s} \right)^{d/2} \mathrm{e}^{-(c^2 \tau^2 +
\mathbf{x}^2)/4 \hbar s} \, \mathrm{e}^{-s m^2 c^2/\hbar} ,$$ while the continuum action (\[Scont\]) becomes $$\label{relS}
S_0 = \int \mathrm{d}  \tau \, \mathrm{d}^{d-1} x \; \varphi \left( -
\hbar^2 \frac{\partial^2}{c^2\partial \tau^2} - \hbar^2\nabla^2 + m^2 c^2
\right)\varphi,$$ where now the spacetime coordinates are denoted by $x_\mu = ({\bf x},c
\tau)$, with $\tau$ the imaginary time and $\mathbf{x}=(x_1,
\cdots,x_{d-1})$ the spatial coordinates.

Besides the number of worldlines $z_b(n, n')$ on the spacetime lattice connecting the lattice sites $n$ and $n'$ in $b$ steps, also the probability $P_b(n, n')$ for a particle to move from $n$ to $n'$ in $b$ steps will become important below. The probability is given by the ratio $$\label{Pn}
P_b(n, n') = z_b(n,n')/z_b$$ of $z_b(n, n')$ and the number of paths of $b$ steps starting at $n$ and ending at an arbitrary lattice site, $$\label{zb}
z_b \equiv \sum_{n'} z_b(n,n').$$ The denominator in Eq. (\[Pn\]) is included so that $P_b$ is a genuine probability, taking values between 0 and 1. To evaluate it, imagine the particle taking a first step to one of its nearest neighbors $n
\pm\hat{\mu}$. It then has only $n-$1 steps still available to reach the final destination $n'$ with probability $P_{b-1}(n 