Extraordinary Physics

CHAPTER 4

EXTRAORDINARY PHYSICS

    Maxwell's Lost Unified Field Theory
    About the time of the U.S. Civil War, James Clerk Maxwell succeeded in unifying magnetism and electricity.  Actually he did far more than that, in his theory as originally written.
    In fact, he had produced a theory which also captured the free interchange between electromagnetic energy and gravitational energy, but no one - including Maxwell himself - realized it at the time.
    Maxwell wrote his original theory in quaternion and quaternion-like mathematics.

Wrong! In the original article from 1865 Maxwell does not use any quaternions. The revised version from 1873 contains quaternions only in some main equations. All calculations are executed in component representation without referring to quaternion calculus, cf. [3].

The modern form of vector mathematics had not yet been finalized by Gibbs and Heaviside.  It is most instructional to examine some of the fundamental differences between a vector and a quaternion.
    In a conventional 3-dimensional vector, one may have three vector components, such as
                   v = ai + bj + ck                          (4-1)

where i, j, k are unit vectors in the directions of the x, y, and z axes respectively and a, b, and c are constants.

    Obviously if the vector components of vector v are zero, then

                   v = 0                                          (4-2)

     We shall be interested in the "vector product" of two identical
vectors v, where

                   v X v = AA sinØ = 0                  (4-3)

Error: Type mismatch! The left hand side of (4-3) is a vector, the middle part is single number.

and A is the length (magnitude) of vector v, Ø is the angle between
the two vectors (in this case zero), and 0 is the zero vector.

    Now let us look for a moment at the quaternion situation. 

First, in addition to the three vector components, a quatemion also has a separate scalar component, w.  So the quatemion q for this
situation is
                    q = w + ai + bj + ck                   (4-4)

     Now when this quatemion is multiplied times itself, the vector
part zeros, just as it did for the vector expression.  However, the scalar part does not go to zero.  Instead, we have

                     q X q = A² = a² +b² +c²          (4-5)

Error: The definition of the quaternions product is

            (α,a) (β,b) : = (αβ − a·b, a×b + αb + βa)

This yields

            q q = (w,v) (w,v) = (w² − v·v, 0 + 2 wv),

where q = (w,v) = w + v = w + ai + bj+ ck.

Even if we remove the scalar w in the definition (4-4) of the quaternion q, i.e. we consider q' = (0,v), then we obtain

            q' q' = (0,v) (0,v) = (− v·v, 0) = − (a² + b² + c²),

that means that, at least, (4-5) has a wrong sign.

      There is a very good physical interpretation of this result.  It is a square of the amplitude, hence for the vector part of a wave, it is directly proportional to the energy density of the vacuum, as a function of time, at the particular position.  . . .

What follows are several lines of Bearden's verbal speculations containing no equations and can hence be skipped. Then Bearden returns to his consideration:

       But to return to our vector/quaternion examples. 

Note also that the two vectors

                      v₁= ai + bj + ck,
                      v₂ = -ai - bj - ck                                  (4-6)

sum to zero vectorially when added, such that

                   v₁ + v₂ = 0                                              (4-7)

    However, quaternions may behave quite differently, even under addition.  For example, the two quaternions

                   q₁ = w + ai + bj + ck,
                   q₂ =w - ai -bj - ck                                   (4-8)

    sum their vector parts to a vector zero resultant, but do not sum to a scalar zero as well.  Instead, they sum to

                  q₁ + q₂ = 2w                                            (4-9)

Evidently, Bearden's idea is that each physical field vector v should be replaced with a quaternion q = (−v²,v) (we have added a minus sign to the scalar part due to our preceeding remark), where the scalar part is proportional to the energy density of the vector field shown in the vector part. Then the addition of two quaternions of that type, q₁ = (−v₁²,v₁) and q₂ = (−v₂²,v₂) would give the energy density of the sum field v₁ + v₂, especially in the case v := v₂ = − v₁, which would yield the quaternion sum

(− v², v) (− v², − v) = (− 2 v² , 0).

The question for the physical meaning of that calculation arises.

Is there any well-known physical phenomenon, which could be interpreted in the sense of the above equation?

There is the phenomenon of destructive interference of two electromagnetic waves one could think of. We have analyzed a realistic experimental device, a so-called beam splitter for generating destructive interference of two electromagnetic waves in [2]. We consider a (loss-free) halftransparent mirror, a λ/4-plate, that is spotlighted with two linear polarized laser beams from two sides under 45° angle of incidence, the phase difference of which is π/2. Then on one side of the plate the transmitted and the reflected beams respectively are totally extinguished due to destructive interference. And one can calculate the energy density contained in that zeroed field.

   

   

   

   

   

   

   

   

   

   

On one side of the plate we have total destructive interference. The total energy that is radiated in by the arriving beams runs off completely on the opposite side of the plate.

Thus, the energy conservation law yields zero energy for the extinguished field.

Therefore destructive interference cannot be used as an example of vector zero resultant fields in the sense of quaternions.

Since the rest of Bearden's article contains no further concrete examples of vector zero resultant quaternion fields we close our Commentary here.