DG - Clifford Algebra / Differential Forms

[Mike706]

Hello Everyone,

I'm currently working through a differential geometry book that uses Clifford's algebra instead of differential forms. If anybody has knowledge of both, would you please explain what the differences between the approaches are, and what (if any) are the advantages of each type?

Thanks for your help,
Mike

 

[quasar987]

Have you read the preface/introduction/... of your book? I would think that the author would have justified his preference of Clifford algebra over diff. forms at length in there.

 

[arkajad]

I do not think you can avoid differential forms in differential geometry. I am sure your book makes also use of them. I would be really surprised if it does not.

 

[Mike706]

I have read it, but I was wondering if anybody could be more specific or elaborate on some of the advantages differential forms might have over Clifford Algebra. The book is "A New Approach to Differential Geometry Using Clifford's Geometric Algebra," by John Snygg. The following excerpts are from the preface:

"That is because in this book, Clifford algebra replaces the more complicated and less powerful formalism of differential forms. ... Using Clifford algebra, it becomes unnecessary to discuss mappings back and forth between the space of tangent vectors and the space of differential forms. With Clifford algebra, everything takes place in one space."

"The advantage that quaternions have for dealing with rotations in 3 dimensions can be generalized to higher dimensions using Clifford Algebra. This is important for dealing with the most important feature of a surface in any dimension - namely its curvature."

"The components of the Riemann tensor, used to measure curvature, are somewhat abstract in the usual formalism. Using Clifford algebra, the components of the Riemann tensor can be interpreted as components of an infinitesimal rotation operator that indicates what happens when a vector is 'parallel transported' around an infinitesimal loop in curved space."

 

[arkajad]

I would not take too seriously these statements. Distinction between covariant and contravariant vectors is useful. When there is a metric, the two spaces can indeed be identified, no Clifford algebra is needed for that. Riemann tensor components interpretations is a standard in books on Riemannian geometry, also no Clifford algebra needed.

If metric is not specified, for instance when it is a dynamical field of your theory, you do not have a fixed Clifford algebra, but you are still able to conteplate Lagrangians etc.

Clifford algebra makes certain calculations easier (once you learn the technique), but it restricts your field of applications.

 

[Mike706]

Thanks again for your reply.

Which fields do Clifford algebras not have applications in that differential forms do (and vice versa)? Also, does the differential forms approach not make the distinction between contravariant and covariant vectors?

 

[arkajad]

Thanks again for your reply.

Which fields do Clifford algebras not have applications in that differential forms do (and vice versa)? Also, does the differential forms approach not make the distinction between contravariant and covariant vectors?

It depends whether you are interested in physics or just in mathematics.

As I wrote, when you have your metric[/fixed], then you can only work with vector fields ond operators on them, like div, etc. You often consider grad as a vector field, not as a form. That happens often in older texts. Clifford algebra is not needed. It certainly makes things more "elegant" (once you have the right kind of a taste). Sometimes it is convenient to use the Hodge operator as well. This is also elegant. But there are fanatics of Clifford algebras, and they want to rewrite evrything in this language. Sometimes, like with theories of gravitation, such a rewriting becomes essentially unreadable. Devotion is needed.

For instance there are theories of gravitation based on the concept of an affine connection instead of that of a metric. Clifford algebra is inapplicable in this case. Differential geometry is.