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\begin{document}



 \subsection{A Non--Commutative Quantum Observable Algebra is a Clifford Algebra}

\begin{definition}
Let us briefly define the notion of a \emph{Clifford algebra}. Thus, let us consider first a pair $(V, Q)$, where $V$ denotes a real \textcolor{HI}{\uline{\mbox{vector}}} space and
$Q$ is a quadratic form on $V$~. Then, the \emph{Clifford algebra associated to $V$}, is denoted here as
$\Cl(V) = \Cl(V, Q)$, is the \emph{algebra over $\bR$ 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, $v^2 = -2Q(v,v)$~.
\end{definition}

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


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

\end{document}
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