The Superposition Principle of Waves Not Fulfilled
under M.W. Evans' O(3) Hypothesis

Erhard Wielandt
Institute of Geophysics, Stuttgart University, Germany

ew@geophys.uni-stuttgart.de, e.wielandt@t-online.de

1. M.W. Evans' O(3) hypothesis

In the following text quotations from M.W. Evans' GCUFT book [1] appear with equation labels [1.nn] at the left margin.

The assertion of O(3) symmetry is at the center of M.W. Evans' considerations since 1992. He claims that each plane circularly-polarized electromagnetic wave is accompanied by a constant longitudinal field B⁽³⁾, a so-called "ghost field". In addition to numerous papers, with far reaching implications as e.g. in [2] and [3], M.W. Evans is author of five books on "The Enigmatic Photon" dealing with the claimed O(3)-symmetry of electromagnetic fields. His hypotheses have met many objections. The reader will find a historical overview written by A. Lakhtakia in (5; Sect.5].

M.W. Evans considers a circularly polarized plane electromagnetic wave propagating in z-direction, cf. [1; Chap.1.2]. Using the electromagnetic phase

[1.38]                 Φ = ω t − κ z ,

where κ = ω /c , he describes the wave relative to his complex circular basis [1.41] derived from the cartesian basis vectors i, j, k. The magnetic field is given as

[1.43/1]                 B⁽¹⁾ = ¹/_sqr(2) B⁽⁰⁾ (i − i j) e^{i Φ},

[1.43/2]                 B⁽²⁾ = ¹/_sqr(2) B⁽⁰⁾ (i + i j) e^{− i Φ} ,

[1.43/3]                 B⁽³⁾ = B⁽⁰⁾ k ,

and satisfies the "cyclic O(3) symmetry relations"                

[1.44/1]                 B⁽¹⁾ × B⁽²⁾ = i B⁽⁰⁾ B⁽³⁾* ,

[1.44/2]                 B⁽²⁾ × B⁽³⁾ = i B⁽⁰⁾ B⁽¹⁾* ,

[1.44/3]                 B⁽³⁾ × B⁽¹⁾ = i B⁽⁰⁾ B⁽²⁾* .

Especially equ.[1.43/3] defines the "ghost field" B⁽³⁾ which is coupled by the relations [1.44] with the transversal components B⁽¹⁾ and B⁽²⁾ .

M.W. Evans' Cyclic Theorem is the statement that each plane circularly polarized wave [1.43/1-2] is accompanied by a longitudinal field [1.43/3], and the associated fields fulfil the cyclic equations [1.44]. M.W. Evans considers this O(3) hypothesis as a Law of Physics.

2. Checking the superposition property of the O(3) hypothesis

Instead of [1.38] we consider a phase shifted wave with the more general phase function

(2.1)                 Φ_α(t,z) = ω t - κ z + α = Φ(t,z) + α

which can be understood as a time shifted wave where the time shift is t_o := - ^α/_ω:

(2.2)                 Φ_α(t,z) = Φ(t-t_o,z) .

We use the phase Φ_α in [1.43] to obtain the time-shifted magnetic field

(2.3)                 B⁽¹⁾ = ¹/_sqr(2) B⁽⁰⁾ (i−ij) e^i(Φ+α) ,

(2.4)                 B⁽²⁾ = ¹/_sqr(2) B⁽⁰⁾ (i+ij) e^−i(Φ+α),

(2.5)                 B⁽³⁾ = γ B⁽⁰⁾ k ,

where we have introduced a coefficient γ that should equal 1 following M.W. Evans while in classical electrodynamics γ =0 .

Now we consider the wave generated by the superposition of two waves with the phase functions Φ_α and Φ_−α , respectively, and α such that cos α < 1 . According to the superposition principle the total field is then

(2.6)                 B⁽¹⁾ = ¹/_sqr(2) B⁽⁰⁾ (i−ij) [e^{i (Φ+α)} + e^{i (Φ−α)}] = ¹/_sqr(2) B⁽⁰⁾ (i−ij) e^{i Φ} 2 cos α ,

(2.7)                 B⁽²⁾ = ¹/_sqr(2) B⁽⁰⁾ (i+ij) [e^−i(Φ+α) + e^{−i(Φ−α)}] = ¹/_sqr(2) B⁽⁰⁾ (i+ij) e^−iΦ 2 cos α ,

(2.8)                 B⁽³⁾ = 2 γ B⁽⁰⁾ k .

Considering the first two (transversal) components we recognize that the superposition yields the original wave [1.43/1-2] multiplied by the factor 2 cos α . Hence, according to M.W. Evans' O(3) hypothesis [1.43/3] it should be accompanied by a longitudinal component 2 γ cos α · B⁽⁰⁾ k with γ=1 . The superposition principle, however, yields B⁽³⁾ = 2 γ B⁽⁰⁾ k (Eq. (2.8)). Since we assumed cos α < 1 , this is a contradiction. Only the classical case γ=0 is compatible with the superposition principle, and M.W. Evans' "ghost field" cannot exist.

M.W. Evans' cyclical O(3)-hypothesis is incompatible with the superposition principle of waves.

Remark without detailed proof:

References

[1] M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics; Arima 2006

[2] M.W. Evans e.a., Classical Electrodynamics without the Lorentz Condition: Extracting Energy from the Vacuum;
      Physica Scripta Vol. 61, No. 5, pp. 513-517, 2000
      Abstract: http://www.physica.org/xml/article.asp?article=v061a00513.xml
      Full text: http://www.physica.org/asp/document.asp?article=v061p05a00513

[3] M.W. Evans, On the Nature of the B⁽³⁾ Field, Physica Scripta, Vol.61, 287--291, 2000,
      Abstract: http://www.physica.org/xml/v061a00287.xml
      Full text: http://www.physica.org/asp/document.asp?article=v061p03a00287

[4] G.W. Bruhn and A. Lakhtakia, Commentary on Myron W. Evans' paper "The Electromagnetic Sector ...",
      http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html

[5] G.W. Bruhn, Refutation of Myron W. Evans’ B(3) field hypothesis,
      http://www2.mathematik.tu-darmstadt.de/~bruhn/B3-refutation.htm

[6] A. Lakhtakia, Is Evans' longitudinal ghost field B⁽³⁾ unknowable?
      Foundations of Physics Letters Vol. 8 No. 2, 1995

[7] E. Wielandt, The Superposition Principle of Waves Not Fulfilled under M.W. Evans' O(3) Hypothesis
      http://arxiv.org/pdf/physics/0607262

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