Geometric and Algebraic Topological Methods in Quantum Mechanics

TL;DR

This paper explores advanced geometric and topological methods that underpin modern quantum mechanics, emphasizing the algebraic language and schemes of quantization influenced by recent developments in the field.

Contribution

It provides a comprehensive guide to differential geometric and topological techniques used in quantum theory, highlighting their algebraic foundations and applications in modern quantization schemes.

Findings

Integrates geometric and algebraic methods in quantum mechanics

Highlights the role of topology in modern quantization

Provides a framework for advanced geometric techniques in quantum physics

Abstract

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The present book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.