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What is a Clifford Algebra



Hi all-
On Wed. August 28 John Denker wrote:
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.


Not according to my book.

Here are the axioms of a Clifford Algebra according to one text:
Let V(n,(s)) s<=n, be an n-dimensional vector space over the real numbers,
with inner product (v|w) and a basis (e_i) [read this as "e sub i"]
such that
(e_i|e_j) =0 i notequal j
(e_i|e_i) =1 i = 1,..,s
(e_i|e_i) =-1 i = s+1,...n

Introduce a product vw of vectors in V(n,(s)) which is associative and
distributive with respect to addition, and which satisfies the condition:
vw+wv = 2(v|w)

The resulting algebra of all possible sums and products is called the
Clifford algebra C(V) of V.

The Clifford algebra is a linear space of dimension 2^n. The basis is the
set of all ordered products of the e_i.

Note: V(1,0) is the set of complex numbers; V(2,0) is the quaternion
algebra. The authors state that the Dirac algebra in d-dimensions is the
Clifford algebra of V(n,(1)), but I have not investigated this.






--
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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
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