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Kluwer Academic Publishers eBooks

The field B[(3)]

Myron Wyn Evans, Jean Pierre Vigier

Published Jan 1, 1994• 0 citations
OpenAlex

Abstract

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.

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

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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.

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More variations on Aharonov-Bohm

A. I. Ershkovich

The article about Aharonov-Bohm effects is interesting and comprehensive. The primary and best-known effect shows that the vector potential A of the electromagnetic field is a physical reality rather than a mathematical artifice. That reality was implicit in the Schrödinger equation, as the Hamiltonian H depends on A instead of the electric field E and magnetic field B = ∇ × A. But the same statement also refers to the Hamilton-Jacobi equation in classical mechanics, so one may expect that a similar effect exists in classical physics as well.Indeed, classical mechanics is governed by the fundamental Hamilton-Jacobi equation for the action S: δS/δt + H = 0, which naturally follows from William Hamilton’s principle of least action. Both H and S depend on A (even with E = B = 0). Thus it is quite possible that the vector potential is also a physical reality in classical physics. Erwin Schrödinger arrived at his famous equation in 1926 by using a mechanics-optics analogy, the so-called eikonal equation. Hamilton in 1834 proved that eikonal and Hamilton-Jacobi equations are equivalent, so that the Schrödinger equation actually follows from Hamilton-Jacobi.Had Hamilton known that classical mechanics does not always hold, quantum mechanics might have appeared a century earlier. 1 1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . The analogy between the equations of classical and quantum mechanics provides a reason to search for experiments that might prove whether a classical analogue of the Aharonov-Bohm effect exists.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . Google Scholar© 2010 American Institute of Physics.

Physics Today•Apr 1, 2010•2 citations

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More variations on Aharonov-Bohm

Herman Batelaan, Akira Tonomura

Batelaan and Tonomura reply: Werner Ehrenberg and Raymond Siday did propose the magnetic version of what is now called the Aharonov-Bohm (AB) effect, 1 1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. as Peter Sturrock and Timothy Groves point out. We had included this reference in the early versions of our manuscript. However, limited space directed the focus of the paper to the “effect without a force” discussion, rather than a historic perspective.Alexander Ershkovich notes that in the Hamilton-Jacobi formulation of classical mechanics, both the action and the Hamiltonian depend on the vector potential; he ponders whether the AB effect might have a classical manifestation. Newton’s formulation of classical mechanics is equivalent to the Hamilton-Jacobi formulation. Because the absence of a field means the absence of a force in Newton’s formulation, classical trajectories are unaffected. That result is not expected to change in the equivalent Hamilton-Jacobi formulation. Thus the AB effect is usually considered to be a pure quantum effect. On the other hand, we may interpret “classical mechanics” in a broader sense, such as in general relativistic classical mechanics. In electrodynamics classical trajectories are not affected by a localized magnetic field through which they do not pass. However, considering that the energy content of a current-carrying solenoid is larger than that of one without current, the trajectory is clearly affected gravitationally, at least in principle. Though not due to the AB effect, that result elucidates that a generalized description may lead to other insights. The Hamilton-Jacobi equation may be an example of a theoretical vehicle by which to explore generalizations such as relativistic effects, separation of variables, multiple particle effects, or the classical limit of the de Broglie-Bohm theory. C. Alden Mead recollects interesting statements made by Bohm. We agree fully with Bohm’s statement that “it would be much more revolutionary for this effect to be wrong than for it to be right.” Attempts to disprove the AB effect should be seen for what they are, outright attempts at finding limits to the validity of quantum mechanics itself. And although quantum mechanics is unfinished with respect to, say, de-coherence theory and quantum gravity, the AB effect appears to be well within its validity range.We do not share Bohm’s astonishment that, as Mead relates, “certain physicists refused to accept the AB effect and even went to great lengths to try to disprove it.” Rather, to risk overusing a platitude, extraordinary phenomena should be exposed to extraordinary scrutiny. Failed attempts to disprove an idea often provide insight into its fundamental character. In that context, we reiterate the main message of our article. Many facets of the AB effect—for example, the electric version, the dispersionless nature, relativistic momentum conservation, the relation to the Mott-Schwinger effect, and the AB effects for other than electromagnetic gauge-invariant theories—need exploration. We predict a bright future for the AB effect, with many surprises to come.REFERENCESection:ChooseTop of pageREFERENCE <<1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. Google ScholarCrossref© 2010 American Institute of Physics.

Physics Today•Apr 1, 2010•1 citation
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.
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.
Geared toward advanced undergraduate and graduate students of physics, this text provides readers with a background in relativistic wave mechanics and prepares them for the study of field theory. The treatment originated as a series of lectures from a course on advanced quantum mechanics that has been further amplified by student contributions.An introductory section related to particles and wave functions precedes the three-part treatment. An examination of particles of spin zero follows, addressing wave equation, Lagrangian formalism, physical quantities as mean values, translation and rotat
The article about Aharonov-Bohm effects is interesting and comprehensive. The primary and best-known effect shows that the vector potential A of the electromagnetic field is a physical reality rather than a mathematical artifice. That reality was implicit in the Schrödinger equation, as the Hamiltonian H depends on A instead of the electric field E and magnetic field B = ∇ × A. But the same statement also refers to the Hamilton-Jacobi equation in classical mechanics, so one may expect that a similar effect exists in classical physics as well.Indeed, classical mechanics is governed by the fundamental Hamilton-Jacobi equation for the action S: δS/δt + H = 0, which naturally follows from William Hamilton’s principle of least action. Both H and S depend on A (even with E = B = 0). Thus it is quite possible that the vector potential is also a physical reality in classical physics. Erwin Schrödinger arrived at his famous equation in 1926 by using a mechanics-optics analogy, the so-called eikonal equation. Hamilton in 1834 proved that eikonal and Hamilton-Jacobi equations are equivalent, so that the Schrödinger equation actually follows from Hamilton-Jacobi.Had Hamilton known that classical mechanics does not always hold, quantum mechanics might have appeared a century earlier. 1 1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . The analogy between the equations of classical and quantum mechanics provides a reason to search for experiments that might prove whether a classical analogue of the Aharonov-Bohm effect exists.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA (1980), p.484 . Google Scholar© 2010 American Institute of Physics.
Batelaan and Tonomura reply: Werner Ehrenberg and Raymond Siday did propose the magnetic version of what is now called the Aharonov-Bohm (AB) effect, 1 1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. as Peter Sturrock and Timothy Groves point out. We had included this reference in the early versions of our manuscript. However, limited space directed the focus of the paper to the “effect without a force” discussion, rather than a historic perspective.Alexander Ershkovich notes that in the Hamilton-Jacobi formulation of classical mechanics, both the action and the Hamiltonian depend on the vector potential; he ponders whether the AB effect might have a classical manifestation. Newton’s formulation of classical mechanics is equivalent to the Hamilton-Jacobi formulation. Because the absence of a field means the absence of a force in Newton’s formulation, classical trajectories are unaffected. That result is not expected to change in the equivalent Hamilton-Jacobi formulation. Thus the AB effect is usually considered to be a pure quantum effect. On the other hand, we may interpret “classical mechanics” in a broader sense, such as in general relativistic classical mechanics. In electrodynamics classical trajectories are not affected by a localized magnetic field through which they do not pass. However, considering that the energy content of a current-carrying solenoid is larger than that of one without current, the trajectory is clearly affected gravitationally, at least in principle. Though not due to the AB effect, that result elucidates that a generalized description may lead to other insights. The Hamilton-Jacobi equation may be an example of a theoretical vehicle by which to explore generalizations such as relativistic effects, separation of variables, multiple particle effects, or the classical limit of the de Broglie-Bohm theory. C. Alden Mead recollects interesting statements made by Bohm. We agree fully with Bohm’s statement that “it would be much more revolutionary for this effect to be wrong than for it to be right.” Attempts to disprove the AB effect should be seen for what they are, outright attempts at finding limits to the validity of quantum mechanics itself. And although quantum mechanics is unfinished with respect to, say, de-coherence theory and quantum gravity, the AB effect appears to be well within its validity range.We do not share Bohm’s astonishment that, as Mead relates, “certain physicists refused to accept the AB effect and even went to great lengths to try to disprove it.” Rather, to risk overusing a platitude, extraordinary phenomena should be exposed to extraordinary scrutiny. Failed attempts to disprove an idea often provide insight into its fundamental character. In that context, we reiterate the main message of our article. Many facets of the AB effect—for example, the electric version, the dispersionless nature, relativistic momentum conservation, the relation to the Mott-Schwinger effect, and the AB effects for other than electromagnetic gauge-invariant theories—need exploration. We predict a bright future for the AB effect, with many surprises to come.REFERENCESection:ChooseTop of pageREFERENCE <<1. A. Tonomura, Electron Holography, 2nd ed., Springer, New York (1999), p. 63. Google ScholarCrossref© 2010 American Institute of Physics.