Transforming quaternion equations

Subject: transforming quaternion equations
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/05/22
Message-Id: <EALF7r.1AH@world.std.com>
Newsgroups: sci.physics.research

Hello:

It is common for an explorer like myself to work under the delusion
that the math they employ is "different" from what is commonly
used. In this post I hope to substantiate the difference I perceive by
examining how terms in quaternion equations change under
inversion, time reversal and reflection. Hopefully I will show that
quaternions do not form covariant equations (but that's OK :-)

Let me take a snapshot of the status quo from Byron and Fuller's
"Mathematics of Classical and Quantum Physics, vol. 1" p. 14:

     In summary then, the invariance of a physical law under
     orthogonal transformation of the spatial coordinate system
     requires that all the terms of the equation be tensors of the
     same rank.  We say then that the terms are _covariant_
     under orthogonal transformations, i.e., they "vary together."

Quaternions are designed to be at odds with this guiding principle.
For example, the quaternion for an event, t + x i + y j + z k combines a
tensor of rank zero (the scalar t) with a tensor of rank one (the
vector x i + y j + z k). Scalars and vectors are not covariant. In fact,
the difference can be the basis of their definitions.

Let's examine the operator equation that generates the E and B fields:

     (d/dt + Del)(phi + A) = d phi/dt - div A + dA/dt + Grad phi + Curl A

Inversion of the coordinates makes i -> -i, j -> -j and k -> -k. The
terms dA/dt and Grad phi will flip signs, the rest will remain
unchanged. The terms in this quaternion operator equation do not
all vary together. Instead, quaternion equations form a well-
balanced hodgepodge of terms, whose transformations are known.
Consider what happens under time reversal. The d phi/dt and
dA/dt terms should change their signs, but the rest are unaffected.
Here again, the terms do not all vary together. Finally, consider a
mirror reflection, what happens if a left-handed 3-coordinate vector
basis substitutes for a right-handed one. Only the Curl A term will
change signs. In summary:

           d/dt Del | phi A | d phi/dt div A dA/dt Grad phi Curl A
inversion        -  |     - |                  -         -
timerev.    -       |       |      -           -
reflection          |       |                                   -

Each of the three quaternions which form this equation are
composites of old, familiar friends. The operator d/dt, the potential
phi, d phi/dt and div A are scalars. The operator Del, the vector
potential A, dA/dt and Grad phi are polar vectors. Curl A is an axial
vector. There will not be a universal rule for how quaternions
transform because the transformation depends on what terms are
contained within the quaternion. Yet each of the three quaternions
involved in this equation is uniquely defined by this set of
transformations. The potential quaternion is affected only by
inversion. The differential operator is affected by inversion and time
reversal. The quaternion with E and B is the only one affected by
reflection. Note: a reflection can be mimicked by reversing the order
of the differential and potential quaternions.

There are a great many things one can struggle with in physics. In
my graduate-level classes, I was left with the impression that there
was no way one could question the basic structure of terms in
equations as described by Byron and Fuller. Therefore my gut
feeling is that quaternion physics is fundamentally different. Please
correct my delusion (or join in the hunt :-)


Doug Sweetser
http://world.std.com/~sweetser

Trying to get quaternions to leap from
the nineteenth century to the twenty-first.