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Re: Geometric Algebra (was Re: Intro to Differential Forms)



On 12/02/2003 06:44 PM, Larry Smith wrote:
>
> aren't these differential forms isomorphic to geometric (or
> Clifford) algebra?

They're definitely not isomorphic.

I think the Venn diagram is the most general diagram
you can have with two things:

____________________
| |
| |
| B |
| |
_________|___________ |
| | | |
| | A+B | |
| |___________|________|
| |
| A |
| |
|_____________________|


Region A is algebra without differentiation. You
can't differentiate anything using just algebra,
Clifford or otherwise. One example in this region
is using geometric products to represent torque,
angular momentum, magnetic fields, rotation
operators, etc.... no calculus required. Down
with cross products!

Another example in this region (IMHO, although not
everybody recognizes it as such) is introductory
quantum mechanics, in the form of matrix mechanics,
such as discussing polarization vectors and Stern-Gerlach
machines, where it is quite useful to distinguish
bras from kets ... but still no calculus involved.

Region B is differentiation without Clifford algebra.
People have been doing this for eons. Example: gradient
vector as a pointy vector. If you have a nice Cartesian
metric, you can freely convert back and forth between
row vectors and column vectors, i.e. between one-forms
and pointy vectors, so you don't need to bother learning
the distinction. What might have been a contraction
between a row vector and a column vector becomes just
a dot product between two vectors.

Region A+B is where geometric algebra sheds light on
differentiation and vice versa. In particular, if
you do not have a nice Cartesian metric, the gradient
operator produces a one-form that you cannot in
general convert into a pointy vector. The perfect
example of this is thermodynamics: If E is a
state function E(V,S) you can't take the dot
product between one (V,S) vector and another. You
can take the gradient of E in (V,S) space but you
have to treat the gradient as a row vector (one form)
i.e. you can't convert it to a pointy vector. I find
it tremendously liberating (i.e. simple and powerful)
to think of dS as an exact one-form and T dS as a
non-exact one-form.

Another example in the A+B region is general relativity,
where it is often appropriate to take gradients, but
it would be a nightmare to convert the gradient to a
pointy vector because funny stuff is happening with
the metric.

=========

Often people first encounter one-forms in region A+B,
which muddies the water.