Bounds on variable-length compound jumps

TL;DR

This paper explores bounds on variable-length jumps in different spaces, revealing connections between geometric inequalities and physical systems like relativity and scattering, with implications for velocity composition and particle interactions.

Contribution

It establishes a unified framework linking geometric bounds to diverse physical phenomena through the application of polygon inequalities in various spaces.

Findings

Bounds on velocity composition in relativistic space

Constraints on transfer matrices in scattering

Relations between geometric inequalities and physical systems

Abstract

In Euclidean space there is a trivial upper bound on the maximum length of a compound "walk" built up of variable-length jumps, and a considerably less trivial lower bound on its minimum length. The existence of this non-trivial lower bound is intimately connected to the triangle inequalities, and the more general "polygon inequalities". Moving beyond Euclidean space, when a modified version of these bounds is applied in "rapidity space" they provide upper and lower bounds on the relativistic composition of velocities. Similarly, when applied to "transfer matrices" these bounds place constraints either (in a scattering context) on transmission and reflection coefficients, or (in a parametric excitation context) on particle production. Physically these are very different contexts, but mathematically there are intimate relations between these superficially very distinct systems.