From Newton to the Path Integral

Abstract

Physically meaningful laws arise as controlled limits of composable local refinements. This paper develops that thesis across classical and quantum theory. We begin with Newton's polygonal approximation of central-force motion and its limit to continuous dynamics, then re-express the same logic in modern variational form through additive action functionals. The path integral is treated as a composition law over refined time slices rather than an isolated quantum postulate, and deformation quantization and renormalization are framed as two mathematically distinct control mechanisms for limit consistency. The narrative is constructive: each stage retains the previous one as a limiting or compatibility condition. Within this architecture we reserve a dedicated role for point-like (Dirac-supported) probes in weak formulations of the action principle, specifying where they are mathematically valid and where regularization is mandatory. The result is a staged program from Newtonian limit methods to quantum amplitudes in which the classical equations are recovered as stationary limits of a broader compositional framework.

Keywords