 Solution

Use $\hslash =1$. Taking the trace of the imaginary-time time-evolution operator identifies the initial and final position, giving the path integralThe measure includes integration over the common endpoint. The paths are real periodic bosonic paths on the imaginary-time circle, not paths with fixed zero endpoints and not only classical trajectories. The kinetic measure is defined by time slicing; typical contributing paths need not be differentiable. For the free expansion assume $m,\kappa >0$. The interacting partition function requires a potential bounded below; otherwise the following formulas have the meaning of a formal perturbation series about the free harmonic oscillator.