This should provide a good alternative derivation of Kirchhoff diffraction theory, however it will still suffer from the practical limitation of the function only being approximately known on the boundry[@Jackson]. Alternatively, this equation could be physically interpreted as a factored form of the time-independent quantum [Pauli equation]{} (see recent exposition by Adler[@Adler]). However, one is then ontologically constrained by the quantum principle which states that only the modulus $\rho=\phi^{\dagger}\phi$ of the wavefunction $\phi=\sqrt{\rho}\exp(i\theta)$ can be measured; the absolute quantum phase $\theta$ cannot be known. Hence it will be impossible to perform a Cauchy boundry integral except for very special situations. This has influenced the way that physicists approach or “formulate” quantum problems.

The Propagating Kernel
----------------------

One common scenario is to a physical system’s initial configuration, [*even though it cannot be directly measured*]{}, \[e.g. $\Psi(\vector{X},t^{\prime})$ known for all $\vector{X}$ at $t^{\prime}=0$\], and then to ask how the system must change with time. The [*Time Evolution Operator*]{} $U(t;t^{\prime})$ is defined, $$\Psi(\tpo{X}{t})=
U(t;t^{\prime})\ \Psi(\tpo{X}{t^{\prime}}), \eqno(4.1a)$$ $$U(t,t^{\prime})=U(t-t^{\prime})=
\exp{[-(t-t^{\prime})\ (\vector{\nabla} - \basis{4} m)]}, \eqno(4.2b)$$ where $\vector{\nabla}=\basis{4}\wedge\Box = \basis{4}\basisu{j}\partial_j$ is the gradient. It will be assumed that $t>t^{\prime}$, otherwise one would be talking about ‘retrodiction’, a different problem. An integral representation of eq. (4.1a) is, 