Generalised Brownian Motion and Second Quantisation Mm Add Alin Gut A

A new approach to the generalised Brownian motion introduced by M. Bo_ zejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor FV of the category of Hilbert spaces with contractions mapping a Hilbert space H to a symmetric Hilbert space FV (H) with the same symmetry as the species V. A general framework for annihilation and creation operators on these spaces is developed and shown to give vacuum expectations as prescribed by Bo_ zejko and Speicher. The existence of the second quantisation as functor from Hilbert spaces to von Neumann algebras with completely positive maps is investigated. For a certain one parameter interpolation between the classical and the free Brownian motion it is shown that the \\eld algebras" ?(K) are type II1 factors when K is innnite dimensional.