Space-Time Geodesics and the Derivation of Schr\"odinger's equation

TL;DR

This paper derives Schr"odinger's equation from space-time geodesics and Feynman's path integral, linking geometric paths to quantum wave functions in a novel way.

Contribution

It introduces a geometric approach to derive Schr"odinger's equation using space-time geodesics and Feynman's path integral framework.

Findings

Derives Schr"odinger's equation from geometric principles.

Establishes a connection between space-time geodesics and quantum wave functions.

Provides a new perspective on quantum mechanics foundations.

Abstract

Using the essence of Feynman's path integral and the space-time geodesics, an infinity of differentiable paths that follow the geometry of a continuous geodesic are constructed, and a wave function is associated to each path as a probability amplitude identical to the Feynman's probability amplitude for each path. We prove that each probability amplitude obeys to the Schr\"odinger's equation for a non relativistic physical system moving in a time independent potential, starting from the Jacobi-Hamilton equation.