$$\Psi(\tpo{X}{t}) =
\int d^3\vector{X}^{\prime}\ F(\tpo{X}{t};\tpo{X^{\prime}}{t^{\prime}})
\ \Psi(\tpo{X^{\prime}}{t^{\prime}}),\eqno(4.3a)$$ $$F(\tpo{X}{t};\tpo{X^{\prime}}{t^{\prime}})=
U(t,t^{\prime})\ \delta(\vector{X}-\vector{X^{\prime}}),\eqno(4.3b)$$ where the [*Dirac Delta Function*]{} $\delta(\vector{X})$ is a distribution of measure one at the origin and zero elsewhere. The kernel $F({\bf x,x}^{\prime})$ of eq. (4.3b) is known in physics as the [*transformation function*]{}[@Abers], or sometimes the [*Hyugen Propagator*]{} since eq. (4.3a) is a description of Hyugen’s principle: [each point on a wave front may be regarded as a new source of waves]{}. The integral is over the hypersurface defined by $t^{\prime}=$constant, however it could be generalized to any [*spacelike*]{} surface, i.e. a surface made of points that are not [*causally connected*]{} to each other, and further that $t$ of eq. (4.11a) is greater than the $t^{\prime}$ of each point on the hypersurface. This amounts to requiring that the surface cannot have any “kinks” in it that would have the surface locally slip inside the “light cone”. The mathematician would equivalently say it must be a [*Lipschitz surface*]{}.

By inspection of eq. (4.3b), the kernel is function only of the of the coordinates: $F(\tpo{X}{t};\tpo{X^{\prime}}{t^{\prime}})=F({\bf x},{\bf x^{\prime}})=
F({\bf x}-{\bf x^{\prime}})$. Substituting eq. (3.9a) for the delta function in equation (4.3b) gives a closed integral form, $$F(\tpo{X}{t};\tpo{X^{\prime}}{t^{\prime}})=
\int d^3\vector{P}\ \eigen{\Psi}(\tpo{X}{t})
\ \eigen{\Psi^\dagger}(\tpo{X^{\prime}}{t^{\prime}}), \eqno(4.4a)$$ 