Generalised Brownian Motion and Second Quantisation

✍ Scribed by Mădălin Guţă; Hans Maassen

• Book ID: 102588333
• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 265 KB
• Volume: 191
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3855

✦ Synopsis

A new approach to the generalised Brownian motion introduced by M. Boz ˙ejko 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 C t from Hilbert spaces to noncommutative probability spaces is investigated for functions t with the multiplicative property. For a certain one parameter interpolation between the fermionic and the free Brownian motion it is shown that the field algebras C(K) are type II 1 factors when K is infinite dimensional.