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The retarded Green's function $G_R(x,x^\prime)$ is usually defined by using the Wightman function $$W(x,x^\prime) = \langle0| \varphi(x) \varphi(x^\prime) |0\rangle\,,$$ by $$ G_R(x,x^\prime) = \Theta(x^0 - x^{\prime\,0}) W(x,x^\prime).$$ The defining condition of a Green's function, however, is that it fulfills $$ D_x G(x, x^\prime) = \delta(x - x^\prime)\,, $$ where $D_x$ is a differential operator. Moreover, the corresponding Wightman function fulfills $$ D_x W(x, x^\prime) = 0 \,, $$ What's the defining equation of a retarded Green's function? By using the equations given above, we can calculate \begin{align} D_x G_R(x, x^\prime)&= D_x\Theta(x^0 - x^{\prime\,0}) W(x,x^\prime) \\ &= \Big( D_x\Theta(x^0 - x^{\prime\,0})\Big) W(x,x^\prime) + \Theta(x^0 - x^{\prime\,0}) \Big( D_xW(x,x^\prime)\Big)\\ &= \delta(x^0 - x^{\prime\,0}) W(x,x^\prime) . \end{align} In a more general context, we replace the Wightman function by the kernel $K(x,x^\prime)$ of the corresponding equation: $$ D_x G_R(x, x^\prime)=\delta(x^0 - x^{\prime\,0}) K(x,x^\prime).$$ Is this the correct defining equation of a retarded Green's function? And if yes, how can we understand it?

jak
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  • I should expect that the equal time kernel should yield $\delta^{d-1}(x-x')$. – Jon Oct 28 '19 at 14:02
  • Ah yes. In the equal time limit the Kernel becomes the delta distribution. In addition, i believe the correct equation is given here https://physics.stackexchange.com/questions/123587/analytical-problems-with-greens-function – jak Oct 28 '19 at 16:01

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