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Consider spacetime to be homogeneous and isotropic. Then, the Green's function for the wave equation satisfies \begin{equation} \square G(x^{\mu}) = \delta^{n+1}(x^{\mu}).\tag{1} \end{equation}

In $3+1$ dimensions, this can be solved to get the retarded Green's function \begin{equation} G(x^{\mu}) = \frac{\delta(t-r)}{4\pi r},\tag{2} \end{equation} or, in a covariant form, \begin{equation} G(x^{\mu}) = \frac{1}{2\pi} \theta(t)\delta(x^{\mu}x_{\mu}).\tag{3} \end{equation}

Can we get a similar covariant expression in $n+1$ dimensions? If so, how do we go about doing it?

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Ishan Deo
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    Since this is link-only, it is nor an answer, but you might be interested in this paper. It does not make the calculation you wan, but it gives the general steps and shows how one can use the result in $d$ dimensions to obtain the result in $d+2$ dimensions. It should be mentioned that the expressions tend to be considerably different in $d\neq 3+1$ (and are particularly nasty in odd-dimensional spacetimes) – Níckolas Alves Mar 31 '22 at 17:22
  • Unfortunately, I was looking if there was an explicit covariant expression for the Green's function, while this paper both leaves it as a recurrence relation, and the covariance is not immediately obvious. – Ishan Deo Mar 31 '22 at 17:51

1 Answers1

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Well, there is a recursion formula $$G_{d+2}(r^2,t)~=~-\frac{1}{\pi} \frac{\partial G_d(r^2,t)}{\partial (r^2)}, \qquad d~\geq~2,\tag{29'} $$ which preserves the covariant structure, cf. Ref. 1. The first few read $$G_1(t)~=~\theta(t)t~=~\max(t,0),$$ $$G_2(r^2,t)~=~\frac{1}{2}\theta(t)\theta(t^2-r^2)\tag{31'}$$ $$G_3(r^2,t)~=~\frac{\theta(t)\theta(t^2-r^2)}{2\pi\sqrt{|t^2-r^2|}}+\text{sing. terms}\tag{31"} $$ So e.g. combining eqs. (29') & (31') leads to OP's covariant eq. (3).

In eq. (31") the singular terms [which appear in odd spacetime dimension $d$] have support on the light-cone $\{(\vec{r},t)\in\mathbb{R}^d | r^2=t^2\}$. This is related to failure of Huygens' principle, cf. e.g this Phys.SE post and this Math.SE post.

The singular terms require care to be mathematically well-defined. For more details, see e.g. my Math.SE answer here.

References:

  1. H. Soodak & M.S. Tiersten, Wakes and waves in $N$ dimensions, Am. J. Phys. 61 (1993) 395.
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