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  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) = B0/21/2 compared with the real magnetic flux amplitude B0 of the circularly polarized plane wave. Two years later, in a paper  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 , where a kind of PMM, the MEG, is justified by means of the B(3) field.
1. Evans’ circular basis (taken from  , 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)/21/2 e(2) = (i + i j)/21/2 e(3) = k .
This means that a certain unitary coordinate transform is executed.
We suppose the coordinates ax, ay, az of all vectors a = ax i + ay j + az k to be real. Then from
ax i + ay j + az 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 (ax + i ay), a(2) = 2−1/2 (ax − i ay), a(3) = az.
Evidently the coordinates fulfil the equation
(1.3) |a(1)|2 +|a(2)|2 +|a(3)|2 = ax2 + ax2 + ax2 = |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)
(1.7) a1 = a2*, a2 = a1*, a3 = a3*.
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  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 B0 > 0 is given by the equations
(2.1) Bx = B0 cos ω(t−z), By = + B0 sin ω(t−z) , Bz = 0
Here the sign + in By 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 B0
the components of the Maxwellian B relative to Evans’ circular basis can be written as
(2.3) B(1) = e(1) B(0) eiω(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) eiω(t-z)
in case of right circular polarization.
The hypothesis of Evans’ paper  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 . Here he starts with the statement that the magnetic flux vector B of each circularly polarized plane wave that he had equipped in  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) BL = B0 [i cos ω(t−z) − j sin ω(t−z)]
and the right circularly polarized wave
(4.2) BR = B0 [i cos ω(t−z) + j sin ω(t−z)],
we obtain the linearly polarized wave
(4.3) B = 2B0 i cos ω(t−z), i.e. Bx = 2B0 cos ω(t−z), By = Bz = 0.
Due to Evans both circularly polarized waves should be accompanied by ghost fields BR(3) and BL(3) which give the resulting sum field
(4.4) B(3) = BR(3) + BL(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) = + 21/2 B0 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) = 21/2 (Bx + i By) = 21/2 B0 cos ω(t−z), B(2) = B(1)* = 21/2 B0 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 B02 cos2 ω(t−z) i k ≠ 0 = i B(0) B(3) .
Cases B(3) = + 21/2 B0
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 B02 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 B0(2) cos2 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.
 M. W. Evans: The elementary static magnetic field of the photon, Physica B 182 (1992) 227-236.
 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.
 M. W. Evans: The Enigmatic Photon, Vol. 5, Kluwer Academic Publishers 1999, ISBN 0-7923-5792-2 .
 M. W. Evans e.a. : Explanation of the Motionless Electromagnetic Generator (MEG) with O(3) Electrodynamics; Foundations of Physics Letters, Vol. 14 No. 1 (2001)