Newsgroups: sci.physics.research

Subject: The Maxwell
equations from quaternions

Summary:

Expires:

Sender:
Doug Sweetser<sweetser@alum.mit.edu>

Followup-To:

Distribution:

Organization: The World Public Access UNIX,
Brookline, MA

Keywords:

Cc:

Hello:

I've been auditing a graduate level course on

electromagnetism. We've been working our way through

Jackson's "Classical Electrodynamics." You can only say you

are a real physics nerd (in the positive sense, or a Ph. D.

for that matter :-) once you've worked your way through this

impressive body of work.

The professor has remarked that after assuming Maxwell's

equations and the Lorentz force - the first equations in the

book - the rest of the text explores their mathematical

consequences, excluding the way nature dictates sources and

boundary conditions of real systems. And what an impressive

collection of mathematical consequences, with over 200 pages

on statics alone!

But there still remains the question: Why these four

equations? To be logically consistent with the rest of the

mammoth work, the equations should be a mathematical

consequence of something. Consider the following proposal:

The four Maxwell
equations are generated by

one quaternion
wave equation.

Quaternions were a purely mathematical invention of Hamilton.

Since quaternions are a mathematical field, it makes sense to

form a second order partial differential wave equation. With

the right choices of letters, one quaternion wave equation

will look like the four Maxwell equations. I said "letters"

because I am playing a symbolic game with quaternions, hoping

to clone a twin of physics :-)

Two players are needed, the quaternion partial differential

operator:

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

and the quaternion-valued potential function:

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

Construct a second order partial differential operator

that generates the four Maxwell equations (I spent a month

playing around till I found it :-)

[(d/dt c +
Del)^2 + (d/dt
c - Del)^2](phi + A)/2

-----Gauss' law------
--No Monopoles--

= d^2 phi/dt^2
c^2 - del^2 phi -
Div(Curl A) +

-----
the B law with no name----

d^2 A/dt^2 c^2 + Grad(div A) + Curl(Curl A)

--Faraday's law--

Curl(Grad phi)

= 0
for a
vacuum free of charges and currents

or = -
4 pi (rho + J/c)
for sources in a vacuum

or = -4 pi (rho -
div P + J/c + Curl M)
macroscopic media

or = whatever sources and interactions constructed by nature.

The four Maxwell equations in the Lorentz gauge are imbedded

in this one second order quaternion partial differential

equation.

technical notes: Gaussian units are employed. The square of

an operator means that it should act twice on the potential.

To confirm that the operator math is correct requires a few

insights. There should be two second order time derivatives

and two del^2's. Only one Curl Curl operation is properly

defined. There are 6 pairwise combinations of Div, Grad and

Curl, but only 3 are well defined. The reason two operators

are needed is to eliminate the 3 terms of the form Del d/dt.

The structure of the operator that generates the Maxwell

equations is interesting in light of the recent discussion

concerning John Baez's puzzle about bosons and fermions. I

posted a response to a quaternion variant of that puzzle,

where "fermionic stuff" might be generated by

g(v, w) = (v* w + w* v)/2 .

Stretch the notion of g(v, w) a bit further to allow it to

consist of operators which can act on a quaternion-valued

function. Let

v = d/dt -
Del
and w = d/dt + Del .

Then the Maxwell equations for sources in a vacuum are

g(v, w)(phi + A)
= -4 pi (rho + J/c)
.

I don't grok what this all means, but it works and I think

it's cool anyway.

Doug
Sweetser

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

The quaternions
of choice

` t + x
I + y J + z K
E + px I +
py J + pz K
phi + Ax I + Ay J +
Ax K`

for a new generation.

Back to: SPR posts

Home Page | Quaternion Physics | Pop Science | The Bike | Lindy Hop | Contact Doug