Historical research guide · 0790 · 2008
Topological Foundations of Electromagnetism
Terence W. Barrett
In one minute
Maxwell’s equations are the most useful piece of nineteenth-century physics still in daily service, and Terence Barrett’s book is not an attack on them. It is an argument that the familiar form of them is a special case: the case where the region the field occupies has no holes, no twists and no imposed structure. Barrett collects the electromagnetic effects that the familiar form does not account for, and asks what has to be added to cover them. His answer is topology and a larger gauge symmetry — keep track of the shape of the space the field lives in, and of the potentials rather than only the force fields, and effects that looked anomalous become ordinary consequences. The Aharonov-Bohm effect, in which an electron beam responds to a magnetic field it never touches, is the standing laboratory proof that potential and topology do real physical work. The Sagnac effect of the rotating interferometer gets the same treatment, chapter by chapter.
Connection
Chapter 10 is about what an electromagnetic field can do when you control its structure rather than its strength, and this book is the mathematics under that idea — the vector potential and the topology of the field as physical quantities. Chapter 4 gets its lever, because the same author is the co-author who took this formalism to Froning’s proposal that a specially conditioned field could couple to the metric, and chapter 13 gets a candidate for how electromagnetism joins the rest of the picture.
From the source
The book’s premise is stated in the title of its longest chapter: there are electromagnetic phenomena not explained by Maxwell’s equations in their conventional form. The conventional form is treated as a special case rather than as the whole of the theory.Chapter 1, Electromagnetic Phenomena Not Explained by Maxwell’s Equations, pages 1 to 94
The topology of the region a field occupies is physically consequential: whether that region is simply or multiply connected changes what the field does, which is why the potentials carry physical meaning and not merely computational convenience. The Aharonov-Bohm effect, where an electron beam responds to a magnetic flux it is shielded from, is the standing laboratory instance.Chapter 3, Topological Approaches to Electromagnetism, pages 141 to 181
The Sagnac effect — the fringe shift seen in a rotating interferometer — is presented as a consequence of conservation of action arising from gauge field global conformal invariance in a multiply joined topology of coherent fields. A settled interferometric measurement is given a topological rather than a kinematic explanation.Chapter 2, pages 95 to 140, chapter title and treatment
Running through the volume is Barrett’s extension of Maxwell theory from the usual U(1) gauge symmetry to a higher SU(2) symmetry, which is the piece of mathematics his collaborators name when they describe what specially conditioned electromagnetic fields are supposed to be.Chapters 1 and 3, pages 1 to 94 and 141 to 181, applied in Froning and Barrett 1998 and in Froning, Barrett and Miley 2012
What to watch is the experiment that the applied side of this work names: condition a beam so that it carries the higher gauge symmetry, then measure an effect on gravitation, on the vacuum state or on a fusion cross-section that tracks the conditioning rather than the delivered power. Barrett supplies the formalism; the measurement is the open item.Read against Froning and Barrett, AIP Conference Proceedings 420, pages 1449 to 1454, 1998
The way in
https://doi.org/10.1142/6693How to cite it
Terence W. Barrett (2008) Topological Foundations of Electromagnetism doi:10.1142/6693