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Our replies displayed in **blue: EW: ...**

under M.W. Evans' O(3) Hypothesis

Institute of Geophysics, Stuttgart University, Germany

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

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**−*i***j**) e^{i(Φ+α)} ,

(2.4)
**B**^{(2)}
= ^{1}/_{sqr(2)} B^{(0)}
(**i**+*i***j**) 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**−*i***j**) [e^{i (Φ+α)} + e^{i (Φ−α)}]
= ^{1}/_{sqr(2)} B^{(0)}
(**i**−*i***j**) e^{i Φ} 2 cos α ,

(2.7)
**B**^{(2)} = ^{1}/_{sqr(2)} B^{(0)}
(**i**+*i***j**) [e^{−i(Φ+α)} + e^{−i(Φ−α)}]
= ^{1}/_{sqr(2)} B^{(0)}
(**i**+*i***j**) e^{−iΦ} 2 cos α ,

(2.8)
**B**^{(3)} = 2 γ B^{(0)} **k** .

We have for each wave:

1)
**B**_{1}^{(1)} × **B**_{1}^{(2)} =
½ B^{(0)2}(**i**−*i***j**)×(**i**+*i***j**)
e^{i(Φ+α)} e^{−i(Φ+α)} =
*i* γB^{(0)} **B**^{(3)}* ,

2)
**B**_{2}^{(1)} × **B**_{2}^{(2)} =
½ B^{(0)2}(**i**−*i***j**)×(**i**+*i***j**)
e^{i(Φ−α)} e^{−i(Φ−α)} =
*i* γB^{(0)} **B**^{(3)}* ,

**γ=1 in each case**

**EW:** We have inserted a missing factor γB^{(0)} (γ=1 possibly)
in your eqs. 1) and 2) and then agree so far.
But you don't talk about the **superposition** of that two phase shifted waves.

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.

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://www.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html

[5]
G.W. Bruhn,
Refutation of Myron W. Evans’ B(3) field hypothesis,

http://www.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

[8]
E. Wielandt, The Superposition Principle of Waves Not Fulfilled
under M.W. Evans' O(3) Hypothesis

http://www.mathematik.tu-darmstadt.de/~bruhn/arebuttalofwielandt.pdf

and

http://aias.gq.nu/pdf/arebuttalofwielandt.pdf