Interpretation of Feynman formalism of quantum mechanics in terms of probabilities of paths

TL;DR

This paper reexamines non-relativistic Feynman path integrals by reframing the quantum evolution in terms of probabilities of individual paths and by drawing parallels with Huygens optics and diffusion. It derives a pathway to interpret transition probabilities as sums over path probabilities in special cases, and shows how the Born approximation for scattering emerges within this framework, yielding a positive probability $P\propto|\int d\mathbf{x}\, e^{i\Delta\mathbf{k}\cdot\mathbf{x}} V(\mathbf{x})|^{2}$ without invoking the wavefunction postulate. The work emphasizes that positivity is not guaranteed for generic potentials, and that the nonlocal character of the potential’s action and the absence of a direct Heisenberg uncertainty description complicate a literal particle-or-path ontology. Overall, it offers an ontology-leaning interpretation of path integrals as a calculational and conceptual alternative to wavefunction-based quantum mechanics, with concrete links to scattering theory and classical limits.

Abstract

Feynman path integrals formalism for non-relativistic quantum mechanics is revisited. A comparison is made with the cases of light progagation (Huygens principle) and Brownian motion. The difficulties for a physical model behind Feynman formalism are pointed out. It is proposed a reformulation where the transition probability from one space-time point to another one is the sum of probabilities of the possible paths. The Born approximation for scattering is derived within the formalism, which suggests an interpretation in terms of particles, without the need of Born assumption that the modulus squared of the wavefunction is a probability density.