Uncertainty Relations in Self-Similar Convergent Trajectories

TL;DR

This paper explores the relationship between self-similar fractal trajectories, like the Koch curve, and quantum uncertainty, deriving an alternative perspective on Heisenberg's uncertainty principle through fractal geometry.

Contribution

It introduces a novel approach linking fractal self-similar trajectories to quantum uncertainty, providing an alternative derivation of Heisenberg's relation.

Findings

Demonstrates the unbounded growth of momentum uncertainty as position uncertainty diminishes.

Connects fractal dimension of trajectories to quantum indeterminacy.

Proposes an alternative geometric derivation of the uncertainty principle.

Abstract

The Koch curve is a self-similar object whose length grows unboundedly when the measuring unit by which is calculated diminishes. If this curve is considered to be the trajectory of a point corpuscle of mass m (a particle) rendering it in a time t, while the measuring unit in the kth scale is associated with the indetermination in the position of the corpuscle, then it is possible to demonstrate that when the indetermination of the corpuscle position diminishes, the indetermination in its linear momentum grows unboundedly. Based on the concept of similarity dimension of a corpuscle trajectory, from the before stated line of reasoning an alternative deduction of Heisenberg's uncertainty relation (Delta)x(Delta)p(sub)x (aprox)h is developed and discussed.