Historical research guide · 0657 · 2006
The Superposition Principle of Waves Not Fulfilled under M. W. Evans' O(3) Hypothesis
Erhard Wielandt
In one minute
Erhard Wielandt, a geophysicist at Stuttgart, takes two pages of algebra to test a specific proposal about light. Since 1992 Myron Evans had argued that every circularly polarised electromagnetic wave carries, besides its two ordinary transverse components, a third constant field pointing along the direction of travel — the B(3) or ghost field — and that the three are locked together by a cyclic relation he treated as a new law of electromagnetism. Wielandt asks whether that survives the most-checked property electromagnetic waves have: linear superposition, the principle that makes interference work. He adds two copies of the same wave shifted in phase by plus and minus alpha. The transverse parts add up to the original wave scaled by two times the cosine of alpha, as they must; the longitudinal part adds up to twice itself, unscaled. The two do not track, and the only value that keeps them consistent is the classical one, with no longitudinal field at all.
Connection
Chapter 10 is where this site takes longitudinal structure, scalar waves and the vector potential seriously, so it needs the constraint every such proposal has to clear stated exactly once and stated cleanly: whatever you add to a plane wave has to add the way waves add, because interference measures that directly. Chapter 1 gets a small model of the evidence ladder working — a claim, a two-page test with an unambiguous outcome, and a named measurement that decides it.
From the source
The hypothesis under test, stated in the author’s own terms. Evans writes a circularly polarised plane wave travelling along z in a complex circular basis, giving two transverse magnetic components carrying the phase factors for plus and minus the electromagnetic phase, and adds a third component that is a constant vector along the direction of propagation. The three satisfy cyclic relations in which each pair’s cross product returns the conjugate of the third, and the B Cyclic Theorem is the statement that every plane circularly polarised wave is accompanied by that longitudinal field.Section 1, M. W. Evans’ O(3) hypothesis; Equations [1.43] and [1.44]
The test is superposition, and the reason it is the right test is that superposition is one of the best-checked laws of electrodynamics: it is what interference measures, and its mathematical equivalent is the statement that electric and magnetic fields are vectors, which add.Abstract; Section 1, opening paragraphs
The arithmetic. Wielandt introduces a coefficient that is one under the O(3) hypothesis and zero in classical electrodynamics, then superposes two waves whose phases are shifted by plus and minus alpha, with the cosine of alpha smaller than one. The two transverse components combine to give the original wave multiplied by twice the cosine of alpha. The longitudinal components, being constants, simply add to twice the coefficient times the original amplitude — with no cosine factor. Under the hypothesis the sum should have been accompanied by a longitudinal field scaled by that same cosine, so the two results disagree.Section 2, Checking the superposition property of the O(3) hypothesis; Equations (1) to (8)
The conclusion the algebra forces: only the classical value of the coefficient, zero, is compatible with the superposition principle. Wielandt states it as a single line — the cyclical O(3) hypothesis is incompatible with the superposition principle of waves.Section 2, closing statement
A corollary about wave packets, offered as a remark without detailed proof. The transverse components of a circularly polarised wave oscillate in space and time, so a Fourier sum of them can be arranged to cancel outside a chosen region — that is how one builds a localised packet. A longitudinal component that is a constant vector for every Fourier term cannot do that: summing constant vectors returns another constant vector. So under the hypothesis the longitudinal field of a wave packet must either vanish or fill the whole universe at all times, which amplifies an objection Akhlesh Lakhtakia had already raised in 1995.Section 2, Remark without detailed proof
The way in
https://arxiv.org/abs/physics/0607262How to cite it
Erhard Wielandt (2006) The Superposition Principle of Waves Not Fulfilled under M. W. Evans' O(3) Hypothesis doi:10.1088/0031-8949/74/5/009