Generalised Brownian Motion and Second Quantisation

TL;DR

This paper introduces a new symmetry-based approach to generalized Brownian motion using tensorial species, establishing a second quantisation framework and analyzing the resulting operator algebras.

Contribution

It develops a symmetry-based method for generalized Brownian motion and proves the existence of second quantisation as a functor under certain conditions.

Findings

Second quantisation exists as a functor Gamma_t under multiplicative t.

Field algebras are type II_1 factors for infinite-dimensional Hilbert spaces.

The approach unifies fermionic and free Brownian motions through interpolation.

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.