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Re: left/right symmetry, manifest or not



Larry Smith wrote:

Is this differential form wedge product the same thing as the Geometric
Algebra wedge product?

Wow, what a good question. I've been wrestling with that.
If you'd asked a week ago I wouldn't have had a good answer.

Answer: The axioms are the same, but the physical intepretation is
slightly different.

Certainly geometric algebra is not the same as differential topology.

Geometric algebra revolves around the geometric product, which
is something more than the wedge product. In fact for plain
(grade-1) vectors, the rule is
A B = A dot B + A wedge B
where the geometric product, as is customary, is written by
juxtaposition without any special operator symbol.

The typical textbooks that apply differential topology to
physics (notably e.g Misner/Thorne/Wheeler) have not heretofore
unified the dot product with the wedge product, so in some
sense they don't do enough. They miss the full power and
simplicity of Clifford Algebra.

On the other hand, they typically associate the wedge product
with differential forms and nothing else, so in some sense
they make it more complicated than it needs to be.

To repeat: I consider Clifford Algebra (dot plus wedge) to
be _simpler_ than wedge products alone. The rules for
multiplying wedge products are just tricky enough that I
often find it easier to grind out the entire geometric product
and then pick out the pieces that correspond to the wedge
product. See equations 18 and 20 in Harke to see how this
fits into an even grander scheme of things.
http://www.harke.org/ps/intro.ps.gz

Beware that the physical intepretation of one-forms is related
to but backwards from the physical interpretation of vectors,
and similarly for two-forms versus bivectors.
-- a strong magnetic field bivector A wedge B is visualized
as a large patch of area in the AB plane.
-- a strong magnetic field two-form A wedge B is visualized
as a large amount of flux-lines (actually flux-tubes)
crossing _per unit area_ in the AB plane.

That means you can't trivially switch back and forth from
the geometric-algebra representation and the differential-forms
representation. The dimensional-analysis is upside-down.

But the axiomatic properties of the wedge product are the same
in both cases.

Other than this upside-down interpretation, if you learn
Clifford Algebra you won't need to unlearn anything when you
get to differential topology.