The Electromagnetic Duality Formulation of Geometric Phases

Jesús Liñares, María C. Nistal

Mustapha Maamache, Nadir Chaabi, J. R. Choi

J. Anandan, Yakir Aharonov

The geometric phase in quantum mechanics is formulated for charged particles in a gauge-invariant, geometric manner. It is then extended to an evolution resulting from a sequence of measurements as in the work of Pancharatnam and Aharonov and Vardi. Its close connection to the Feynman formulation of quantum mechanics is pointed out. The geometric angles, which are generalizations of the classical, adiabatic angles introduced by Hannay and the quantum, adiabatic angles introduced by Anandan and Stodolsky in their group-theoretic treatment of Berry's phase, are studied in quantum and classical physics. The geometric phase for a quantum spin in a magnetic field due to a second particle is obtained using the quantum reference frame defined by the latter. The question of whether the geometric phase and angles are local or nonlocal and their relationship to the electromagnetic and gravitational phases are also discussed.

Ravi Baijulal Bhandari

Reinhold A. Bertlmann, Nicolai Friis

Abstract Chapter 9 discusses the rigorous incorporation of electromagnetism in quantum mechanics via minimal substitution and the inclusion of spin, which leads to the Pauli equation. The gauge invariance of this formulation is inspected more closely, which leads us to understanding quantum mechanics as a gauge theory, and on to fascinating geometric effects such as the Aharonov-Bohm effect and the discovery of geometric phases such as the Berry phase and the Aharonov-Anandan phase. To give a better context to these effects, a compact introduction to the mathematical background of differential geometry is provided, introducing concepts such as differentiable manifolds, tangent and cotangent spaces, and maps between them, as well as from their elements to the real numbers, arriving at tensors and differential forms. With these tools at hand, we first define fibre bundles before formulating the previously discussed geometric phases as well as the so-called Dirac monopole in this language

Ludwig-Maximilians-Universität München, Detlef Dürr