of the trajectory traced out by the point particle, for which $$\left(\frac{\mathrm{d} x_\mu(\tau_\mathrm{p})}{\mathrm{d}
\tau_\mathrm{p}} \right)^2=1.$$

To make this evident, we write the integrand in the second line of Eq. (\[continuum\]) as a Feynman path integral[@StoneThomas] $$\begin{gathered}
\label{pathintegral}
\left( \frac{1}{4 \pi s} \right)^{d/2} \mathrm{e}^{-(x-x')^2/4s} \,
\mathrm{e}^{-s m^2} \\ = \int^{x(s) = x'}_{x(0) = x} \mathrm{D}x(s') \,
\exp \left\{-\int_0^{s}\mathrm{d} s' \left[ \tfrac{1}{4} {\dot
{x}}^2(s') + m^2 \right]\right\},\end{gathered}$$ where the so-called Wiener measure is defined by $$\int^{x(s) = x'}_{x(0)=x} \mathrm{D} x(s') \equiv  
\lim_{b\to \infty} 
A^b \prod_{i=1}^{b-1} \int_{-\infty}^\infty \mathrm{d}  x_i \, ,$$ with the normalization factor $$A = \left( \frac{d}{2 \pi a^2} \right)^{d/2}.$$ Here, it is understood that the limit $b \to \infty$ is taken concurrently with the continuum limit $a \to 0$ in a way such that $b
a^2 \to \mbox{const}$. The classical action of the point particle in Eq. (\[pathintegral\]), tracing out its trajectory in spacetime is the continuum limit of the lattice action present in Eq. (\[discrete\]): $$\begin{aligned}
\label{W0}
W_0 &=& \frac{d}{2} \sum_{i=1}^b \left(\frac{x_{i-1} - x_i}{a}\right)^2 +
\frac{m^2 a^2 b}{2d} \nonumber \\ &=& \sum_{i=1}^b \frac{a^2}{2d}
\left[\frac{1}{4} \left(\frac{x_{i-1} - x_i}{a^2/2d}\right)^2 + m^2
\right] \nonumber \\ &\to& \int_0^{s} \mathrm{d} s' \left[\tfrac{1}{4}
\dot{x}^2(s') + m^2 \right],\end{aligned}$$ where in the last step the identification (\[identification\]) 