Hopf maps and Wigner's little groups

TL;DR

This paper explores the mathematical relationship between Hopf maps and Wigner's little groups, providing explicit formulas, simplified forms, and invariant Lagrangians to deepen understanding of their geometric and physical significance.

Contribution

It introduces explicit formulas linking Hopf maps with Wigner's little groups and discusses their invariant Lagrangians and reductions, offering new insights into their mathematical structure.

Findings

Explicit formulas relating Hopf maps to Wigner's little groups.

Simplified forms for the third Hopf fibration.

Invariant Lagrangians associated with these structures.

Abstract

We present the explicit formulae relating Hopf maps with Wigner's little groups. They, particularly, explain simple action of group on a fiber for the first and second Hopf fibrations, and present most simplified form for the third one. Corresponding invariant Lagrangians are presented, and their possible reductions are discussed.