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 , Failed to parse (unknown function "\Cl"): {\displaystyle \Cl[\mathcal{H}]} , which admits a canonical representation on Failed to parse (unknown function "\bF"): {\displaystyle \mathcal L(\bF (\mathcal{H}))} the bounded linear operators on the Fock space Failed to parse (unknown function "\bF"): {\displaystyle \bF (\mathcal{H})} of ${\mathcal {H}}$, (as in Plymen and Robinson, 1994), and hence one has a natural sequence of maps Failed to parse (unknown function "\lra"): {\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 Failed to parse (unknown function "\Cl"): {\displaystyle \Cl(V) = \Cl(V, Q)} , is the algebra over Failed to parse (unknown function "\bR"): {\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 Failed to parse (unknown function "\lra"): {\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 Failed to parse (unknown function "\Cl"): {\displaystyle \phi : \Cl(V) \lra W} such that the diagram

Failed to parse (unknown function "\xymatrix"): {\displaystyle \xymatrix{&&\hspace*{-1mm}\Cl(V)\ar[ddrr]^{\phi}&&\\&&&&\\ V \ar[uurr]^{\Cl} \ar[rrrr]_&&&& W}}

Commutes. (It is in this sense that Failed to parse (unknown function "\Cl"): {\displaystyle \Cl(V)} is considered to be `universal').

Then, with the above notation, one has the precise definition of the $C^{*}$-Clifford algebra as Failed to parse (unknown function "\Cl"): {\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 $Cliff(Q,V)$.