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Clifford algebra (Definition)

A Non–Commutative Quantum Observable Algebra is a Clifford Algebra

Definition 0.1   Let us briefly define the notion of a Clifford algebra. Thus, let us consider first a pair $(V, Q)$, where $V$ denotes a real vector space and $Q$ is a quadratic form on $V$ . Then, the Clifford algebra associated to $V$, is denoted here as ${\rm Cl}(V) = {\rm Cl}(V, Q)$, is the algebra over $\mathbb{R}$ generated by $V$, where for all $v, w \in V$, the relations: $v \cdot w + w \cdot v = -2 Q(v,w)~,$ are satisfied; in particular, $v^2 = -2Q(v,v)$ .

If $W$ is an algebra and $c : V {\longrightarrow}W$ is a linear map satisfying $c(w) c(v) + c(v) c(w) = - 2Q (v, w)~, $ then there exists a unique algebra homomorphism $\phi : {\rm Cl}(V) {\longrightarrow}W$ such that the diagram

$\xymatrix{&&\hspace*{-1mm}{\rm Cl}(V)\ar[ddrr]^{\phi}&&\\ &&&&\\ \hspace{1mm} V \ar[uurr]^{{\rm Cl}} \ar[rrrr]_{c}&&&& W\hspace{1mm}}$
commutes. (It is in this sense that ${\rm Cl}(V)$ is considered to be `universal').



"Clifford algebra" is owned by bci1.

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Keywords:  Clifford algebra

Cross-references: commutes, diagram, homomorphism, relations, vector space
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This is version 5 of Clifford algebra, born on 2008-10-19, modified 2009-02-25.
Object id is 316, canonical name is CliffordAlgebra.
Accessed 536 times total.

Classification:
Physics Classification: 03.65.Fd (Algebraic methods )
 03.65.Ta (Foundations of quantum mechanics; measurement theory )
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