is used. Written as a path integral, the correlation function (\[continuum\]) thus reads $$\left\langle \varphi^\alpha_x \varphi^\beta_{x'} \right\rangle = \delta_{\alpha,
\beta} \int_0^{\infty} \mathrm{d} s \, 
\int^{x(s) = x'}_{x(0) = x} \mathrm{D}x(s') \, \mathrm{e}^{- W_0} .$$ It forms the continuum limit of the sum over all possible particle trajectories on the spacetime lattice in Eq. (\[Sigma\]). Note that the proper time parameter $s$ can take any positive value, in agreement with the lattice representation (\[Sigma\]) of the correlation function where the worldlines can be arbitrarily long. The Boltzmann factor $\exp (-s m^2)$ exponentially suppresses, however, large proper times as long as $m^2>0$. When we set $$s \equiv \tau_\mathrm{p}/2m ,$$ the continuum action (\[W0\]) yields $m$ times the arc length of the path, $$\int_0^{s} \mathrm{d} s' \left[\tfrac{1}{4}
\dot{x}^2(s') + m^2 \right] = m \tau_\mathrm{p},$$ which is the standard form of the classical action of a relativistic free scalar particle of mass $m$. This demonstrates that $s$ can indeed be thought of as the proper time.

For later use, we record the second line in Eq. (\[continuum\]) with Planck’s constant and the speed of light reinstalled: $$\label{chbar}
\left\langle \varphi^\alpha_x \varphi^\beta_{0} \right\rangle =
\delta_{\alpha, \beta} \, c \int_0^{\infty} \mathrm{d} s \, \left(
\frac{1}{4 \pi \hbar 