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Re: left/right symmetry, manifest or not



Well, I'm having trouble understanding what John means. But
first, let's get the language straight. There is, strictly speaking, a
single algebra that is called "the Clifford Algebra" (Choquet-Bruhat &
DeWitt Morette, 1982 - pp 64 ff)with many sub-algebras; I prefer to refer
to each sub-algebra as a Clifford algebra. The simplest Clifford Algebra
(1-dimension) provides a basis for the complex numbers.
A 2-dimensional Clifford algebra is the algebra of quaternions.
We must distinguish, when talking Clifford, (see my Pauli matrix
example for notation) between a basis vector
s_{j}, j=1,2,3
and a basis bivector
is_{j}.

However, the product -i times a bivector gives a vector.
Now to the point of the RH Rule. The 3-d Clifford algebra
generated by the 3 Pauli matrices has an ordering of the elements labeled
by the numbers 1,2,3, as given in my example (or in Hestenes Eq. 25). The
choice or RH or LH coordinate systems comes in identifying those elements
with coordinate directions. In by first example the x and y directions
were respectively denoted by the vectors (or bivectors) with the labels 1
and 2. In my second example the labeling was reversed. The ordering is
denoted by (I think it's called the "tensor density" or something)
\epsilon_{123} which switches sign under an odd permutation of the
indices 1,2,3.

A further remark: The fact that the equations of classical physics
are chirally invariant, does not imply that there are not chiral solutions
to these equations. Otherwise right-handedness would not be a dominant
human trait. So our mathematics should easily describe object with a
definite chirality.
Which way does the electron drift in a J. J. Thompson apparatus?


On Wed, 28 Aug 2002, John S. Denker wrote:

Jack Uretsky wrote:

The truth is that the RH Rule is built into the Clifford algebra.
An easy example is with the Pauli matrices, usually defined as:
(read s_{x} as "sigma sub-x")

s_{x}s_{y}=is_{z} s_{i} squared =1, s_{i}s_{j}=-s_{j}s_{i},i not equal j

This corresponds to a RH- coordinate system. One could just as
well have written:
s_{y}s_{x}=is_{z}
which would correspond to a LH-coordinate system.

1) I wouldn't have said that. If you want to say that the RH
rule is built into the Pauli algebra, fine. But it is not
built into the Clifford Algebra.

2) The axioms of Clifford Algebra sometimes permit but
never require the construction of chiral critters such
as the unit pseudoscalar (i).

The axioms state that each of the basis vectors anticommutes
with each of the others, and that's all that need be said.

Indeed the axioms don't even say how many basis vectors
there are. You therefore can't take for granted that it is
even possible to construct chiral objects. If you doubt
me, just try it in D=2 for starters.

3) In particular, you can do all of classical electrodynamics
using Clifford Algebra without mentioning (i) or any other
chiral object, and without imposing any ordering on the
basis vectors.

4) Presumably you need to specify the handedness of the basis
vectors in order to describe weak nuclear interactions.

5) If you insist on expressing the magnetic field as a
pseudovector, you will need (i) or some similar chiral
combination of basis vectors in order to undo the mischief
and convert the magnetic field to the desired bivector
representation, for instance in the expression
F = (E + i B) gamma_0
but this doesn't repeal item (3) ... you can (and IMHO
should) describe the magnetic field as a bivector from
the beginning, to save yourself a lot of trouble, and to
make manifest the left/right symmetry of the physical laws.


--
"But as much as I love and respect you, I will beat you and I will kill
you, because that is what I must do. Tonight it is only you and me, fish.
It is your strength against my intelligence. It is a veritable potpourri
of metaphor, every nuance of which is fraught with meaning."
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