Problems with the proposed gauge transformation

Subject: Re: gauge transformations w/quaternions
From: eillihca@drizzle.stanford.edu ( Achille Hui, the Day Dreamer )
Date: 1997/04/22
Message-Id: <eillihca.97042122353012581@drizzle.Stanford.EDU>
Newsgroups: sci.physics.research

+---In <E8xwzq.1CL@world.std.com>---
| sweetser@alum.mit.edu (Doug B Sweetser) writes...
+---------
...
| 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 .
|
...
| 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.

You identification of Grad L -> J doesn't look right. One thing you have to
explain is why your J is irrotational (i.e. Grad L <-> J => Curl J = 0).

There is another aspect I don't like about your gauge transform. If your
are going to use quarternion in you differnetial operator and gauge potential.
What is the underlying reason your gauge transformation is generated by a
single scaler L instead of a generic quarternion valued function:

L(x) = L_0(x) + L_1(x) I + L_2(x) J + L_3(x) K

where L_i(x) are functions in (x_0,x_1,x_2,x_3) (Note, if you try to define
quaternion analytic function over a single unknown quaternion x, this is
the generic form [1].

-achille

Hint: For any quaternion x, consider y - I y I - J y J - K y K where
y = x, I x, J x, K x, respectively.