Re: [Phys-L] physics without cross products and (!) without right-hand rules

[John Denker <jsd@av8n.com>]

Cross products are an obvious source of difficulty, and getting rid of them is
a worthy goal unto itself.  However, hiding behind that is a less obvious but
even more profound problem, namely the whole idea of a right-hand rule.  If
we play our cards right, we solve both problems at once.

We pick up the discussion in the context of:

> > The faces of a bivector can be indistinguishable. They can be transparent.
> > All we need is a sense of circulation around the edge.

That's still true.  For the bivector A∧B you get the correct circulation if
you first follow A, then B ... for any vectors A and B.

> > As a super-important example, the volume of a parallelepiped is given
> > by the trivector A∧B∧C which is equal to A×B·C ... definitely not A×B×C.

On 08/25/2016 08:35 AM, Moses Fayngold wrote:

> This misses a vitally important piece of information. Trivector A^B^C
> is more than just volume (and shape) of the respective
> parallelepiped. It also decrees the way we go around the
> parallelepiped along its edges.

Yes, indeed it does.  You get the correct orientation if you first follow
A, then B, then C ... for any vectors A, B, and C.

>  In fact, this piece is explicitly formulated in the following (correct!) statement about bivectors:

> > All we need is a sense of circulation around the edge.

That's still true.

> But this is in conflict with: 

> > The faces can be indistinguishable. They can be transparent.

The latter is still true, but there is no conflict.

> Marking two faces differently is equivalent to distinguishing between
> two opposite senses of circulation around the edge.

Those two ideas (circulation and face-coloration) are not equivalent
in general.  They can be related to one another *if and only if* you
impose the right-hand rule.  I have been trying to point out that it
is advantageous to /not/ impose the right-hand rule.

In any case, the bivector physics requires the direction of circulation
and does not require the faces to be colored.

As a matter of choice, you may freely /choose/ to impose the right hand
rule and use it to color the faces, but the physics does not require it.


> The reason is that a bivector is not just a parallelogram made of
> segments, but a parallelogram made of directed segments (vectors!)
> and in a way (tail to tip!) creating the sense of rotation.

We agree that the bivector is directed.

> Therefore making the faces transparent would only mask the existing
> handedness (chirality), but not eliminate it.

The physics of electromagnetism is not chiral.  Using the intrinsic non-chiral
orientation of the bivector A∧B, namely first A then B is quite sufficient to
keep the physics happy.  The gory details are here, with no right-hand rules
anywhere:
  https://www.av8n.com/physics/maxwell-ga.htm

>  Transparent sides will not eliminate the difference between A^B and B^A! 

True but irrelevant.  A∧B is -B∧A, necessarily, for any vectors A and B,
whether you distinguish the faces or not, whether you impose a right-hand
rule or not.