Topological quantum mechanics and the first Chern class

TL;DR

This paper explores the intrinsic topological nature of quantum systems, revealing that the first Chern class, linked to the wave function's properties, underpins many topological effects in quantum mechanics.

Contribution

It introduces a new perspective on topological quantum mechanics by connecting the first Chern class to the wave function through gauge potential decomposition.

Findings

The first Chern class is inherent in Schrödinger systems.

The Chern class relates to the Hopf index and Brouwer degree of the wave function.

This relationship explains many topological effects in quantum systems.

Abstract

Topological properties of quantum system is directly associated with the wave function. Based on the decomposition theory of gauge potential, a new comprehension of topological quantum mechanics is discussed. One shows that a topological invariant, the first Chern class, is inherent in the Schr\"odinger system, which is only associated with the Hopf index and Brouwer degree of the wave function. This relationship between the first Chern class and the wave function is the topological source of many topological effects in quantum system.