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Re: vector products



Larry Smith wrote:

I do recommend that others look at it too. The claims the advocates make
for it are pretty wild, but I haven't found any place yet where they
haven't delivered.

:-)

That was my experience exactly. The sequence went
-- Huh??? Not understanding a word of it.
-- These claims are preposterous -- too good to be true.
-- I can't find anything wrong.
-- I can reproduce old results easily.
-- I can produce new results easily.
-- Rapidly forgetting there ever was any other way to do things.

the papers I read seem to explicitly
assume a right-handed coordinate system.

OK, OK, you need a RH rule for some *legacy* applications. In
particular, if you want to write the electromagnetic field as
F = (E + i B) gamma0
there's a RH rule involved in the inherited definition of B,
so there sure better be a RH rule in the definition of the
unit pseudoscalar (i) or the whole thing doesn't make sense.

On the other hand, this RH rule is not required by the physics
of the electromagnetic field; it is _only_ required if we
want to invoke the correspondence principle and make contact
with the old-fashioned magnetic pseudovector, B.

That is, if we just write down Maxwell's equations in the nice
form
del F = 4pi J
it is easy to find solutions, for instance
F = (-gamma0 gamma1 + gamma1 gamma3) sin(k z - w t + ph)
represents a linearly-polarized wave polarized in the gamma1
direction and propagating in the gamma3 direction. No RH
rule need be mentioned. In fact, since gamma2 does not appear
in the question or the answer, this is a perfectly good
solution for an electromagnetic wave in D=2 flatland. I'll
bet you didn't even know it was possible to do electrodynamics
in flatland! I didn't know that, until I wrote down that
solution about a week ago and my eyes bugged out of my head.
But why not? The equation permits it, as long as you don't
befuddle yourself with any nasty cross products. Since gamma2
isn't mentioned, you couldn't define a RH rule if you wanted to!

To pound on this point just a little more: the "magnetic"
piece of the aforementioned wave is represented by the _bivector_
gamma1 gamma3. You need a RH rule if and only if you want to
convert that bivector to a pseudovector. The only reason I've
ever seen for doing such a conversion is to establish correspondence
with nasty legacy pseudovectors.

There may be other reasons (perhaps involving neutral K-mesons
or some such) but I very much doubt there is anything in
classical mechanics, classical mechanics, or even quantum
electrodynamics that requires a RH rule. The reasons for my
doubts are summed up here:
http://www.nobel.se/physics/laureates/1957/press.html

I'm quite interested in exploring this question of introducing Cliffor
Algebra at the earliest (freshman) levels. Most of the papers I'm reading
are trying to convince other physicists to switch to Clifford Algebra so
the examples are junior-undergrad level to grad-school level.

Hestenes proposed pushing it all the way down to the
high-school level. At first I thought that was hopelessly
impractical ... I made a list of about ten reasons why it
couldn't be done ... but I've decided I was wrong about most
of those reasons. Bottom line: Why not? I don't see any
reason why not. Maybe there is a reason, but I'm not seeing it.

John, how
would you like to write a short piece about how it should be introduced in
first-year calc-based physics

I'll tell you right now: In non-calc-based high-school physics,
add the torque*time bivector to the angular-momentum bivector to
describe gyroscopic precession !!without!! any nasty cross products.

Just add 'em edge-to-edge as depicted at
http://www.monmouth.com/~jsd/physics/gif48/add-bivectors.gif
where the green bivector is the initial angular momentum,
the small purple bivector is torque*time, and the yellow
bivector is the new angular momentum, which is in a new
orientation due to precession.

And how about in the math curriculum? I'm teaching Calculus III
(multivariable) this semester and we hit cross products pretty soon; should
I teach it the "right" way instead?

I don't see why not. It's just one step up from complex
numbers. If you can add real-parts and imaginary-parts,
you can add scalars and vectors and bivectors.