Snyder and their representation with creation and annihilation operators

TL;DR

This paper introduces a novel operator representation of Snyder space-time using creation and annihilation operators, inspired by Schwinger's angular momentum representation, leading to quantized geometric properties.

Contribution

It proposes a new representation of Snyder space-time with creation and annihilation operators and demonstrates how to obtain quantized geometric quantities.

Findings

Representation of Snyder space-time via creation and annihilation operators

Matrix representation of $SO(3)$ or $SU(2)$ Lie algebra for spatial sector

Quantized expectation value of a sphere's area

Abstract

Inspired by the Schwinger's representation of angular momentum, we propose a representation of certain operators where we use the algebra of the annihilation and creation operators. In particular, we propose a representation of the Snyder space-time with the help of the annihilation and creation operators, which create and annihilate quantum of space. In addition, we show that by using a matrix representation of the $SO(3)$ or $SU(2)$ Lie algebra it is possible to obtain a representation of the spacial sector of Snyder space-time. Finally, we obtain a quantized expectation value of the area of a sphere.