Clifford Algebra for 2D spaces I am studying Clifford and geometric algebra.  Specifically, decomposing the line element by finding the matrices that satisfy the necessary algebraic relations.  I know that for 4D spacetime, the Dirac matrices are the elements used for this, and for three dimensions, it is the Pauli matrices.  But what about 2D spaces?  Certainly I can also use the Pauli Matrices, but any two of them will do, leaving the choice ambiguous. Is there a specific, unique choice of geometric basis elements for 2D spaces?  Are the dimension of these matrices related to the dimension of the space?
Hope this makes sense, Thanks.
 A: There is not a unique choice, in the same way that a linear transformation isn't uniquely specified by a matrix; it's basis-dependent at the very least. There isn't a unique choice for the spacetime algebra either.
The canonical way of representing Clifford algebras as matrices is talked about here. There's some relationship with spatial dimension, but I don't know that that is explicitly.
If you're interested in really getting into Clifford/geometric algebra and working with it effectively, I would urge you to try to stop thinking about it in terms of matrices, and rather as a construction on vectors themselves; for example, the geometric product can be thought of as a product on the exterior algebra. Two excellent books which introduce geometric/Clifford algebra without matrix representations are Chris Doran and Anthony Lasenby's Geometric Algebra for Physicists; and Pertti Lounesto's Clifford Algebras and Spinors. Doran and Lasenby's book does not talk about matrices at all I believe, but Lounesto's does have a chapter all about the canonical matrix representations.
Probably any book giving a more mathematician-oriented treatment of Clifford algebras, such as Ian Porteous' Clifford Algebras and the Classical Groups, would also talk about these canonical matrix representations, though Porteous' book is a much harder read than the other two I mentioned.
