Free-particle propagator

For a free particle of mass $m$ on the line,

Completing the square in the integral over an intermediate position shows that two free-particle propagators compose to the propagator for the sum of their positive time intervals. For equal slices, the quadratic identity isThe Fresnel integral contributes a factor proportional to $\sqrt{k/(k+1)}$; multiplying successive factors gives $1/\sqrt{N}$. A consistent convergence regulator fixes the phases of these square roots.

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.