Topological Foundations of Electromagnetism: tt.jpg

Topological Foundations of Electrodynamics (World Scientific Series in Contemporary Chemical Physics)
By Terence W. Barrett




Publisher: World Scientific Publishing Company
Number Of Pages: 196
Publication Date: 2008-03-13
ISBN-10 / ASIN: 9812779965
ISBN-13 / EAN: 9789812779960
Binding: Hardcover



Topological Foundations of Electromagnetism seeks a fundamental understanding of the dynamics of electromagnetism; and marshals the evidence that in certain precisely defined topological conditions, electromagnetic theory (Maxwell’s theory) must be extended or generalized in order to provide an explanation and understanding of, until now, unusual electromagnetic phenomena. Key to this generalization is an understanding of the circumstances under which the so-called A potential fields have physical effects. Basic to the approach taken is that the topological composition of electromagnetic fields is the fundamental conditioner of the dynamics of these fields. The treatment of electromagnetism from, first, a topological perspective, continuing through group theory and gauge theory, to a differential calculus description is a major thread of the book. Suggestions for potential new technologies based on this new understanding and approach to conditional electromagnetism are also given.

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter 1: Electromagnetic Phenomena Not Explained
by Maxwell’s Equations 1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Prolegomena A: Physical Effects Challenging a Maxwell
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 3
Prolegomena B: Interpretation of Maxwell’s Original
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 6
B.1. The Faraday–Maxwell formulation . . . . . . . . . 6
B.2. The British Maxwellians and the Maxwell–
Heaviside formulation . . . . . . . . . . . . . . . . 7
B.3. The Hertzian and current classical formulation . . . 9
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. What is a Gauge? . . . . . . . . . . . . . . . . . . . . . . 16
3. Empirical Reasons for Questioning the Completeness
of Maxwell’s Theory . . . . . . . . . . . . . . . . . . . . 18
3.1. Aharonov–Bohm (AB) and Altshuler–Aronov–
Spivak (AAS) effects . . . . . . . . . . . . . . . . . 18
3.2. Topological phases: Berry, Aharonov–
Anandan, Pancharatnam and Chiao–Wu phase
rotation effects . . . . . . . . . . . . . . . . . . . . 27
3.3. Stokes’ theorem re-examined . . . . . . . . . . . . . 36
3.4. Properties of bulk condensed matter —
Ehrenberg and Siday’s observation . . . . . . . . . . 38
3.5. The Josephson effect . . . . . . . . . . . . . . . . . 39
vii
viii Topological Foundations to Electromagnetism
3.6. The quantized Hall effect . . . . . . . . . . . . . . . 42
3.7. The de Haas–van Alphen effect . . . . . . . . . . . 45
3.8. The Sagnac effect . . . . . . . . . . . . . . . . . . . 46
3.9. Summary . . . . . . . . . . . . . . . . . . . . . . . 49
4. Theoretical Reasons for Questioning the Completeness of
Maxwell’s Theory . . . . . . . . . . . . . . . . . . . . . . 50
5. Pragmatic Reasons for Questioning the Completeness of
Maxwell’s Theory . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Harmuth’s ansatz . . . . . . . . . . . . . . . . . . . 56
5.2 Conditioning the electromagnetic field into altered
symmetry: Stokes’ interferometers and Lie algebras 60
5.3 Non-Abelian Maxwell equations . . . . . . . . . . . 70
6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 74
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter 2: The Sagnac Effect: A Consequence
of Conservation of Action Due to Gauge Field Global
Conformal Invariance in a Multiply Joined Topology
of Coherent Fields 95
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
1. Sagnac Effect Phenomenology . . . . . . . . . . . . . . . 96
1.1. The kinematic description . . . . . . . . . . . . . . 98
1.2. The physical–optical description . . . . . . . . . . . 101
1.3. The dielectric metaphor description . . . . . . . . . 105
1.4. The gauge field explanation . . . . . . . . . . . . . 106
2. The Lorentz Group and the Lorenz Gauge Condition . . . 115
3. The Phase Factor Concept . . . . . . . . . . . . . . . . . 116
3.1. SU(2) group algebra . . . . . . . . . . . . . . . . . . 118
3.2. A short primer on topological concepts . . . . . . . 122
4. Minkowski Space–Time Versus Cartan–Weyl Form . . . . 129
5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 134
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Contents ix
Chapter 3: Topological Approaches to
Electromagnetism 141
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
1. Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
2. Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3. Polarization Modulation Over a Set Sampling Interval . . 156
4. The Aharonov–Bohm Effect . . . . . . . . . . . . . . . . 168
5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 181

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Topological Foundations of Electromagnetism

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