Comments on Myron W. Evans’ "REFUTATION"[6]

Gerhard W. Bruhn, Darmstadt University of Technology

(Quotations from M.W. Evans handwritten "REFUTATION" [6] in black, my remarks in blue.)

Checking Evans' claims in [6]

Check 1
Let's quote some eqs. of my paper [5]. Due to the literature (e.g. [7; p.188], [8; p.702 f.]) a vector
                                B = i B_x + j B_y
of left (+ = L) and right (− = R) circular polarization around the z-axis (direction k) is given by choosing
(2.1)                 B_x = B₀ cos Φ                 and                 B_y = B₀ sin Φ .
Then due to Evans' basic equations (A.3) and (A.5) we obtain the components of B relative to the cyclic basis of the vectors e^(r) (r=1,2,3):

(2.3)      B_R⁽¹⁾ = e⁽¹⁾ B⁽⁰⁾ e^iΦ,      B_R⁽²⁾ = e⁽²⁾ B⁽⁰⁾ e^−iΦ
and
(2.4)      B_L⁽¹⁾ = e⁽¹⁾ B⁽⁰⁾ e^−iΦ,      B_L⁽²⁾ = e⁽²⁾ B⁽⁰⁾ e^iΦ

Result 1
There is no doubt about it, the equations (2.3) and (2.4) describe right and left handed circular polarization correctly.

Evans' Claim 1 is erroneous.

Check 2
Preliminary remark (of minor importance):
In (2) a factor 2 is missing. Hence we have correctly
                B_L⁽¹⁾ + B_R⁽¹⁾ = B⁽⁰⁾ 2^1/2 (i − i j) cos Φ                                                                   (2')
We shall use (2') instead of (2) below.

Due to the rule (A.7), a⁽²⁾ = a⁽¹⁾*, we obtain from (2')
                B_L⁽²⁾ + B_R⁽²⁾ = B⁽⁰⁾ 2^1/2 (i + i j) cos Φ
while B⁽³⁾ = 0 holds by assumption.

Hence from B = B⁽¹⁾ + B⁽²⁾ + B⁽³⁾  (cf. (A.2)) we obtain
                B = 2 B⁽⁰⁾ 2^1/2 i cos Φ = i 2 B₀ cos Φ .

Result 2
B is a real vector in constant direction i oscillating proportionally to cos Φ. Hence B is clearly linear polarized.

Evans' Claim 2 has been refuted.

Check 4
By an argumentation analogously to Check 3. we see that for B_x ≠ 0 equ. (5) would imply
            i − i j   | |   i
which is a contradiction, regardless of the rest of Evans' claim.

Result 4
Equ. (5) is inconsistent.

Evans' Claim 4 has proved to be erroneous.

Check 5
By an argumentation analogously to Check 3. we see that (6) would imply
            i − i j   | |   i cos²Φ + j sin²Φ
which is a contradiction since the left side has an imaginary part while the right side is real.
An analoguos argumentation holds for equ. (7).

Concerning Evans' last remark we quote the equations (4.1-2) from our refutation [5]:

(4.1)      B_R = B₀ (i cos Φ + j sin Φ)        and      (4.2)     B_L = B₀ (i cos Φ − j sin Φ).

Equ. (4.1) describes a right polarized wave and (4.2) describes a left polarized wave.

Of course, the equations (4.1-2) are in accordance with the literature (cf. [7; p.188], [8; p.702 f.]).

Result 5
The equs. (6) and (7) are inconsistent.

Evans' Claim 5 has proved to be erroneous.

Check 5
As I have just shown Evans' equs. (3) and (4) are erroneous while my (4.1) and (4.2) are correct since being in accordance with the literature.

Result 6

Evans' Claim 6 is false.

Check 7
As we have remarked above the equs. (4.1) and (4.2) give right and left polarization in accordance with the literature. And the equations
(2.3)      B_R⁽¹⁾ = e⁽¹⁾ B⁽⁰⁾ e^iΦ,      B_R⁽²⁾ = e⁽²⁾ B⁽⁰⁾ e^−iΦ
and
(2.4)     B_L⁽¹⁾ = e⁽¹⁾ B⁽⁰⁾ e^−iΦ,      B_L⁽²⁾ = e⁽²⁾ B⁽⁰⁾ e^iΦ
are nothing but the components of B_R and B_L relative to the basis e⁽¹⁾, e⁽²⁾, as can be shown by an elementary calculation. Hence there cannot be any contradiction between these equations.

Result 7
An elementary calculation would show no contradiction between these equations.

Evans' Claim 7 is false.

Check 8
Evans' quotation of equ. (4.5) is incorrect: The correct quote is

(4.5)           B⁽³⁾ = 0           OR           B⁽³⁾ = + 2^1/2 B⁽⁰⁾ k

Hence (4.5) is valid, since Evans' assures the first alternative of that OR-statement to be true.

Result 8
Evans' Claim 8 is due to an incorrect quotation.

Evans' Claim 8 is unjustified.

Check 8

That claim is at least dubious. Since in 1994 Evans had declared in [2; p.69];

Equ. [2; (2)] are Evans’ cyclic equations that were extended herewith by M. Evans himself to «waves in vacuo» [2, p. 69] without any restriction. And lateron he did never qualify that statement. Thus: Evans’ cyclic equations could be thought to be valid for the superposition of circularly polarized plane waves too. This was what I have checked in my refutation [5] − with negative result.

Now in his "REFUTATION" [6] from 2004 Evans declares as a reply on my provocing paper [5] with restrictive intention

i.e. Evans' cyclic equations do not hold for linear polarization as a superposition of right and left handed circular polarization.

Result 9
Now, in 2004, M. Evans agrees with the result of my refutation [5].

Much ado about nothing!

Conclusion

All claims in Myron W. Evans' "REFUTATION" [6] have turned out to be erroneous or dubious at least. Therefore publications of that author should be read with greatest caution and scepticism.

Appendix 1   Evans’ Basics

Circular basis (Cartesian unit vectors i, j, k, imaginary unit i) :
(A.1)               e⁽¹⁾ = (i − i j)/2^1/2,       e⁽²⁾ = (i + i j)/2^1/2,       e⁽³⁾ = k .

Coordinate representations of all vectors a:

(A.2)             a_x i + a_y j + a_z k = a = a⁽¹⁾ e⁽¹⁾   + a⁽²⁾ e⁽²⁾  + a⁽³⁾ e⁽³⁾

Transformation rule for coordinates

(A.3)             a⁽¹⁾ = 2^−1/2 (a_x + i a_y),          a⁽²⁾ = 2^−1/2 (a_x − i a_y),          a⁽³⁾ = a_z.

(A.4)             |a⁽¹⁾|² +|a⁽²⁾|² +|a⁽³⁾|² = a_x² + a_y² + a_z² = |a|² .

Vector components of a relative to the circular basis

(A.5)             a⁽¹⁾ = a⁽¹⁾ e⁽¹⁾,            a⁽²⁾ = a⁽²⁾ e⁽²⁾,            a⁽³⁾ = a⁽³⁾ e⁽³⁾.

The suffix * denotes the conjugate complex of the quantity where it is attached.

(A.6)             e⁽¹⁾* = e⁽²⁾,         e⁽²⁾* = e⁽¹⁾,         e⁽³⁾* = e⁽³⁾

(A.7)             a⁽¹⁾* = a⁽²⁾,        a⁽²⁾* = a⁽¹⁾,         a⁽³⁾* = a⁽³⁾

(A.8)             a⁽¹⁾ = a⁽²⁾*,         a⁽²⁾ = a⁽¹⁾*,         a⁽³⁾ = a⁽³⁾*.

(A.9)             e⁽¹⁾ × e⁽²⁾ = ie⁽³⁾*,         e⁽²⁾× e⁽³⁾ = i e⁽¹⁾*,         e⁽³⁾× e⁽¹⁾ = i e⁽²⁾*.

Remark The rules (A.1-8) have an important consequence that was not explicitly pointed out in Evans' book [3]: The vector coordinates a_x, a_y and a_z have to be reals, in other words, the vectors a subject to the above rules must be real: Due to (A.2) and (A.7) we obtain

                         a* = a⁽¹⁾* + a⁽²⁾* +a⁽³⁾* = a⁽²⁾ + a⁽¹⁾ +a⁽³⁾ = a

which implies the reality of the coordinates a_x, a_y, a_z by using (A.2) again.

Appendix 2

Appendix 3

Some remarks on

J. D. Jackson's Chapter 7 from [9] on polarized plane waves

We should realize that (7.19) is not the original wave vector which is real: The imaginary part was supplemented in order to get arithmetical advantages and has no physical meaning.

Examples

For the pupose of comparison with Evans' formulas we replace Jackson's notation with that one of Evans:
                                e₁ → i,         e₂ → j,         e₋ → e⁽¹⁾,         e₊ → e⁽²⁾,         E → B, . . .
A linearly polarized wave is given if the quotient B₊/B₋ is real, in which case, by changing the coordinate system, we can attain the standard form
                                                                B_lin = B₀ i e^{− iΦ}
with some real constant B₀. Circularly polarized waves can be described by means of the complex basis vectors e⁽¹⁾ and e⁽²⁾:
                                                                B_L⁽¹⁾ = B₀ e⁽¹⁾ e^{− iΦ}
is Jackson's standard form of a left circularly polarized wave, strictly speaking, its e⁽¹⁾-component, while
                                                                B_R⁽²⁾ = B₀ e⁽²⁾ e^{− iΦ}
is Jackson's standard form of a right circularly polarized wave, more exactly, its e⁽²⁾-component. The e⁽¹⁾-component of B_R can be obtained by taking the conjugate complex expression, i.e.
                                                                B_R⁽¹⁾ = B₀ e⁽¹⁾ e^iΦ .

For a comparison we have to take into account that the last representations are not the original real wave vectors but are the result of complex supplementation. However, it is easy to reconstruct the original physical wave vectors: We merely have to take the corresponding real parts. By doing so we obtain
                B_lin^phys = B₀ i cos Φ for linear polarization,

                B_L^phys = Re {B₀ e⁽¹⁾ e^{− iΦ}} = B₀ (i cos Φ − j sin Φ)/2^1/2 for left circular polarization
and
                B_R^phys = Re {B₀ e⁽²⁾ e^iΦ} = B₀ (i cos Φ + j sin Φ)/2^1/2 for right circular polarization.

We have to compare Jackson's standard forms of polarized waves with that ones given by Evans in his "REFUTATION" equs. (3-5):

            Evans                                                           Jackson
            B_L⁽¹⁾ = 2^−1/2 (B_x i − i B_y j) e^iΦ ,                 B_L⁽¹⁾ = B₀ e⁽¹⁾ e^{− iΦ}

            B_R⁽¹⁾ = 2^−1/2 (B_x i + i B_y j) e^iΦ ,                 B_R⁽¹⁾ = B₀ e⁽¹⁾ e^iΦ .