Reversing the Geometric Product

[Ian McCreath]

Concerning Garret Sobczyk’s book Matrix Gateway to Geometric Algebra, Spacetime, and Spinors, in Chapter 6, eqn 6.10 describing the coordinate expansion of the square of a multivector, a sign function (-1)^r-k (r is the grade of the multivector and k is the cardinality of the intersection of the indices) is given for the reverse of the geometric product of the orthonormal r-blades.

 Following the prescription set out in the beginning of section 6.1, the reverse of a geometric number is obtained by reversing the order of all of the geometric products without reversing the r-vectors themselves. I expanded the square of a geometric number (the one at the bottom of p. 105) which is not a blade. The square of this g-number is -8 – 4e(1236) + 4e(1256) + 4e(1245)+4e(1234). I carried out the expansion (using Maple) to determine all 64 geometric products of the square to check if eqn 6.10 is (anecdotally) valid when the g-number is not a blade and found that this expression yielded the correct result.

While expressions for the reverse of the contraction and exterior product are commonly stated in texts for GA, I have never seen a general expression describing the reverse of the geometric product, ie.  If AB is the product of r-vectors, then AB=(-1)^(r-p)BA.  Usually the product of r-vector basis elements is the contraction, but this expression seems to work with geometric products (of the same grade).  Are there any references where I could get a better grounding of the context of this  useful relation.

As a general comment, this book is the only source I have been able to find which provides any detailed discussion for testing when a general r-vector is an r-blade using the Plucker relations. I found it interesting that it is possible, using Maple’s solve operation to solve the Plucker relations and determine the range of integer coefficients (in some cases) that define when a general r-vector is a blade.  This suggests the possibility of some correspondence to the cycle-cocycle concept from homology theory.

Is there any general reference for the reversal of the geometric product suggested by eqn 6.10.

[Lanco]

To every algebra A is in the same linear space associated an opposite algebra with the reverse product. In case of Clifford algebras there exists a natural anti-isomorphism between the two algebras.

You may look at e.g.
https://math.stackexchange.com/questions/175042/opposite-clifford-algebra
or chapter 4 in https://ctz.dk/wp-content/uploads/2016/10/Direct-Construction-of-Clifford-and-Geometric-Algebras-and-their-basic-algebraic-Structures-v.2.0.pdf

[Manfred]

> Following the prescription set out in the beginning of section 6.1, the reverse of a geometric number is obtained by reversing the order of all of the geometric products without reversing the r-vectors themselves.

You appear to be misinterpreting the statement at the bottom of p.100: "The reverse g^† of a geometric number g ∈ G_n is obtained by reversing the order of all the geometric products of the vectors which define g."  You need to understand "vectors" as "1-vectors" here.  The switching of the product order in (6.10) is not the same thing at all, and is not called the reverse.  Here the order of the product of two *orthonormal basis* r-blades is being switched, with the sign function apparently being used without elaboration, since it is pretty much self-evident.

[Ian McCreath]

Thank you greatly for your responses.  I knew that I was misinterpreting something, but I would have flailed about for a long time without the insight about the opposite algebra. This is obviously one of those areas where clifford algebra and geometric algebra present a slightly different picture, so I have to go back and look at the geometric product to see what I have missed previously.

[Manfred]

> This is obviously one of those areas where clifford algebra and geometric algebra present a slightly different picture

I'm puzzled by this statement.  A geometric algebra is just a Clifford algebra, and the geometric product and the Clifford product is one and the same thing.  Those who call it 'geometric algebra' like to use different terms and to define a bunch of additional products, but that is largely cosmetic.