Euclidean free-particle bridge amplitude ($K_E(q,T;q,0)=\sqrt{M/(2\pi \hslash T)}$)

Wick rotation of the free-line propagator gives the displayed coincident-endpoint kernel for positive Euclidean duration $T$. Equivalently it is the path integral over Dirichlet fluctuations about the straight classical path, with its time-sliced Gaussian measure included. It is a propagator density, not a full thermal partition function: that trace additionally integrates position and, on a multiply connected configuration space, sums the winding sectors. For an angular coordinate replace $M$ by moment of inertia.