# PlanetPhysics/C Clifford Algebra

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

Given a general Hilbert space ${\displaystyle {\mathcal {H}}}$ , one can define an associated ${\displaystyle C^{*}}$ -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 ${\displaystyle {\mathcal {H}}}$ , (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 ${\displaystyle C^{*}}$ -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 ${\displaystyle (V,Q)}$ , where ${\displaystyle V}$  denotes a real vector space and ${\displaystyle Q}$  is a quadratic form on ${\displaystyle V}$ ~. Then, the Clifford algebra associated to ${\displaystyle V}$  , denoted here as $\displaystyle \Cl(V) = \Cl(V, Q)$ , is the algebra over $\displaystyle \bR$ generated by ${\displaystyle V}$  , where for all ${\displaystyle v,w\in V}$ , the relations: ${\displaystyle v\cdot w+w\cdot v=-2Q(v,w)~,}$  are satisfied; in particular, ${\displaystyle v^{2}=-2Q(v,v)}$ ~.

If ${\displaystyle W}$  is an algebra and $\displaystyle c : V \lra W$ is a linear map satisfying ${\displaystyle c(w)c(v)+c(v)c(w)=-2Q(v,w)~,}$  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 ${\displaystyle C^{*}}$ -Clifford algebra as $\displaystyle \Cl[\mathcal{H}]$ when ${\displaystyle {\mathcal {H}}=V,}$  where ${\displaystyle V}$  is a real vector space, as specified above.

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