C*-Clifford algebra | Physics Library

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, ${\rm Cl}[\mathcal{H}]$ , which admits a canonical representation on $\mathcal L(\mathbb{F}(\mathcal{H}))$ the bounded linear operators on the Fock space $\mathbb{F}(\mathcal{H})$ of $\mathcal{H}$ , (as in Plymen and Robinson, 1994), and hence one has a natural sequence of maps $\mathcal{H} {\longrightarrow}{\rm Cl}[\mathcal{H}] {\longrightarrow}\mathcal L(\mathbb{F}(\mathcal{H}))~.$

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.

Definition 0.1 Let us briefly recall the notion of a Clifford algebra with the above notations and auxiliary concepts. Consider first a pair $(V, Q)$

, where

denotes a real vector space and $Q$

is a quadratic form on $V$

. Then, the Clifford algebra associated to $V$

, 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').

Then, with the above notation, one has the precise definition of the $C^*$ -Clifford algebra as ${\rm Cl}[\mathcal{H}]$ when

$\displaystyle \mathcal{H} = V,$

where

is a real vector space, as specified above.

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

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