Topological Foundations of Electromagnetism By Terence W Barrett

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Topological Foundations of Electromagnetism By Terence W Barrett

Preface

Maxwell’s equations are foundational to electromagnetic theory. They
are the cornerstone of a myriad of technologies and are basic to the
understanding of innumerable effects. Yet there are a few effects or
phenomena that cannot be explained by the conventional Maxwell
theory. This book examines those anomalous effects and shows that
they can be interpreted by a Maxwell theory that is subsumed under
gauge theory. Moreover, in the case of these few anomalous effects,
and when Maxwell’s theory finds its place in gauge theory, the conventional
Maxwell theory must be extended, or generalized, to a non-
Abelian form.
The tried-and-tested conventional Maxwell theory is of Abelian
form. It is correctly and appropriately applied to, and explains, the
great majority of cases in electromagnetism. What, then, distinguishes
these cases from the aforementioned anomalous phenomena? It is
the thesis of this book that it is the topology of the spatiotemporal
situation that distinguishes the two classes of effects or phenomena,
and the topology that is the final arbiter of the correct choice of group
algebra — Abelian or non-Abelian — to use in describing an effect.
Therefore, the most basic explanation of electromagnetic phenomena
and their physical models lies not in differential calculus or group
theory, useful as they are, but in the topological description of the
(spatiotemporal) situation. Thus, this book shows that only after the
topological description is provided can understanding move to an
appropriate and now-justified application of differential calculus or
group theory.

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
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
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