A review of differential geometry methods in classical and quantum mechanics through quantization

R. Aldrovandi

Elisa Ercolessi

Elisa Ercolessi

These notes collect the lectures given by the author to the "XXIII International Workshop on Geometry and Physics" held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoretical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by "Feynman Path Integral Technique", is the method which is most widely used in quantum field theory. Finally, we will describe the "Weyl Quantization Approach" which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.

GIOVANNI GIACHETTA, Luigi Mangiarotti, Gennadi Aleksandrovich Sardanashvily

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.

K. R. W. Jones

GIOVANNI GIACHETTA, Luigi Mangiarotti, Gennadi Aleksandrovich Sardanashvily

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric 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 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.