= Euclidean free-particle bridge amplitude
{c}
{title2=$K_E(q,T;q,0)=\sqrt{M/(2\pi\hbar T)}$}
<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>.
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