We investigate a one-parameter family of paths, xₙ (t) =t^n+1 for t0, 1, within the framework of the Feynman path integral. These paths fill the unit square in a layered, onion-like structure and serve as a concrete laboratory to study the mechanism of phase cancellation. We compute their classical action Sₙ = (n+1) ²/2 (2n+1) and show that Sₙ n/4 for large n, leading to increasingly rapid oscillations of the phase e^iSₙ/. This behavior provides a direct visualization of the stationary phase approximation: paths with large n are suppressed in the semiclassical limit. More significantly, we prove that the normalized second derivative (curvature) of these paths converges in the sense of distributions to a Dirac delta at the endpoint: xₙ/ (n+1) (t-1). This demonstrates a singular concentration of curvature, a deterministic analogue of the fractal properties of Brownian paths. We also analyze the embedding of this family in the infinite-dimensional path space, noting that it constitutes a set of measure zero under the Wiener measure. The paper connects these geometric insights with Feynman's postulates, provides a rigorous derivation of the Schrodinger equation, and discusses applications to simple potentials. Extensions to more general families and numerical comparisons are presented, highlighting the pedagogical and mathematical value of this construction.
Arnaldo Adrian Ozorio Olea (Sat,) studied this question.