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arXiv:math-ph/0607052 · International Journal of Quantum Chemistry

Some geometric properties of quantum phases and calculation of phase formulas

Paul Bracken

Published Jan 1, 2008• 2 citations
arXivHTMLPDF

Abstract

Abstract 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 is formulated by defining an appropriate connection on a specific fiber bundle. The holonomy element gives the phase. By introducing another connection, the Pancharatnam phase formula is derived following a different procedure. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

DOI
10.1002/qua.21693

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Abstract 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 is formulated by defining an appropriate connection on a specific fiber bundle. The holonomy element gives the phase. By introducing another connection, the Pancharatnam phase formula is derived following a different procedure. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
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.
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
T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.
Usually one uses completely different mathematical descriptions to formulate classical and quantum mechanics. Classical theory may be nicely formulated in terms of symplectic geometry, and the quantum one in terms of algebraic objects related to a complex Hilbert space. However, it turns out that standard, nonrelativistic quantum mechanics possesses natural geometric structure that is even richer than that found in classical mechanics. This section reveals the beauty of the geometric approach to quantum theory and stands as a basis for the elegant geometrical ideas of Pancharatnam and, later on, of Aharonov and Anandan. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Geometric phases form a rich subject in mathematical physics, lying at the intersection of differential geometry, functional analysis, and topology. This combination of ideas can make the subject difficult to approach for advanced undergraduate students, even though it is very rewarding. The aim of this thesis is to give a self-contained introduction to the mathematical framework behind geometric phases. Assuming only basic differential geometry, we develop the theory of vector bundles, connections, curvature, and holonomy, and use it to describe the Berry and Wilczek–Zee phases as holonomies of natural connections on complex vector bundles. We also discuss the Aharonov–Bohm effect as an example of a flat connection with globally nontrivial holonomy.