feb 28 at 16:48
  • $\begingroup$ Surely you can adapt this. $\endgroup$ – Cosmas Zachos Feb 28 at 17:02
  • $\begingroup$ ... or this. $\endgroup$ – Cosmas Zachos Feb 28 at 17:08
  • $\begingroup$ ...or THIS one. Can you fold it into your question? $\endgroup$ – Cosmas Zachos Feb 28 at 17:50
  • $\begingroup$ Umm, I'm not sure how to do "fold" it into my question. But let me check the link, if it's good then I guess the problem is resolved. $\endgroup$ – K. Sadri Feb 28 at 18:32

OK, by the OP's invitation, the two papers addressing his question are


The idea is to express the N-dimensional Laplacian in polar coordinates, $\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \left(r^{N-1} \frac{\partial }{\partial r} \right)$.

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