THE INFINITE DIMENSIONAL COMPLEX ORTHOGONAL AND SPIN GROUPS

The overall reference for this material is a preprint (with title “The infinite complex spin groups”) by John Palmer and myself. Background on the general area is provided by the articles of Araki [1], Ruijsenaars [25,26] (note that Araki refers to the Clifford algebra as a ‘self-dual CAR algebra’) and Palmer [13] although I will try to make this exposition self-contained except for proofs. An interesting reference containing some of the results described here is [34] chapter 12 (which I first read after writing the draft for these notes).

The work discussed here is all inspired by the papers of the Kyoto School [31]. From a mathematical viewpoint the main difficulty with their work is the free use, for infinite dimensional spaces, of results whose proofs are only established in the finite dimensional case. In the latter the methods of proof are purely algebraic and may be found in the papers cited earlier in this paragraph. The generalisation of these finite dimensional results is of interest in its own right, independent of any desire to make rigorous the work of the Kyoto School. In fact it offers the prospect of going beyond their results within the purely infinite dimensional framework.