TITLE: Are Clifford algebras and differential forms equivalent frameworks for differential geometry?
QUESTION [27 upvotes]: I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based on differential forms. Is this true?
More generally, are these frameworks strictly equivalent? I have heard that geometric algebra is only a suitable approach once a metric tensor has been chosen. Thus, it would seem that this approach is in fact less powerful.
Putting aside all personal and aesthetics preferences, which framework is more general (if any)?

REPLY [7 votes]: I think that two things must be distinguished here. One is the "traditional"/mainstream Clifford algebras, and the other, Hestenes'. 
The traditional definition (which you'll find in pretty much any book) takes a vector space V, and builds an algebra out of its tensor algebra. In particular, it takes a quotient of the tensor agebra by the relation $v\otimes v = |v|^2 I$. Now, from looking at this definition, it is obvious that the vector space must come with a chosen quadratic form. A global version of this is the requirement of a metric on the manifold (if you're considering the tangent bundle, of course). On the other hand, because of the dependence of the algebraic structure on the metric, pretty much every "tangential" object lives in this algebra. For example, vectors and one-forms are naturally expressed as representations of the same element in different bases (cf. Snygg: Clifford Algebra, a Computational Tool for Physicists). Computationally, this is very useful: look, for example, at the definition of the Hodge dual in both the differential form formalism, and the Clifford formalism.
Mor formally, while the alternating and Clifford algebras are not isomorphic themselves, the alternating algebra is isomorphic to the graded algebra associated with a filtration of the Clifford algebra. This means that they are indeed very close to each other, the main difference being this: take vectors $v\in V$. While in $\Lambda^*(V)$ we necessarily have $v\wedge v=0$ (identifying $V$ with its dual with the metric), in $CL(V)$ this is not so. What happens is that taking the gradation implies to quotient out lower order terms. Another formulation of this fact relates to the multiplication of orthogonal vs non-orthogonal elements of $V$.
Now, Hestenes, in his book, has stripped the Clifford algebras to their bare algebraic properties, taking an axiomatic, rather than constructive approach. It is true that from this point of view, no assumption is made on the existence of a metric. But it is also true that it is not clear how to build such an object without choosing one (i.e., a construction different from the one above). As long as you ignore this point, his formalism can be very interesting.