In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "non-flat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and Laplace-Beltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...

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The constuction of models of non-relativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space \Gamma of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on \Gamma, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on R^d.

We derive the precise relation of the internal differential geometry of Poisson spaces developed in [1] to the corresponding external geometry given by the associated Fock structure. In particular, we identify the (internal) Laplace-Beltrami operator H \Gamma ß oe on Poisson space as the second quantization of the Laplacian on IR d . Furthermore, the diffusion process generated by H \Gamma ß oe , which is a kind of distorted Brownian motion on the configuration space, is described explicitly. All this generalizes to the case where the underlying Euclidean space IR d is replaced by an (infinite-dimensional) manifold. L'op'erateur de Dirichlet canonique et le mouvement Brownien tordu sur les espaces de Poisson R'esum'e --- Une relation pr'ecise est 'etablie entre la "g'eom'etrie diff'erentielle intrins`eque" des espaces de Poisson d'evelopp'ee dans la note pr'ec'edante [1] et la "g'eom'etrie ext'erieure" associ'ee `a la structure de Fock correspondante. En particulier, l'op'era...

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