Results 1 
2 of
2
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
Abstract

Cited by 64 (2 self)
 Add to MetaCart
Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation 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 qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian 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 qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Free Calculus
, 1998
"... .4> F(H) := C\Omega \Phi M n1 H\Omega n ; where\Omega is a distinguished unit vector, called vacuum. 2) The vacuum expectation is the state A 7! h\Omega ; A\Omega i: 1 2 ROLAND SPEICHER 3) For each f 2 H we define the (left) annihilation operator l(f) and the (left) creation operator ..."
Abstract
 Add to MetaCart
.4> F(H) := C\Omega \Phi M n1 H\Omega n ; where\Omega is a distinguished unit vector, called vacuum. 2) The vacuum expectation is the state A 7! h\Omega ; A\Omega i: 1 2 ROLAND SPEICHER 3) For each f 2 H we define the (left) annihilation operator l(f) and the (left) creation operator l (f) by l(f)\Omega = 0 l(f)f 1\Omega \Delta \Delta \Delta\Omega f n = hf; f 1 if 2\Omega \Delta \Delta \Delta\Omega f n and l (f)f<F