$$=\int {d^3 \vector{P} \over {(2\pi)^3 E}}
\ \exp{[-\phat\ p^{\mu}(x_{\mu}-x^{\prime}_{\mu})]}
\ {{\bf p}\ \basis{4}}, \eqno(4.4b)$$ $$= -(\Box +m)\ \Delta ({\bf x - x^{\prime}})\ \basis{4}, \eqno(4.4c)$$ $$\Delta({\bf x})=
\int {d^3\vector{P}\over {(2\pi)^3 E}} \ \sin{(p^{\mu}x_{\mu})}. \eqno(4.4d)$$ From eq. (4.4a) it is clear that: $F^\dagger({\bf x,y}) = F({\bf y,x})$. The function $\Delta({\bf x})$ is given by Bjorken[@Drell65] to be related to a regular Bessel function inside the light cone. Clearly it is a solution everywhere to the Klein Gordon eq. (3.1), and so it follows from eq. (4.4c) that the propagator kernel is [*left meta-monogenic*]{}, $$(\Box - m)\ F({\bf x,x^{\prime}})=0, \eqno(4.5a)$$ $$(\Box + m)\ F({\bf x^{\prime},x})=0, \eqno(4.5b)$$ where $\Box$ operates on ${\bf x}$, but not on ${\bf x^{\prime}}$. What we have derived is a special case of the [*Cauchy kernel*]{}. The usual closed boundry integral reduces to eq. (4.3a) when one assumes that $\Psi(\vector{X},t)\rightarrow 0\ {\rm for:}
\ \nnn{X}\rightarrow \infty\ {\rm or}\ t\rightarrow \infty$. In quantum theory, one sidesteps the unmeasurable phase by assuming an ‘initial’ solution at $t^{\prime}\rightarrow -\infty$ which is a coherent plane wave. This approximates for example the collimated beam of electrons shooting down the linear accelerator at Stanford.

Green Function
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Another common situation is to know the function over 