Statistical deviation from stationary action to Schr\"odinger equation

TL;DR

This paper proposes that deviations from stationary action, distributed exponentially, can lead to the Schr"odinger equation, suggesting a new stochastic foundation for quantum mechanics based on classical action principles.

Contribution

It introduces a hypothesis that non-exact stationary action deviations follow an exponential law, deriving the Schr"odinger equation from this stochastic deviation model.

Findings

Dynamics satisfy Schr"odinger equation under specified deviation conditions

Particle behavior can be interpreted as guided by a virtual wave

Deviation distribution leads to quantum-like behavior

Abstract

We discuss the dynamics of single particle by laying a hypothesis that the Hamilton's principle of stationary action is not exact. We then postulate that the deviation of the action with sufficiently short time interval from the stationary action is distributed along a sufficiently long trajectory according to an exponential law. We show that the dynamics of the ensemble of trajectories satisfies the Schr\"odinger equation with Born interpretation of wave function if the average deviation is given by $\hbar/2$ and if two opposite signs of deviation occur equally probably. The particle thus behaves as if it is guided by a virtual wave satisfying the Schr\"odinger equation.