into the expression for $\mathcal{H} (X(0),P(0))$ in (\[Eq3\]) and using the commutator $[X(0),X(\tau)]$ we rewrite each term of $\mathcal{H}$ in a time ordered form with all operators $X(\tau)$ to the left and all operators $X(0)$ to the right;

(iii)

:   with such an ordered hamiltonian, equation (\[Eq3\]) can be readly cast into the form $$\label{Eq5}
    i\hbar\frac{\partial}{\partial\tau} K(x,x^{\prime};\tau) =
     F(x,x^{\prime};\tau)K(x,x^{\prime};\tau),$$ with $F(x,x^{\prime};\tau)$ being an ordinary function defined as $$\label{Eq6}
    F(x,x^{\prime};\tau)=\frac{\langle x,\tau |\mathcal{H}_{ord}
    (X(\tau),X(0))| x^{\prime},0\rangle}
     {\langle x,\tau|x^{\prime},0\rangle}.$$ Integrating in $\tau$, the Feynman propagator takes the form $$\label{Eq7}
    K(x,x^{\prime};\tau)=C(x,x^{\prime})
     \exp\left\{\!\!-\frac{i}{\hbar}\!\! \int^{\tau}
      \!\!\!\!\! F(x,x^{\prime};\tau^\prime)d\tau^{\prime}\!\right\} ,$$ where $C(x,x^{\prime})$ is an integration constant independent of $\tau$ and $\int^{\tau}$ means an indefinite integral;

(iv)

:   last step is concerned with the evaluation of $C(x,x^{\prime})$. This is done after imposing the following conditions $$\begin{aligned}
    \label{CondicaoP(tau)}
    -i\hbar\frac{\partial}{\partial x}\langle
     x,\tau|x^{\prime},0\rangle &=&
     \langle x,\tau|P(\tau)|x^{\prime},0\rangle \, ,  \\
    i\hbar\frac{\partial}{\partial x^{\,\prime}}\langle
    x,\tau|x^{\prime},0\rangle &=&
     \langle x,\tau|P(0)|x^{\prime},0\rangle\, , \label{CondicaoP(0)}\end{aligned}$$ as well as the initial condition $$\label{CondicaoInicial}
    \lim_{\tau\rightarrow 0^+}
     K(x,x^{\prime};\tau) = \delta(x-x^{\prime})\ .$$

Imposing conditions (\[CondicaoP(tau)\]) and (\[CondicaoP(0)\]) means to substitute in their left hand sides the expression for $\langle x,\tau|x^{\prime},0\rangle$ given by (\[Eq7\]), while in their right hand sides the operators $P(\tau)$ and $P(0)$, respectively, written in terms of the operators $X(\tau)$ and $X(0)$ with 