# Geometry of Uncertainty Relations for Linear Combinations of Position and Momentum

## TL;DR

This paper develops a geometric framework for uncertainty relations involving multiple linear combinations of position and momentum, revealing bounds based on their incompatibility and phase space configurations.

## Contribution

It introduces a geometric approach to uncertainty relations for multiple observables, including sum and product bounds dependent on their incompatibility.

## Key findings

- Bounds depend on the area of a parallelogram in coefficient space
- Maximal incompatibility corresponds to regular polygons in phase space
- Conjectures a generalized entropy-based uncertainty relation

## Abstract

For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by $N$ linear combinations of position and momentum observables. The state-independent bounds depend on their degree of incompatibility defined by the area of a parallelogram in an $N$-dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bounds the sum of the position and momentum Shannon entropies of the particle.