# PlanetPhysics/C Clifford Algebra

### Preliminary data for the definition of a C*-Clifford algebra

Given a general Hilbert space ${\mathcal {H}}$ , one can define an associated $C^{*}$ -Clifford algebra , $\displaystyle \Cl[\mathcal{H}]$ , which admits a canonical representation on $\displaystyle \mathcal L(\bF (\mathcal{H}))$ the bounded linear operators on the Fock space $\displaystyle \bF (\mathcal{H})$ of ${\mathcal {H}}$ , (as in Plymen and Robinson, 1994), and hence one has a natural sequence of maps $\displaystyle \mathcal{H} \lra \Cl[\mathcal{H}] \lra \mathcal L(\bF (\mathcal{H}))~.$

The details and notation related to the definition of a $C^{*}$ -Clifford algebra , are presented in the following brief paragraph and diagram.

### A non--commutative quantum observable algebra (QOA) is a Clifford algebra.

Let us briefly recall the notion of a Clifford algebra with the above notations and auxiliary concepts. 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 $V$  , denoted here as $\displaystyle \Cl(V) = \Cl(V, Q)$ , is the algebra over $\displaystyle \bR$ generated by $V$  , where for all $v,w\in V$ , the relations: $v\cdot w+w\cdot v=-2Q(v,w)~,$  are satisfied; in particular, $v^{2}=-2Q(v,v)$ ~.

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

$\displaystyle \xymatrix{&&\hspace*{-1mm}\Cl(V)\ar[ddrr]^{\phi}&&\\&&&&\\ V \ar[uurr]^{\Cl} \ar[rrrr]_&&&& W}$

Commutes. (It is in this sense that $\displaystyle \Cl(V)$ is considered to be `universal').

Then, with the above notation, one has the precise definition of the $C^{*}$ -Clifford algebra as $\displaystyle \Cl[\mathcal{H}]$ when ${\mathcal {H}}=V,$  where $V$  is a real vector space, as specified above.

Also note that the Clifford algebra is sometimes denoted as $Cliff(Q,V)$ .