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Re: left/right symmetry, manifest or not



I wrote:

Now, suppose I define a particular vector product such that
its magnitude
represents a "circulation" in the "direction of A then in
the direction of
B". In what way is this equivalent to the wedge product?
How about the
cross product?

to which John S. Denker responded:

<nit> The word "magnitude" muddies the waters in ways that
probably were not intended. I will ignore it. Wedge products
(bivectors) and cross products (pseudovectors) have directions
as well as magnitudes. </nit>

Understood. I didn't quite know how to write it (I considered "its
magnitude represents the circulation and its direction represents the
direction of A then in the direction of B" but I thought there were too many
"directions" in there).

Certainly that is the meaning of the wedge product: circulation
in the direction of A then B.

Good - I am not as confused as I thought.

Note that I'm assuming that A and B are physically-significant
vectors in the problem at hand.

Yes.

As a concrete example: consider a linearly-polarized electromagnetic
wave with a certain polarization vector K propagating in the direction
L. Then the magnetic piece of the electromagnetic field can be
written as K wedge L. That can be represented as a little patch of
area in the KL plane, with a direction-of-circulation marked on it
(K then L) so that K wedge L is the negative of L wedge K.

The cross product can't be constructed without invoking the Right-Hand
rule. And if you draw it as a vector (perpendicular to the KL plane)
it's just a skinny little vector, with no obvious sense of
circulation.
You can't interpret it as a circulation without a second invocation
of the RH rule.

So IMHO if you want to describe a circulation, write K wedge L and
be done with it -- save yourself from a lot of extra work.

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?

____________________________________________
Robert Cohen; rcohen@po-box.esu.edu; 570-422-3428; http://www.esu.edu/~bbq
Physics, East Stroudsburg Univ., E. Stroudsburg, PA 18301