over all possible classical paths: $x^{\mu}=x^{\mu}(\tau)$, where $p^{\mu}=m \frac{dx^{\mu}}{d\tau}$ and $\tau$ is an affine parameter called the [*proper time*]{}. Classical paths require $\frac{dx^4}{d\tau}>0$, i.e. paths in which particles go backward in time are excluded.

Because of the second order time derivative, the Klein-Gordon eq. (3.1) does not easily lend itself to [PIF]{}. The first order Dirac eq. (3.2a) is a better candidate. However, the standard derivation of eq. (4.8a) from the Schrödinger wave equation[@Abers] required that each eigenfunction have the same generator of quantum phase, a commuting $i$ which is now unavailable to us in the real multivector theory. In fact, we have seen from eq. (3.5a) that each eigenfunction (as seen from the left side) has its own unique quantum phase generator $\phat$, which makes the [PIF]{} derivation problematic. In the restricted case of plus parity eigenfunctions however, we saw from eq. (3.7a) that they all have the generator $\basis{4}$ when viewed from the side. We exploit the advantage of multivector wavefunctions (over standard Dirac column spinors) in that we can now consider operations. Hence we introduce here a new alternative to eq. (4.3a), the [*dextrad propagator*]{}, $$\Psi(\tpo{X}{t})= \int d^3\vector{X}^{\prime}
\ \Psi(\tpo{X^{\prime}}{t^{\prime}})
\ H(\tpo{X}{t};\tpo{X^{\prime}}{t^{\prime}}), \eqno(4.9a)$$ $$H({\bf x,x^{\prime}})=H({\bf x-x^{\prime}})=
\int{ 