where there must be attention to its placement. Equivalently we state (without proof) that the unit four-velocity $\phat$ may be used \[inspection of eq. (4.6e) shows this to be consistent\].

Inspection of eq. (4.6e) shows that $\overline{G}({\bf x})=
G(-{\bf x})$, hence one can show that this Green function also satisfies the adjoint equation: $G({\bf x})(\Box - m) =\delta^4({\bf x})$, where $\Box$ is now understood to operate to the left. Hence we may derive the general integral solution, $$\Psi({\bf x}^{\prime})=
\oint d\Sigma^{\mu}\ G({\bf x^{\prime},x})
\ \basis{\mu}\ \Psi({\bf x}), \eqno(4.7)$$ where ${\bf x}$ is an interior point. Actually, its abit more complicated than that, one must add a homogeneous term to eq. (4.6a) such that the boundry conditions are met.

Path Integral Formulation
-------------------------

The Feynman [PIF]{} (Path Integral Formulation) of non-relativistic quantum mechanics[@Feynman] provides an elegant means of bridging the interpretation gap between classical mechanics and quantum formulations. In the non-relativistic case, it asserts that the propagator kernel of eq. (4.3a) can be written, $$F({\bf x,x^{\prime}})=
\sum{\exp\left(\frac{i}{\hbar} {\cal S}
({\bf x,x^{\prime}})\right)}, \eqno(4.8a)$$ $${\cal S}({\bf x,x}^{\prime})=\int_{\bf x^{\prime}}^{\bf x}
p^{\mu}dx_{\mu}. \eqno(4.8b)$$ The [*classical action*]{} ${\cal S}$ is evaluated over the path from ${\bf x^{\prime}}$ to ${\bf x}$. The sum in eq. (4.8a) is 