# Geometric Phases in Classical and Quantum Mechanics

@inproceedings{Chruciski2004GeometricPI, title={Geometric Phases in Classical and Quantum Mechanics}, author={Dariusz Chruściński and Andrzej Jamiołkowski}, year={2004} }

Preface - Mathematical background - Adiabatic phases in quantum mechanics - Adiabatic phases in classical mechanics - Geometric approach to classical phases - Geometry of quantum evolution - Geometric phases in action - A. Classical matrix Lie groups and algebras - B. Quaternions - Bibliography - Index

## 230 Citations

Geometric Aspects of Quantum Mechanics and Quantum Entanglement

- Physics
- 2006

It is shown that the standard non-relativistic Quantum Mechanics gives rise to elegant and rich geometrical structures. The space of quantum states is endowed with nontrivial Fubini-Study metric…

Introduction to quantum mechanics and the quantum-classical transition

- Physics
- 2007

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…

Remarks on the GNS Representation and the Geometry of Quantum States

- PhysicsOpen Syst. Inf. Dyn.
- 2009

It turns out that the GNS representation provides not only symplectic but even Hermitian realization of a 'quantum Poisson algebra' in the standard non-relativistic quantum mechanics.

Geometric phase in PT-symmetric quantum mechanics

- Physics
- 2010

Unitary evolution in PT-symmetric quantum mechanics (QM) with a time-dependent metric is found to yield an interesting class of adiabatic processes. As an explicit example, a Berry-like phase…

A symmetry approach to geometric phase for quantum ensembles

- Physics
- 2015

We use tools from the theory of dynamical systems with symmetries to stratify Uhlmann's standard purification bundle and derive a new connection for mixed quantum states. For unitarily evolving sys…

Geometric approach to non-relativistic quantum dynamics of mixed states

- Mathematics, Physics
- 2013

In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of…

Some Geometric Properties of Quantum Phases and Calculation of Phase Formulas

- Physics
- 2008

An introduction to several geometrical ideas which are of use to quantum mechanics is presented. The Aharonov-Anandan phase is introduced and without reference to any dynamical equation, this phase…

Quantum information metric and Berry curvature from a Lagrangian approach

- Physics
- 2017

A bstractWe take as a starting point an expression for the quantum geometric tensor recently derived in the context of the gauge/gravity duality. We proceed to generalize this formalism in such way…

On Time-Local Generators of Quantum Evolution

- PhysicsOpen Syst. Inf. Dyn.
- 2014

The properties of time-local generators giving rise to legitimate completely positive trace preserving quantum evolutions are characterized and the analysis of Markovian and non-Markovian quantum dynamics is presented.

Phase-Space Approach to Berry Phases

- PhysicsOpen Syst. Inf. Dyn.
- 2006

This approach sheds a new light onto the correspon-dence between classical and quantum adiabatic phases – both phases are related with the av-eraging procedure: Hannay angle with averaging over the classical torus and Berry phase with averaged over the entire classical phase space with respect to the corresponding Wigner function.