interpret $- \partial_\mu^2 + m^2$ as the Hamilton operator describing the motion of a *nonrelativistic* particle of mass $M=\tfrac{1}{2}$ in a constant potential $V=m^2$ in $d$ dimensions. With the Schwinger proper time $s$ interpreted as Newtonian time, $W_0$ is the corresponding nonrelativistic action, where it is recalled that in natural units $\hbar =1$.

The first line in Eq. (\[Z\_0S\]) is sometimes represented by the one-loop Feynman diagram $$\ln Z_0 = \bigcirc.$$ In the spacetime approach, this diagram stands for all the terms in the sum (\[Z0\]), each consisting of a *single* worldline starting and ending at a given site $n$. That is, the Feynman diagram at the same time denotes the topology of the particle trajectories contributing to $\ln Z_0$.[@GFCM]

The partition function, obtained by exponentiating Eq. (\[Z0\]), can be written in terms of Feynman diagrams as $$\label{loopgas}
  Z_0 = 1 + \bigcirc + \frac{1}{2!} \bigcirc \bigcirc + \frac{1}{3!} \bigcirc 
  \bigcirc \bigcirc + \cdots$$ Each diagram involves a separate integration over loop momentum $k_\mu$. With the right hand of Eq. (\[loopgas\]) taken as picturing the topology of the particle trajectories contributing to the partition function, it follows that $Z_0$ describes a loop gas–a grand canonical ensemble of fluctuation loops of 