## q-Gaussian processes: Non-commutative and classical aspects (1997)

Venue: | Commun. Math. Phys |

Citations: | 64 - 2 self |

### BibTeX

@ARTICLE{Kümmerer97q-gaussianprocesses:,

author = {Burkhard Kümmerer},

title = {q-Gaussian processes: Non-commutative and classical aspects},

journal = {Commun. Math. Phys},

year = {1997},

pages = {154}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We examine, for −1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt = at + a ∗ t – where the at fulfill the q-commutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 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 [FB].