= Solution
A <Euclidean configuration-space transition kernel> is obtained by slicing $\tau$ into $M$ steps $\delta=\tau/M$, inserting position resolutions and using the free Gaussian kernel. With initial $q_i=\pm a$ and final $q_f=a$,
$$
\boxed{\langle q_f|e^{-H\tau/\hbar}|q_i\rangle=\int_{q(0)=q_i}^{q(\tau)=q_f}\!\mathcal Dq\,
\exp\left[-\frac1\hbar\int_0^\tau\left(\frac m2\dot q^2+V(q)\right)dt\right].}
$$
The normalization means the limit of $(m/(2\pi\hbar\delta))^{M/2}\int\prod_{j=1}^{M-1}dq_j$ with exponent $-\sum_j[m(q_{j+1}-q_j)^2/(2\delta)+\delta V(q_j)]/\hbar$, in a consistent time-slice prescription. This fixes the endpoint normalization that an informal continuum symbol alone leaves unspecified. The <Wick rotation> converts oscillatory real-time weight into the positive-potential Euclidean weight; classical extrema of this action organize the semiclassical approximation.
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