From p-adic to real Grassmannians via the quantum

Let F be a local field. The action of GL(n,F) on the Grassmann variety Gr(m,n,F) induces a continuous representation of the maximal compact subgroup of GL(n,F) on the space of L^2-functions on Gr(m,n,F). The irreducible constituents of this representation are parameterized by the same underlying set both for Archimedean and non-Archimedean fields. This paper connects the Archimedean and non-Archimedean theories using the quantum Grassmannian. In particular, idempotents in the Hecke algebra associated to this representation are the image of the quantum zonal spherical functions after taking appropriate limits. Consequently, a correspondence is established between some irreducible representations with Archimedean and non-Archimedean origin.Comment: 24 pages, final version, to appear in Advances in Mathematic

Clifford geometric parameterization of inequivalent vacua

We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras --as Clifford algebras-- by different filtrations resp. induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C^*-algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non-statistical, non-definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U(2)-symmetry producing a gap-equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov-Valatin transformations is given.Comment: Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.