Hopf Map and Quantization on Sphere

TL;DR

This paper explores the geometric and topological structures underlying the quantization of systems constrained to spheres, focusing on the Hopf map and its implications for monopole and instanton gauge fields.

Contribution

It introduces a novel approach to quantization on spheres by utilizing the square root of the on-sphere condition, revealing the fibre bundle structure of the Hopf map for S^2 and S^4.

Findings

Identifies the fibre bundle structure of the Hopf map in quantization.

Provides geometric insights into monopole and instanton gauge structures.

Enhances understanding of quantization constraints on spherical manifolds.

Abstract

Quantization of a system constrained to move on a sphere is considered by taking a square root of the ``on sphere condition''. We arrive at the fibre bundle structure of the Hopf map in the cases of $S^{2} $and $S^{4}$. This leads to more geometrical understanding of monopole and instanton gauge structures that emerge in the course of quantization.