Introduction to Quantum Mechanics and the Quantum-Classical transition

TL;DR

This paper surveys differential geometric formalisms in quantum mechanics, analyzing Schrödinger and Heisenberg frameworks, and discusses the Weyl-Wigner approach and bi-Hamiltonian structures at the quantum level.

Contribution

It introduces a geometric perspective to quantum mechanics, relating different formulations and exploring advanced structures like bi-Hamiltonian systems.

Findings

Unified geometric framework for Schrödinger and Heisenberg pictures

Relation between momentum map and quantum formalisms

Implications of bi-Hamiltonian structures in quantum theory

Abstract

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the action of the unitary group on the Hilbert space allows to relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics and discuss the implications of bi-Hamiltonian structures at the quantum level.