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CRC Press eBooks

The Wave Theory of Geometric Phase

Nicolae Mazilu, Maricel Agop, Ioan Mercheş

Published Sep 12, 2019• 0 citations
Publisher

Abstract

The wave mechanics, with its quantum amendment, brings an important point to fore: a specific intervention of the geometry, in fact, of the differential geometry. The main result of Sir Michael Berry’s work on the geometry required by the existence of arbitrary phase factors makes its mark, showing that the nonstationary Schrödinger equation means much more than usually thought, when taken from the perspective of fundamental knowledge. There is a ‘Berry moment’ of the human knowledge, which carries, among others, the precise meaning of liberating the knowledge from the lock-up of adiabaticity, and this is reflected in the general attributes of this landmark of physics. The electromagnetic fields are, in general, Yang-Mills fields to be described in a geometry of the manifold of positions of Kepler orbits of classical material points. That geometry is a geometry of sl(2,R) Riemannian space. A Hertz material particle can be characterized exactly in the manner in which Schrödinger describes the color continuum.

DOI
10.1201/9780429329050-6

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Topological and geometric phases

Helmut Rauch

Abstract Topological and geometrical effects appear in the solution of the Schrödinger equation due to special geometrical forms of the interaction, or due to interactions which do not result from conservative forces acting on the system (see Section 1.2). A conscientious reader may point out that all these effects could be entirely explained using the Schrödinger equation only. But the geometrical phase formalism is useful because it sometimes plays a role in fundamental quantum phenomena and it shows that additional phase factors exist which are easily overlooked, making the interpretation of measured effects more mysterious than necessary. Thus, our understand-ing of quantum mechanics has deepened since the existence of geometrical phases was explicitly revealed (Shapere and Wilczek 1989). This happened due to the seminal works of Pancharatnam (1956), Berry (1984), Anandan (1992), and Bhandari (1997). It also shows that a wavefunction often carries much more information than is usually extracted in a standard experiment. A typical example is the assumption of adiabaticity, where the magnetic moment of the neutron produces (inside a slowly varying magnetic field) a dynamical phase shift proportional to the path integral ∫ B ds, but when the direction of the field varies slowly an additional, i.e. geometrical, phase also appears. The back reaction of a quantum measurement can also be described as a kind of an induced topological phase (Aharonov et al. 1998).

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During the last few years, considerable interest has been focused on the phase that waves accumulate when the equations governing the waves vary slowly. The recent flurry of activity was set off by a paper by Michael Berry, where it was found that the adiabatic evolution of energy eigenfunctions in quantum mechanics contains a phase of geometric origin (now known as 'Berry's phase') in addition to the usual dynamical phase derived from Schrödinger's equation. This observation, though basically elementary, seems to be quite profound. Phases with similar mathematical origins have been identified

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The geometric phase is the comparison the of a physical quantity due to the changes on a closed cruve in the system state space. These phases are experimentally measurable because of, that the space of a time dependent quantum system may be a curveture, therefore the Holonomy arises from this curveture. The Berry phase has three key features, which makes it very important. First, the Berry phase is gauge invariance. Especial functions are defined by aequal linear equation, hence it has a multiplicative guage freedom in a general phase factor independent of the parameter. Berry phase (forget from on phase factor ) with this phase factor remains unchanged provided that the special function is on a single loop value. This makes the Berry phase to be a physical quantity that is experimentally measureable by inter ference phenomena. Secondly, Berry phase is a geometric concept. that’s mean it can be written as a linear integral of loop in parameter space, which does not depend on the rate of change on the loop. This property makes it mesureable the Berry phase in terms of local geometric quantites in the parameter space. In fact, Berry showed that this phase can be written as a find integral (Berry curve) on the surface. Berry curve plays an essential role in effective dynamics with variables with degrees of slow release. Third, the Berry phase is similar to that of guage field and differential geometry theory. This makes beauty, intuition and the same concept in physics.

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Geometric quantum phase and angles

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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.

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Mathematical Background

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The mathematical background required for the study of geometric phases in classical and quantum mechanics is rather extensive. The aim of this introductory chapter is to provide a background of some basic notions of classical differential geometry and topology. Classical differential geometry is now a well established tool in modern theoretical physics. Many classical theories like mechanics, electrodynamics, Einstein’s General Relativity or Yang-Mills gauge theories are well known examples where the geometrical methods enter in the natural and very effective way. As we shall see throughout this book, also quantum physics shows its intricate beauty when one applies an appropriate geometric framework. All this proves Wigner’s celebrated statement about the “unreasonable effectiveness” of mathematics in natural sciences. 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.

Birkhäuser Boston eBooks•Jan 1, 2004•1 citation

Coherence Quantum Numbers, Holonomy Geometry, and the Coherence Resonance Condition Toward an Emergent Geometric Origin of Quantum States and Particle Stability

Borros Arneth

Quantum mechanics is traditionally formulated on complex Hilbert spaces, while geometric structures are introduced only after the quantum formalism has been established. In this work, we investigate a coherence-geometric reformulation in which geometry is fundamental and Hilbert space emerges as a derived structure. The framework is based on a coherence manifold with local coordinates , where denotes radial coherence and denotes holonomy phase. The wavefunction arises naturally as , and the continuity equation, quantum Hamilton–Jacobi equation, and Schrödinger equation emerge from the differential geometry of the coherence manifold. A second ingredient is a discrete coherence spectrum characterized by quadruples . These integers are interpreted as coherence quantum numbers rather than phenomenological fitting parameters. The resulting framework introduces a coherence operator whose spectrum is determined by a rank-four coherence lattice. We further propose a Coherence Resonance Condition in which stable particle states correspond to rational phase locking between local coherence oscillations and global holonomy cycles. The framework combines concepts from geometric quantum mechanics, geometric quantization, Berry holonomies, Yang–Mills gauge theory, spectral geometry, and representation theory within a common geometric language.

•Jun 13, 2026•0 citations
The wave mechanics, with its quantum amendment, brings an important point to fore: a specific intervention of the geometry, in fact, of the differential geometry. The main result of Sir Michael Berry’s work on the geometry required by the existence of arbitrary phase factors makes its mark, showing that the nonstationary Schrödinger equation means much more than usually thought, when taken from the perspective of fundamental knowledge. There is a ‘Berry moment’ of the human knowledge, which carries, among others, the precise meaning of liberating the knowledge from the lock-up of adiabaticity, and this is reflected in the general attributes of this landmark of physics. The electromagnetic fields are, in general, Yang-Mills fields to be described in a geometry of the manifold of positions of Kepler orbits of classical material points. That geometry is a geometry of sl(2,R) Riemannian space. A Hertz material particle can be characterized exactly in the manner in which Schrödinger describes the color continuum.
Abstract Topological and geometrical effects appear in the solution of the Schrödinger equation due to special geometrical forms of the interaction, or due to interactions which do not result from conservative forces acting on the system (see Section 1.2). A conscientious reader may point out that all these effects could be entirely explained using the Schrödinger equation only. But the geometrical phase formalism is useful because it sometimes plays a role in fundamental quantum phenomena and it shows that additional phase factors exist which are easily overlooked, making the interpretation of measured effects more mysterious than necessary. Thus, our understand-ing of quantum mechanics has deepened since the existence of geometrical phases was explicitly revealed (Shapere and Wilczek 1989). This happened due to the seminal works of Pancharatnam (1956), Berry (1984), Anandan (1992), and Bhandari (1997). It also shows that a wavefunction often carries much more information than is usually extracted in a standard experiment. A typical example is the assumption of adiabaticity, where the magnetic moment of the neutron produces (inside a slowly varying magnetic field) a dynamical phase shift proportional to the path integral ∫ B ds, but when the direction of the field varies slowly an additional, i.e. geometrical, phase also appears. The back reaction of a quantum measurement can also be described as a kind of an induced topological phase (Aharonov et al. 1998).
During the last few years, considerable interest has been focused on the phase that waves accumulate when the equations governing the waves vary slowly. The recent flurry of activity was set off by a paper by Michael Berry, where it was found that the adiabatic evolution of energy eigenfunctions in quantum mechanics contains a phase of geometric origin (now known as 'Berry's phase') in addition to the usual dynamical phase derived from Schrödinger's equation. This observation, though basically elementary, seems to be quite profound. Phases with similar mathematical origins have been identified
The geometric phase is the comparison the of a physical quantity due to the changes on a closed cruve in the system state space. These phases are experimentally measurable because of, that the space of a time dependent quantum system may be a curveture, therefore the Holonomy arises from this curveture. The Berry phase has three key features, which makes it very important. First, the Berry phase is gauge invariance. Especial functions are defined by aequal linear equation, hence it has a multiplicative guage freedom in a general phase factor independent of the parameter. Berry phase (forget from on phase factor ) with this phase factor remains unchanged provided that the special function is on a single loop value. This makes the Berry phase to be a physical quantity that is experimentally measureable by inter ference phenomena. Secondly, Berry phase is a geometric concept. that’s mean it can be written as a linear integral of loop in parameter space, which does not depend on the rate of change on the loop. This property makes it mesureable the Berry phase in terms of local geometric quantites in the parameter space. In fact, Berry showed that this phase can be written as a find integral (Berry curve) on the surface. Berry curve plays an essential role in effective dynamics with variables with degrees of slow release. Third, the Berry phase is similar to that of guage field and differential geometry theory. This makes beauty, intuition and the same concept in physics.
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.
The mathematical background required for the study of geometric phases in classical and quantum mechanics is rather extensive. The aim of this introductory chapter is to provide a background of some basic notions of classical differential geometry and topology. Classical differential geometry is now a well established tool in modern theoretical physics. Many classical theories like mechanics, electrodynamics, Einstein’s General Relativity or Yang-Mills gauge theories are well known examples where the geometrical methods enter in the natural and very effective way. As we shall see throughout this book, also quantum physics shows its intricate beauty when one applies an appropriate geometric framework. All this proves Wigner’s celebrated statement about the “unreasonable effectiveness” of mathematics in natural sciences. 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.
Quantum mechanics is traditionally formulated on complex Hilbert spaces, while geometric structures are introduced only after the quantum formalism has been established. In this work, we investigate a coherence-geometric reformulation in which geometry is fundamental and Hilbert space emerges as a derived structure. The framework is based on a coherence manifold with local coordinates , where denotes radial coherence and denotes holonomy phase. The wavefunction arises naturally as , and the continuity equation, quantum Hamilton–Jacobi equation, and Schrödinger equation emerge from the differential geometry of the coherence manifold. A second ingredient is a discrete coherence spectrum characterized by quadruples . These integers are interpreted as coherence quantum numbers rather than phenomenological fitting parameters. The resulting framework introduces a coherence operator whose spectrum is determined by a rank-four coherence lattice. We further propose a Coherence Resonance Condition in which stable particle states correspond to rational phase locking between local coherence oscillations and global holonomy cycles. The framework combines concepts from geometric quantum mechanics, geometric quantization, Berry holonomies, Yang–Mills gauge theory, spectral geometry, and representation theory within a common geometric language.