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CRC Press eBooks

From Gauge-Photon Wave Mechanics to the Magnetostatic Aharonov-Bohm Vector Potential

Ole Keller

Published Aug 6, 2024• 0 citations
Publisher

Abstract

Elements of a theory which allows one to connect gauge-photon wave mechan- ics to the magnetostatic Aharonov-Bohm vector potential are outlined. In a first step a relativistic calculation gives one the correct identification of the field momentum and energy in classical electrodynamics. By working with manifest covariant expressions, the conservation law for the four-momentum is given on integral form. Only in the special Lorentz frame where the relativistic factor γ = [1 − ( v/c )2] − 1 / 2 equals γ = 1 the integral form reduce to the textbook integrals. In the rest frame of a charge we discuss aspects of photon wave mechanics based on the analytic part of the transverse vector potential, (A (+) ). Hamiltonian-like dynamical evolution equations for the two helicity species of A (+) represent together in bispinor form a theory which holds in a dynamical T (+)(+) structural vacuum, i.e. for E T ± ic B = 0. On the basis of the basic dynamical equations a dynmical version (extension) of the Aharonov-Bohm effect is derived for a linear magnetic tube (often called a string). From a microscopic point of view the string may be a ferrmagnet in which a running (or standing) spin wave is present on top of the time independent ferromagnetic magneti- zation. In the magnetostatic limit the well-known Aharonov-Bohm transverse vector potential is obtained. Chapters 9 – 12 constitute the books SECTION III , a section dealing with conservation laws, Lagrangians, geometric phases and topology.

DOI
10.1201/9781003029458-9

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Elements of a theory which allows one to connect gauge-photon wave mechan- ics to the magnetostatic Aharonov-Bohm vector potential are outlined. In a first step a relativistic calculation gives one the correct identification of the field momentum and energy in classical electrodynamics. By working with manifest covariant expressions, the conservation law for the four-momentum is given on integral form. Only in the special Lorentz frame where the relativistic factor γ = [1 − ( v/c )2] − 1 / 2 equals γ = 1 the integral form reduce to the textbook integrals. In the rest frame of a charge we discuss aspects of photon wave mechanics based on the analytic part of the transverse vector potential, (A (+) ). Hamiltonian-like dynamical evolution equations for the two helicity species of A (+) represent together in bispinor form a theory which holds in a dynamical T (+)(+) structural vacuum, i.e. for E T ± ic B = 0. On the basis of the basic dynamical equations a dynmical version (extension) of the Aharonov-Bohm effect is derived for a linear magnetic tube (often called a string). From a microscopic point of view the string may be a ferrmagnet in which a running (or standing) spin wave is present on top of the time independent ferromagnetic magneti- zation. In the magnetostatic limit the well-known Aharonov-Bohm transverse vector potential is obtained. Chapters 9 – 12 constitute the books SECTION III , a section dealing with conservation laws, Lagrangians, geometric phases and topology.
Abstract A local dynamical evolution equation for a two-spinor photon wave function is established. The vectorial spinor components are those of the two helicity components A T , ± ( + ) of the analytical part of the gauge invariant transverse vector potential. The theory’s Hamilton operator is spatially local and without singularities. Thereby previously established field-based formulations leading to various nonlocal integro-differential equations for the dynamical evolution of the photon wave function are simplified. The dynamical evolution equations for the spinor components can be written in the compact form F ± ( + ) ( r , t ) = 0 , where F ± ( + ) ( r , t ) are the analytical Riemann–Silberstein–Oppenheimer–Bialynicki vectors. The theory after extension to the QED level, allows one to describe the dynamical time evolution of single-photon wave packets. The bridge between the wave mechanical and the QED formulations is an operationally suitable mean position state | R ⟩ ( r , t ) for a transverse photon in Hilbert space. The inner product ⟨ R ( r , t ) ⟩ R ( r , t ) relates to our inability to localize a photon in space. The normalization condition for the two-spinor wave packet is established. A dynamical evolution equation describing the photon-source entanglement is established starting from the Heisenberg equation of motion for the annihilation operators associated to the two helicity species. We show how each released photon is labelled by the transverse current density operator dynamics of the charged source particles. The basic theory is applied to a study of single-photon emission from a rectlinear magnetic dipole tube (string) carrying a rotationally symmetric surface current. Numerical results are obtained for photon wave train emission. The transition from the photon wave mechanical formulation to the magnetostatic Aharonov–Bohm vector potential field is established. The generality of the A T , ± ( + ) -thoery is underlined linking it to the Minimal Coupling Principle, magnetic monopole electrodynamics, spatial photon localization, the perfe
Preface * 1: Wave and Particle * 2: Fundamental Symmetries * 3: The Origins of Wave Mechanics * 4: Interrelation of Field Equations * 5: Transverse and Longitudinal Photons and Fields * 6: Creation and Annihilation of Photons * 7: Experimental Evidence for Ii * 8: The Concept of Photon Mass * 9: Aharonov-Bohm Effects * 10: Modifications of Lagrangian Field Theory * 11: Pseudo Four-Vector Representations of Electric and Magnetic Fields * 12: Derivation of Ii3 from the Relativistic Hamilton-Jacobi Equation of e in Amu * Appendices * References * Index.
The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts; the one is the so-called transverse part represented by a divergenceless vector and the other the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when we use the Helmholtz theorem are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, we choose here the Aharonov-Bohm effect. As for the Aharonov-Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z-axis. We show that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is uniquely determined by the magnetic field and does not alter by a gauge transformation, on the other hand, the longitudinal part depends on the choice of special gauge. We show that the Aharonov-Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.
Abstract The extended Aharonov–Bohm electrodynamics has a simple formal structure and allows to couple the e.m. field also to currents which are not locally conserved, like those resulting from certain non-local effective quantum models of condensed matter. As it often happens in physics and mathematics when one tries to extend the validity of some equations or operations, new perspectives emerge in comparison to Maxwell theory, and some “exotic” phenomena are predicted. For the Aharonov–Bohm theory the main new feature is that the potentials AμA^\mu Aμ A μ become univocally defined and can be measured with probes in which the “extra-current” I=∂μjμI=\partial _\mu j^\mu I=∂μ​jμ I = ∂ μ j μ is not zero at some points. As a consequence, it is possible in principle to detect pure gauge-waves with E=B=0{\textbf{E}}={\textbf{B}}=0E=B=0 E = B = 0 , which would be regarded as non-physical in the Maxwell gauge-invariant theory with local current conservation. We discuss in detail the theoretical aspects of this phenomenon and propose an experimental realization of the detectors. A full treatment of wave propagation in anomalous media with extra-currents and of energy–momentum balance issues is also given.
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.
The Aharonov-Bohm (AB) effect is considered in the context of Generalized Electrodynamics (GE) by Podolsky and Bopp. GE is the only extension to Maxwell electrodynamics that is locally U(1)-gauge invariant, admits linear field equations and contains higher-order derivatives of the vector potential. GE admits both massless and massive modes for the photon. We recover the ordinary quantum phase shift of the AB effect, derived in the context of Maxwell electrodynamics, for the massless mode of the photon in GE. The massive mode induces a correction factor to the AB phase shift depending on the photon mass. We study both the magnetic AB effect and its electric counterpart. In principle, accurate experimental observations of AB the phase shift could be used to constrain GE photon mass.