1996/04/30 by Marek Bozejko, M. Bożejko, Burkhard Kummerer +3 · 2 citations
Mathematics · Physics and Astronomy · #Class (philosophy) #Covariance #Functor #Gaussian #Hilbert space #Markov process #Quantization (signal processing) #Quantum Mechanics and Applications #Random Matrices and Applications #Second quantization #Statistical Mechanics and Entropy #cond-mat #funct-an #hep-th #math.OA #math.QA #q-alg
paper · pdf · doi:10.1007/s002200050084
published as Commun.Math.Phys. 185 (1997) 129-154 · AMS-TeX 2.1
arxiv created 1996/04/30 · openalex publication_date 1997/04/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We examine, for -1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt=at+at^* -- where the at fulfill the q-commutation relations asat^*-qat^*as=c(s,t)⋅ \id for some covariance function c(⋅,⋅) -- equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret \citeFB.