Refutation of Myron W. Evans’ B^{(3)}
field hypothesis

Gerhard W. Bruhn,
Darmstadt University of Technology

**Summary.** In 1992 Myron W.
Evans published a paper [1] where he proposed the hypothesis that each circularly
polarized plane electromagnetic wave – in addition to its Maxwellian
transversal components – should have a longitudinal component of magnitude B^{(3)}
= B_{0}/2^{1/2} compared with the real magnetic flux amplitude B_{0}
of the circularly polarized plane wave. Two years later, in a paper [2] he added
so-called “cyclical relations” that should hold between the components of the
flux **B** relative to a certain complex basis **e**^{(1)}, **e**^{(2)},**
e**^{(3)} for general plane waves. By application to the
superposition of two circularly polarized plane waves to a linearly polarized wave
we show here that Evans’ “cyclical relations” cannot hold generally. The
assumption of a longitudinal **B**^{(3)} field leads to a
contradiction. This affects especially the paper [4], where a kind of PMM, the
MEG, is justified by means of the **B**^{(3)} field.

**
1. Evans’ circular basis** (taken from [3] , p. 7-14, with slight corrections)

Let (x,y,z) denote Cartesian
coordinates with unit vectors **i**,** j**, **k **parallel to the
corresponding axes. Evans assumes the z-direction to be the direction of
propagation of a plane electromagnetic wave. The letter i denotes the
imaginary unit as usual.

Evans
replaces the Cartesian unit vectors **i**,** j**, **k **with** **another system of unit vectors called circular basis

(1.1)** e**^{(1)} = (**i** − i** j**)/2^{1/2} **e**^{(2)} = (**i** + i** j**)/2^{1/2 }**e**^{(3)}
= **k **.

This means that a certain unitary coordinate transform is executed.

We suppose the coordinates a_{x}, a_{y}, a_{z
} of all vectors **a **= a_{x}** i** + a_{y}
**j** + a_{z} **k** to
be real. Then from

a_{x}** i** + a_{y} **j**
+ a_{z} **k** = **a **
= a^{(1)}** e**^{(1)} + a^{(2)} **e**^{(2)} + a^{(3)} **e**^{(3)}

we obtain the transformation rule
for coordinates

(1.2) a^{(1)}** ** = 2^{−1/2}
(a_{x} **+** i a_{y}), a^{(2)}** ** = 2^{−1/2} (a_{x}
**−** i a_{y}), a^{(3)} = a_{z}.

Evidently the coordinates fulfil the equation

(1.3) |a^{(1)}|^{2} +|a^{(2)}|^{2} +|a^{(3)}|^{2} = a_{x}^{2}
+ a_{x}^{2} + a_{x}^{2} = |**a**|^{2} .

Additionally the vector components of **a** relative to the circular basis are defined by

(1.4) **a**^{(1)}** **
= a^{(1)}** e**^{(1)}, **a**^{(2)}** ** = a^{(2)}** e**^{(2)}, **a**^{(3)}** ** = a^{(3)}** e**^{(3)}.

Let …* denote the conjugate complex of
the term where * is attached. Then evidently
we have the symmetry properties

(1.5) ** e**^{(1)}* = **e**^{(2)}
,^{ }**e**^{(2)}* = **e**^{(1)},^{ }**e**^{(3)}* = **e**^{(3)}

(1.6) ** a**^{(1)}* = **a**^{(2)}
,^{ }**a**^{(2)}* = **a**^{(1)},^{ }**a**^{(3)}* = **a**^{(3)}

and

(1.7) a_{1 }= a_{2}*, a_{2}
= a_{1}*, a_{3}
= a_{3}*.

By direct calculation one can obtain
the cyclic cross product rules

(1.8) ** e**^{(1)}× **e**^{(2)}_{
}= i **e**^{(3)}*, **e**^{(2)}× **e**^{(3)}
= i **e**^{(1)}*,
**e**^{(3)}× **e**^{(1)}_{
}= i **e**^{(2)}*.

**
2. Evans B ^{(3)} hypothesis from 1992**

In 1992 Myron W. Evans published a paper [1] where he proposed the hypothesis that a monochrome circularly polarized plane electromagnetic wave should have - in addition to its Maxwellian transversal components - a longitudinal component the size of which will be specified at the end of this section.

We assume the speed c of light to be 1. Then a monochrome Maxwellian circularly
polarized plane wave propagating in z-direction with amplitude B_{0} >
0 is given by the equations

(2.1) B_{x} = B_{0}
cos ω(t−z),
B_{y} = __+__ B_{0}
sin ω(t−z)
, B_{z} = 0

Here
the sign __+__ in B_{y} determines the chirality of the polarization: The + sign
is valid for right (R) circular polarization and the − sign for
left (L) circular polarization.
Introducing the abbreviation

(2.2) B^{(0)} =2^{−1/2} B_{0}

the components of the Maxwellian **B **relative to Evans’
circular basis can be written as

(2.3) **B**^{(1)} = **e**^{(1)} B^{(0)}
e^{iω(t−z)}, **B**^{(2)} = **e**^{(2)} B^{(0)}
e^{−iω(t−z)}_{}

in case of left circular polarization and

(2.4) **B**^{(1)
} = **e**^{(1)} B^{(0)}
e^{−iω(t-z)}, **B**^{(2)} = **e**^{(2)} B^{(0)}
e^{iω(t-z)}_{}

in case of right circular polarization.

The hypothesis of Evans’
paper [1] is that a monochrome circularly polarized plane electromagnetic wave should
have - in addition to its Maxwellian transversal components - a longitudinal
component of magnitude
B^{(3)}
= B^{(0)},
i.e.

(2.5)
**B**^{(3)} = **e**^{(3)} B^{(0)}.

He does not mention whether there
should be a sign dependency on the chirality of the circularly polarized wave.

In summary may be said that the
equations (2.3-5) describe the Evans version of a circularly polarized wave,
while for the Maxwellian circularly polarized wave equation (2.5) has to be
replaced with **B**^{(3)} =
**0**.

**
3. Evans’ Cyclic Relations**

In
1994 Evans supplemented his former hypothesis from 1992 by another paper [2].
Here he starts with the statement that the magnetic flux vector **B** of each circularly polarized plane wave that he had equipped in [1] with
the additional longitudinal component (2.5) fulfils the “cyclic
relations”

(3.1) **
B**^{(1)}
×** B**^{(2)} = i B^{(0)}_{ }**B**^{(3)}* ,

(3.2) **
B**^{(2)} ×** B**^{(3)} = i B^{(0)}_{ }**B**^{(1)}* ,

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

which can be confirmed easily by means of the
equations (2.2-5).

Evans’ new hypothesis of 1994 generalizes the
equations (3.1-3) to general waves in vacuo [2, p. 69]:

*“We assert therefore that in classical
electrodynamics there are three components ***B**^{(1)}*, ***B**^{(2)} *and*_{ }**B**^{(3)}*
of a travelling plane wave in vacuo. These are interrelated in the circular basis
by equations (3.1-3). The third
component, the ghost field*

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

*is real and independent of phase.”*

Hence Evans’
cyclic equations should be valid for the superposition of circularly polarized plane waves
too. This is it what we will check now.

**
4. Superposition of circularly polarized waves**

The superposition of a right circularly
polarized wave with its
left circularly polarized counterpart yields linearly polarized plane waves. If
we superpose the left circularly polarized wave

(4.1)
**B**_{L}** = **B_{0} [**i **cos ω(t−z) − **j **sin ω(t−z)]

and the right circularly polarized wave

(4.2)
**B**_{R}** = **B_{0} [**i **cos ω(t−z) + **j **sin ω(t−z)],

we obtain the linearly polarized wave

(4.3) **B** = 2B_{0} **i **cos ω(t−z),
i.e. B_{x} = 2B_{0} cos ω(t−z), B_{y}
= B_{z} = 0.

Due to Evans both circularly
polarized waves should
be accompanied by ghost fields **B**_{R}^{(3)} and **B**_{L}^{(3)} which
give the resulting sum field

(4.4)
**
B**^{(3)}_{ }
= **B**_{R}^{(3)}
+ **B**_{L}^{(3)}

But due to the indeterminacy of the sign of the additional Evans field for circularly polarized waves we have to discuss all combinations of signs: the cases of constructive and destructive superposition of the corresponding Evans fields. The resulting Evans field for linearly polarized plane waves could be

(4.5)
**B**^{(3)}
= **0**
or
**B**^{(3)}
= __+__ 2^{1/2} B_{0}
**k**.

We have to check whether there is a combination that fulfils the cyclic equations (3.1-3):

Due to the rules (1.2) the linearly polarized wave (4.3) yields

(4.6)
B^{(1)} = 2^{1/2} (B_{x}
+ i B_{y}) = 2^{1/2} B_{0} cos ω(t−z),
B^{(2)} =
B^{(1)}*
= 2^{1/2} B_{0} cos ω(t−z).

Hence we get:

**
Case **B^{(3)} =**
0**

Then the equation (3.1) leads to a contradiction, since we obtain

(4.7)
**B**^{(1)} × **B**^{(2)}
= B^{(1)} B^{(2)}**
e**^{(1)}×** e**^{(2)} = 2 B_{0}^{2}_{ }cos^{2}
ω(t−z) i **k **≠ **0 = **i B^{(0)}
**B**^{(3)} .

**Cases **B^{(3)}
=** **__+__
2^{1/2} B_{0}

Each of the equations (3.2) and (3.3) yields B^{(0)}
= B^{(3)}. Therefore the right side of (3.1) gives

(4.8)
i B^{(0)}_{ }**B**^{(3)}* =** **i B^{(3)}^{2} **k **= 2 i B_{0}^{2} **k**** **.

But for the left side of (3.1) we obtain

(4.9) **B**^{(1)} × **B**^{(2)}
= B^{(1)} B^{(2)}**
e**^{(1)}×** e**^{(2)} = 2 B_{0}^{(2)}_{ }cos^{2}
ω(t−z) i **k **

Hence we have a contradiction again.

Thus Evans’ cyclic equations do not hold for the superposition
of two circularly polarized plane waves to a linearly polarized plane wave.

Hence the validity of Evans’
cyclic equations, of the base of Evans’

O(3) theory, is
refuted.

** **

**References**

[1] M. W. Evans: The
elementary static magnetic field of the photon, Physica B **182 **(1992) 227-236.

[2] M. W. Evans: The
photomagneton B^{^}^{(3)} and its
longitudinal ghost field **B**^{(3)}
of electromagnetism, Foundations of Physics Letters, Vol. **7**, No. 1 (1994) 67 -
74.

[3] M. W. Evans: The
Enigmatic Photon, Vol. **5**, Kluwer Academic Publishers 1999, ISBN 0-7923-5792-2 .

[4] M. W. Evans e.a. : Explanation of the **M**otionless **E**lectromagnetic
**G**enerator (MEG) with O(3) Electrodynamics; Foundations of Physics
Letters, Vol. **14** No. 1 (2001)