Gauge transformations w/quaternions

Subject: gauge transformations w/quaternions
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/04/20
Message-Id: <E8xwzq.1CL@world.std.com>
Newsgroups: sci.physics.research

Hello:

I could use some professional help concerning gauge transformations
in quaternion electromagnetic theory. A gauge transformation of the
quaternion electromagnetic field leaves the E and B fields unchanged,
as usual. There is an additional constraint on the gauge
transformation that arises from the scalar component of the
quaternion, but I will argue that this is a natural consequence of the
continuity equation.

Consider the quaternion partial differential operator

d/dt + Del = d/dt + d/dx I + d/dy J + d/dz K

acting once on the quaternion potential

phi + A = phi + Ax I + Ay J + Az K

like this

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

The divergence, the gradient, the curl and two time derivatives are all
rolled tightly together into one mathematical object (one I call the
"quaternion electromagnetic field" for reasons shown below). This
quaternion has components that transform under inversion (I for -I,
etc.) like a scalar, a polar vector and an axial vector. The polar and
axial vectors are familiar:

- E = Grad phi + dA/dt and B = Curl A

Look at the following gauge transformation with a scalar field L:

phi -> phi' = phi - dL/dt and A -> A' = A + Grad L

Substitute these into the first equation

d phi/dt - d^2 L/dt^2 - divA - div Grad L + Grad phi - Grad dL/dt
+ dA/dt + d Grad L/dt + Curl A + Curl Grad L

= d phi/dt - div A + Grad phi + dA/dt + Curl A
- d^2 L/dt^2- div Grad L

The polar and axial vectors are unchanged by this gauge
transformation, as is typical for gauge transformations of
electromagnetic fields. The scalar portion will remain unchanged only
if

- d^2 L/dt^2- div Grad L = 0 .

What is this additional constraint on L? We could imagine the
following map between L and sources in a vacuum:

dL/dt -> rho and Grad L -> J .

This map then allows the constraint to be interpreted as the continuity
equation:

d rho/dt + div J = 0 .

An arbitrary scalar field L must not only leave the E and B fields
unchanged - and thus the four Maxwell equations - but also may not
alter the continuity equation which is implicit in the Maxwell
equations. Under this interpretation, the only way to add a scalar field
to the Maxwell equations _and_ the continuity equation without
altering their form is to add a scalar field which can be identified with
a charge density or a current density.

Seems like a conservative interpretation to me, but I would be
interested in the opinions of folks more familiar with gauge
transformations.


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