TITLE: Infinite dimensional Clifford algebras?
QUESTION [7 upvotes]: Do infinite dimensional Clifford (and/or Grassmann) algebras exist/makes sense? Do you know good references about them?

REPLY [3 votes]: There are infinite dimensional Clifford C*-algebras, which do make sense and are connect to quantum mechanics. In fact the algebra of Canonical Anticommutation Relations can be thought of as a Clifford algebra over a complex vector space, so that in field theory (or any quantum theory with infinitely many degrees of freedom) one requires Clifford algebras over infinite dimensional spaces.
Usually the algebra is defined over an infinite-dimensional space with well-defined geometric properties, namely a Hilbert space. For a real Hilbert space $H$ the Clifford algebra $Cl(H)$ is the (unique) C*-algebra generated by unitaries $u(f)$, $(f\in H)$ subject to
$$ u(f)u(g)+u(g)u(f)=< f,g >$$
$$ u(f)^2=1, \qquad u(f+ag)=u(f)+au(g).$$
Have a look at 
D. Shale, F. Stinespring "States of the Clifford algebra", Ann. Math. 80 (1964), pp 365-381
for more details and possibly other references
-Ollie