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
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
This thesis reviews fibre-bundle theory for the purpose of showing its equivalence to gauge theory and, in particular, utilizing fibre bundles to explain the origin of a quantum mechanical effect named the Aharonov-Bohm (AB) effect. Initially, the mathematical preliminaries necessary to review fibre-bundle theory are presented. This includes topological manifolds, smooth manifolds, vector fields and 1-forms on smooth manifolds as well as Lie groups and their Lie algebras. Thereafter, fibre bundles are introduced. In particular, principal bundles, connections on principal bundles, parallel transport and associated bundles are studied. The electrons in the setup of the AB effect experience the effect of an electromagnetic potential and form an interference pattern on the screen they hit. The AB effect is described by a principal U(1)-bundle over R2\{0} and an associated complex line bundle with bundle space P xρ C, where P is the bundle space in the principal bundle and ρ a representation of U(1) on C defined ρ(x) = xq, for x in U(1) and q in Z. The vector potential A, on a simply connected subset U ⊂ R2\{0}, is given by A = ∇χU, for a smooth real-valued function χU. The complex scalar field, i.e. wave function, ψU: U → C of an electron with charge q has a local solution ψU = eiqχUψ0, where ψ0 satisfies the free Schrödinger equation. Consequentially, the holonomy is eiqΦ, where Φ is the total flux of the magnetic field.
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.
We discuss the Aharonov-Bohm (A -B) effect and the Dirac (D) monopole of magnetic charge g = 1 2 in the context of bundle theory, which allows to exhibit a deep geometric relation between them.If ξA-B and ξD are the respective U (1)-bundles, we show that ξA-B is isomorphic to the pull-back of ξD induced by the inclusion of the corresponding base spaces.The fact that the A -B effect disappears when the magnetic flux in the solenoid equals an integer number of times the quantum of flux associated with the unit of electric charge, reflects here as a consequence of the pull-back of the Dirac connection in ξD to ξA-B, and the Dirac quantization condition.
We discuss the Aharonov-Bohm (A−B) effect and the Dirac (D) monopole of magnetic charge g=21 in the context of bundle theory, exhibiting a purely geometric relation between them. If ξA−B and ξD are the respective U(1)-bundles, we show that ξA−B is isomorphic to the pull-back of ξD induced by the inclusion of the corresponding base spaces ι:(D02)∗→S2}. The fact that the A−B effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux Φ0=∣e∣2π associated with the electric charge ∣e∣, reflects here as a consequence of the pull-back by ι of the Dirac connection in ξD to ξA−B, and the Dirac quantization condition. We also show the necessary vanishing in ξA−B of the pull-back of the Chern class c1 in ξD.
This special issue celebrates the discovery of two of the most important aspects of quantum mechanics: the Aharonov–Bohm effect and the Berry phase. The issue includes work presented at two conferences, 50 Years of the Aharonov–Bohm Effect, 11–14 October 2009, Tel Aviv University, Israel, and the Aharonov–Bohm Effect and Berry Phase 50/25 Anniversary, 14–15 December 2009, Bristol University, UK. The Aharonov–Bohm (AB) effect and the Berry phase have many things in common, apart from the happy coincidence that they were discovered 25 years apart, making possible such a joint anniversary. First, they were both discovered in the same place, the H H Wills Physics Laboratory, University of Bristol, UK. Second, they both were there for everybody to see from day one, from the very beginning of quantum mechanics. They were, so to say, right on the surface; there was no need for sophisticated mathematics or involved constructions. Yet it took so long for somebody to see them. The explanation, we dare to speculate, is that they are so unusual, they challenge so much our everyday experience and ingrained intuitions about physics that it was difficult to recognise them. Third, both the AB effect and the Berry phase are quintessentially quantum; they both manifest themselves as topological/geometric quantum phases. The AB effect revolutionized our understanding of the role of potentials in physics. For the first time, it was shown that a particle moving in a field-free region could be affected by a field in a disjoint region. In doing so, it shattered one of the strongest held principles of physics. In the case of the Berry phase, the importance lies in its absolute generality—the topological/geometric phase named after Berry is acquired by all physical systems when the external parameters they depend upon undergo an adiabatic cyclic change. Remarkably this phase depends only on the geometry of the cyclic path in the abstract parameter space, a fact which deepens our appreciation of the underlying geometry of quantum mechanics. The AB effect and the Berry phase are ubiquitous in modern physics and have implications ranging from the most fundamental to the very applied. First and foremost, they give a deeper understanding of the nature of quantum mechanics itself. Furthermore, we find them in fields as varied as cosmology, particle physics, non-abelian gauge theories, condensed matter, chemical and molecular physics, and laser dynamics. Generalizations of the AB effect to non-abelian gauge theories, such as the Wilson and `t Hooft loops, are important tools for studying confinement and spontaneous symmetry breaking. They explain charge quantization, the quantum Hall effect and the Josephson junction and also play a crucial role in electron microscope holography. Without them an understanding of many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behaviour, would be impossible. Like good wine, the AB effect and the Berry phase become finer and more appreciated with time. Even though they appear in textbooks and encyclopaedias, the number of researchers thinking and publishing papers on these two effects is increasing annually. The AB effect and the Berry phase keep being observed in new systems and with every day that passes novel applications are routinely found. This special issue is just a small snapshot of the state of the art 50 and 25 years, respectively, from their discoveries.
An introduction to topology in electrodynamics and the geometric phase con- cept is given. The form-invariance of the field-coupled Dirac equation is demon- strated, and it is shown, starting from the free Dirac equation, that demand of local phase invariance of the wave function dictates the presence of an elec- tromagnetic field. Afterwards the Dirac phase factor, Φ( C ), for a fixed-time loop is defined and discussed. It appears from The Principle of Local Phase Invariance that only the transverse part of the vector potential contributes to Φ( C ). The conditions ∇ × A T = 0 and ∇ · A T = 0 [A T being the transverse vector potential] in a certain finite spatial domain Ω do not imply that A T = 0 necessarily in Ω, as a certain integration involving A T over the domain outside Ω shows. The vector potential of a magnetic string with curvature and torsion is derived, and from this potential the result for a rectilinear magnetic string is obtained, In turn the Aharonov-Bohm effect is derived for a magnetic flux tube. Starting from the evolution operator in quantum physics an adiabatic theorem is established, and Berry's adiabatic geometric phase is introduced and discussed. An extended discussion shows that a geometric phase, called the Aharonov-Anandan phase, not restricted by the adiabatic principle can be introduced.