Updated on June 13, 2006: **Appendix**

O(3)-Symmetry Law

Summary

In 1992 M.W. Evans proposed a so-called O(3)-symmetry of electromagnetic fields by adding a
constant longitudinal ghost field to the well-known transversal plane em waves. Evans considered this symmetry
as a new law of electromagnetics. Later on, since 2002,
this O(3)-symmetry became the center of his CGUFT which he recently renamed as ECE Theory.
A law of Physics must be invariant under admissible coordinate transforms, namely under Lorentz transforms.
Therefore, to check the validity of Evans' O(3)-symmetry law, we apply a longitudinal Lorentz transform
to Evans' plane em wave (the ghost field included). As is well-known from SRT and recalled here the transversal amplitude
decreases while the additional longitudinal field remains unchanged. Thus, Evans' O(3)-symmetry
cannot be invariant under (longitudinal) Lorentz transforms: **Evans' O(3)-symmetry
is no valid law of Physics.**

In the following text quotations from Evans' book manuscript [1] appear in **black**
with equation labels (1.nn) at the right margin.

The claim of O(3)-symmetry is a central concern of Evans' considerations since 1992. The reader
will find a historical overview in [2; Sect. 5] written by A. Lakhtakia.
Among a lot of papers Evans has written five books on "The Enigmatic Photon" that deal with the claimed
O(3)-symmetry of electromagnetic fields.

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

Φ = ω t − κ z , (1.38)

where κ = ω/c, Evans describes the wave relative to his complex circular basis [1; (1.41)], see also [3; Appendix 1] and [4]. The magnetic field is given by

**B**^{(1)}
= *B*^{(0)}**q'**^{(1)}
= ^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**) e^{iΦ} ,
^{ }

**B**^{(2)}
= *B*^{(0)} **q'**^{(2)}
= ^{1}/_{sqr(2)} *B*^{(0)}
(**i**+*i***j**) e^{−iΦ} ,
(1.43)

**B**^{(3)}
=
*B*^{(0)} **q'**^{(3)}
=
*B*^{(0)} **k** ,

satisfying Evans' "cyclic O(3)-symmetry relations"

**B**^{(1)} × **B**^{(2)}
= *i**B*^{(0)}**B**^{(3)}* ,

**B**^{(2)} × **B**^{(3)}
= *i**B*^{(0)}**B**^{(1)}* ,
(1.44)

**B**^{(3)} × **B**^{(1)}
= *i**B*^{(0)}**B**^{(2)}* .

Due to M.W. Evans the corresponding electric field is given by

**E**^{(1)}
= − ^{E(0)}/_{sqr(2)}
(*i***i**+**j**) e^{iΦ} ,

**E**^{(2)}
= ^{E(0)}/_{sqr(2)}
(*i***i**−**j**) e^{−iΦ} ,
(1.85)

**E**^{(3)}
=
−*i**E*^{(0)} **k** .

The relation between *E*^{(0)} and *B*^{(0)} is

*E*^{(0)} = c *B*^{(0)} ,
(1.87)

i.e. *E*^{(0)} = *B*^{(0)}, if we assume c=1 below.

Due to
**B**^{(2)} = **B**^{(1)}*
and
**E**^{(2)} = **E**^{(1)}*
the complex fields (1.43) and (1.85) belong to the real fields

**B** = **B**^{(1)} + **B**^{(2)} + **B**^{(3)}
=
*B*_{x} **i** +
*B*_{y} **j** +
*B*_{z} **k**

and

**E** = **E**^{(1)} + **E**^{(2)} + **E**^{(3)}=
*E*_{x} **i** +
*E*_{y} **j** +
*E*_{z} **k** .

Inserting of (1.43) and (1.85) and coefficient matching yields

(1)
*B*_{x} = *B*^{(0)} sqr(2) cos Φ ,
*B*_{y} = *B*^{(0)} sqr(2) sin Φ ,
*B*_{z} = *B*^{(0)} ,

(2)
*E*_{x} = *E*^{(0)} sqr(2) sin Φ ,
*E*_{y} = − *E*^{(0)} sqr(2) cos Φ ,
*E*_{z} = *E*^{(0)} .

Summing the equations in (1) with combination factors 1,__+__*i*
and comparing with (1.43) yields

(3)
(*B*_{x} + *i* *B*_{y}) (**i**−*i***j**)= 2 **B**^{(1)},
(*B*_{x} − *i* *B*_{y}) (**i**+*i***j**) = 2 **B**^{(2)} .

and therefore for further use in rewriting of the first equation of (1.44)

(4)
**B**^{(1)} × **B**^{(2)}
=
½
(*B*_{x} + *i* *B*_{y})
(*B*_{x} − *i* *B*_{y})
*i* **k**
=
½
(*B*_{x}˛ + *B*_{y}˛)
*i* **k** .

while the last equations of (1.43) and (1) yield

*B*^{(0)}**B**^{(3)}*
= *i***k** *B*^{(0)}˛
= *i***k** *B*_{z}˛ .

Thus, one of "Evans' cyclic symmetry relations", the first rule of (1.44), is equivalent to

(5)
½ (*B*_{x}˛ + *B*_{y}˛)
=
*B*_{z}˛ .

The first two equations of (1.43) and (1.85) describe a circularly polarized plane wave
propagating in z-direction. The third equations, however, contain * Evans' O(3)-Law* from 1992,
saying that the well-known plane wave is always accompanied by a constant
longitudinal magnetic "ghost" field

If * Evans' O(3)-Law* were a

Therefore we consider the wave as observed from other coordinate systems S' in constant motion
**v** = v **k** relative to our original Cartesian coordinate system S.
The transformation rules for the electromagnetic field are well-known
(β=v/c, γ=1/sqr(1−β˛) and c=1 assumed)):

(6)
E_{x}' = γ (E_{x} − β B_{y}),
E_{y}' = γ (E_{y} + β B_{x}),
E_{z}' = E_{z} ,

(7)
B_{x}' = γ (B_{x} + β E_{y}),
B_{y}' = γ (B_{y} − β E_{x}),
B_{z}' = B_{z} .

We shall check the first rule of Evans' O(3) symmetry Law (1.44) in our equivalent real formulation (5). Therefore we are now going to transform the wave (1-2) into the coordinate frame S' by means of the transformation rules (6-7) to obtain

(8)
*B*_{x}'
= (1−β)γ *B*^{(0)} sqr(2) cos Φ
= (1−β)γ *B*_{x} ,
*B*_{y}'
= (1−β)γ *B*^{(0)} sqr(2) sin Φ
= (1−β)γ *B*_{y} .
*B*_{z}' = *B*_{z} ,

which shows that due to

(9)
½ (*B*_{x}'˛ + *B*_{y}'˛)
=
^{1−β}/_{1+β} ½ (*B*_{x}˛ + *B*_{y}˛)
=
^{1−β}/_{1+β} *B*_{z}˛
=
^{1−β}/_{1+β} *B*_{z}'˛
<
*B*_{z}'˛
(0 < β < 1)

Evans' first O(3)-symmetry relation (5) in S', the equation
½ (*B*_{x}'˛ + *B*_{y}'˛)
= *B*_{z}'˛,
is **not** fulfilled:

[1]
M.W. Evans, *Generally Covariant Unified Field Theory,
the geometrization of physics*;
Web-Preprint,

http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[2]
G.W. Bruhn and A. Lakhtakia, *Commentary on Myron W. Evans' paper
"The Electromagnetic Sector ..." *,

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

[3]
G.W. Bruhn, *Comments on Myron W. Evans’ "REFUTATION"*

http://www.mathematik.tu-darmstadt.de/~bruhn/Refutation_of_EVANS_REFUTATION.htm

[4]
V.V. Dvoeglazov, *Comment on the 'Comment on the Longitudinal Magnetic Field ...’
by E. Comay ... *,

http://132.236.180.11/pdf/physics/9801024

[5]
M.W. Evans, *On the Application of the Lorentz Transformation in O(3) Electrodynamics*

APEIRON Vol. 7 Nr.1-2, 2000, 14-16

http://redshift.vif.com/JournalFiles/Pre2001/V07NO1PDF/V07N1EV1.pdf

In his article [5: p.14] M.W. Evans tries to prove the Lorentz invariance of the O(3) hypothesis
by referring to the invariance of the vector potential **A** under Lorentz transforms.
That is a good method obtaining the transform of ED fields if one calculates *correctly*.
So first the reader should check that the vector potentials of the transversal components
**B**^{(1)} and **B**^{(2)}
of the plane wave under consideration are given by

(A1)
**A**^{(1)} =
^{1}/_{κ} **B**^{(1)} =
^{1}/_{κ}
^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**) e^{iΦ}
,
**A**^{(2)} =
^{1}/_{κ} **B**^{(2)} =
^{1}/_{κ}
^{1}/_{sqr(2)} *B*^{(0)}
(**i**+*i***j**) e^{−iΦ}

while the vector potential of the longitudinal field **B**^{(3)} is

(A2)
**A**^{(3)} = ½ **B**^{(3)}× (x**i** + y**j**) .

The invariance of **A**^{(3)} yields the invariance of **B**^{(3)} as well.

What Evans has ignored: Frequency ω and wavenumber κ are *no invariants*.
Under *longitudinal* Lorentz transforms we have the well-known Doppler effect:

(A3)
ω' = sqr(^{1−β}/_{1+β}) ω ,
κ' = sqr(^{1−β}/_{1+β}) κ .

Therefore we obtain from (A1) taking into account the invariance of **A**

(A4)
**B**'^{(1)}×**B**'^{(2)} =
κ'² **A**'^{(1)}×**A**'^{(2)}
=
^{κ'²}/_{κ²} κ²
**A**^{(1)}×**A**^{(2)}
=
^{1−β}/_{1+β}
**B**^{(1)}×**B**^{(2)} ,

i.e. the expression does not remain invariant while **B**^{(3)}, as was shown above,
remains invariant. Hence the first equation of (1.44),
if valid in the inertial system S, *cannot be valid* also in the inertial system S'. Thus,
Evans' cyclical symmetry (1.44) is not Lorentz invariant. Thus,