# Poincar\'e Symmetry from Heisenberg's Uncertainty Relations

## TL;DR

This paper explores how the fundamental symmetries of spacetime, including the Poincaré group, can be derived from the Heisenberg uncertainty relations, linking quantum mechanics to spacetime symmetries.

## Contribution

It demonstrates that the Poincaré symmetry naturally emerges from the Heisenberg commutation relations through group contractions and isomorphisms.

## Key findings

- Heisenberg relations encode $Sp(2)$ symmetry related to Lorentz group
- Uncertainty relations for two variables lead to $O(3,2)$ de Sitter group
- Contracting time-like variables yields the Poincaré group

## Abstract

It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the $Sp(2)$ group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the $O(2,1)$ group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group $O(3,2)$, namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in $O(3,2)$, it is possible, to construct the inhomogeneous Lorentz group $IO(3,1)$ which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz covariant world. This $IO(3,1)$ group is commonly known as the Poincar\'e group.