Infinite dimensional Clifford algebras?

Question

Do infinite dimensional Clifford (and/or Grassmann) algebras exist/makes sense? Do you know good references about them?

Answer 1

Sure they make sense, and nothing really differs in the construction.

I'm not handy with the applications themselves, but I can point you to a few references I did read that hint at the applications.

• Zou. Ideal structure of Clifford algebras Advances in Applied Clifford Algebras February 2009, Volume 19, Issue 1, pp 147-153

• Wene. The idempotent structure of an infinite dimensional Clifford algebra Clifford Algebras and their Applications in Mathematical Physics Fundamental Theories of Physics Volume 47, 1992, pp 161-164

• Lounesto & Wene. Idempotent structure of Clifford algebras Acta Applicandae Mathematica July 1987, Volume 9, Issue 3, pp 165-173

My main interest was in the ring theoretic structure of the algebra itself. It's also worth noting that these are talking about the countable dimensional case only. Countable dimension covers most of the ground for applied mathematics, though :)

Answer 2

Infinite dimensional clifford algebras are the setting of David Hestenes' so-called "universal geometric algebra". Hestenes uses this setting to embed vector manifolds---manifolds whose points are vectors in the UGA and thus admit a lot of niceties in terms of vector operations. For instance, if $\mathbf r(x^1, x^2, \ldots, x^n)$ is a vector function of $n$ parameters that defines a vector manifold, then the tangent vectors are indeed $\partial \mathbf r/\partial x^i$ and so on, and this is well-defined in terms of the manifold being embedded in the UGA.

For more about this application of infinite dimensional clifford algebras, see Hestens and Sobczyk's Clifford Algebra to Geometric Calculus.

Answer 3

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 $$ u(f)u(g)+u(g)u(f)= \langle f,g \rangle$$ $$ u(f)^2=1, \qquad u(f+ag)=u(f)+au(g).$$ 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