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

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 $$\Cl(V) = \Cl(V, Q)$$, is the 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, $$v^2 = -2Q(v,v)$$~.

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

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

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

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

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