Generalised Brownian Motion and Second Quantisation

Abstract

A new approach to the generalised Brownian motion introduced by M. Bozejko 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 F_V of the category of Hilbert spaces with contractions. A generalised Brownian motion is an algebra of creation and annihilation operators acting on F_V(H) for arbitrary Hilbert spaces H and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions. The positivity is encoded by a *-semigroup of "broken pair partitions" whose representation space with respect to t is V. The existence of the second quantisation as functor Gamma_t from Hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t. For a certain one parameter interpolation between the fermionic and the free Brownian motion it is shown that the ``field algebras'' Gamma(K) are type II_1 factors when K is infinite dimensional.