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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?

Qmechanic
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Ishan Deo
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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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