Re: left/right symmetry, manifest or not

["John S. Denker" <jsd@MONMOUTH.COM>]

Robert Cohen wrote:
> My problem, it seems, was that I always interpreted the *cross* product as
> "circulation in the direction of A then B" - the RHR is introduced only as a
> way to keep track of the direction (in terms of psuedovectors).  As I now
> understand it, the wedge product means the same thing - it is just that the
> math incorporates the meaning with resorting to the RHR.  Conceptually, they
> are the same, though.
>
> Is this correct?

Sounds good to me.  It sounds like there never was much of a
"problem" -- Robert's only "problem" came from being _too skilled_
with the cross product, that is, if every time you see a cross
product you automatically re-interpret it in terms of circulation,
you're fine.

Indeed, in all cases where the cross product is well defined
(i.e. in D=3 with an established notion of Right-Handedness)
there is an isomorphism between cross products (pseudovectors)
and wedge products (bivectors).  To say the same thing the
other way, you don't have much to gain from from wedge products
unless:
 -- you want to get rid of the RH rule and write the laws of
    physics in a way that is manifestly left/right symmetric, or
 -- you want to work in D=2 or D=1+3 or anything other than D=3.

=================

Historical note:  I stumbled into this right after writing my
note explaining the deep similarities between ordinary rotations
and boosts,
  http://www.monmouth.com/~jsd/physics/rapidity.htm

In that document, I considered only rotations in one plane, and
boosts in one plane.  I wanted to extend it to handle the general
case of combinations of boosts and rotations in all possible
directions in D=1+3.  I thought about Euler angles for about 5
minutes, but it was pretty obvious that they were ugly in D=3
and dead-on-arrival in D=1+3.  I knew how to represent rotations
in terms of quaternions in D=3, but I had no idea how to generalize
that to D=1+3.  Google turned up a bunch of references to Clifford
Algebra and (to my very great surprise) that turned out to be the
right answer.  A happy surprise.

These things are not new, by the way.  William Clifford died in 1879.
Quaternions are older than vectors.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Clifford.html
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html

=============

I saw lots of wedge products in Misner/Thorne/Wheeler _Gravitation_,
but they were all mixed up with differential topology -- I got the
impression that they were complicated, useful only for difficult
problems.  Not true.