We define a natural transformation of certain symmetric probability measures, and show that it brings the Gaussian-law as well as several deformed analogue of it to the Poisson-law or to the corresponding deformed Poisson-law, respectively. An important example is the q-Gaussian-law with its transform being the q-Poisson-law of Saitoh and Yoshida. By analogy we define an even more general class of deformed Poisson distributions, and show that, as in the classical and the q-case, these deformed Poisson-laws can be achieved as the distributions of certain operators.
We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.
Graphical representations of the
q-creation and the
q-annihilation operators and set partition statistics (Infinite Dimensional Analysis and Quantum Probability Theory)
In this note, we introduce acertain graphical representations of the
q-creation the
q-annihilation, the
q-numb er, and the scalar operators on the
q-Fock space.These are given as the cards on which some flow lines are drawn and increasing and decreasing of the number of lines correspond to creation and annihilation, respectively.By the cards arrangement technique with these cards, of which machinery is the same as of the partition function for the lattice spins model in statistical me- chanics, we write the moments of the
q-Poisson random variable using the number of arc crossings, which is the same set partition statistic as restricted crossings investigated by P. Biane.This result suggests another
q-deformed moments-cumulants relations, which interpolates between the usual and the free cases exactly.This is the joint work with Naoko Saitoh at Ochanomizu University.1The
q -Fock space For aHibert space 7{ and
q∈(−1,1) , the
q-Fock space
Fq(H) can be defined as follows: (see, for instance, [BS1], [BKS]) Let
Ffin(H) be the linear span of vectors of the form
ξ1⊗⋯⊗ξn∈H⊗n , where
n varies in
z≥0 and we put
H⊗0≅CΩ for some distinguished vector
Ω called vacuum.We consider the sesquilinear form
⟨⋅∣⋅⟩q on
FJjn(H) given by the sesquilinear extension of
⟨ξ1⊗⋯⊗ξn∣η1⊗⋯⊗ηm⟩q=δn,m∑π∈Snqi(π)⟨ξ1∣ηπ(1)⟩⋯⟨ξn∣ηπ(n)⟩ ,
As I type these words, I have in front of me a collection of notes from a seminar in Quantum Probability run by Luigi Accardi in Roma in 1984-5 where I was at the time based as a research fellow. In those days I was rather meticulous in ordering my jottings and on turning to page 131 in my file I find twenty pages entitled “Representations of Current Groups” by K.R. Parthasarathy. I remember being fascinated by these lectures (I think there were two) in which Partha gave an account of how the Lévy-Khinchine formula arises, naturally from the construction of continuous tensor products of unitary group representations (see e.g. [St], [Ar], [PS]) and then revealed the beautiful Fock space decomposition of infinitely divisible random variables as sums of creation, conservation and annihilation operators. This is in fact one of the many perspectives through which the historian can see the “birth” of quantum probability — as a student of Robin Hudson, I had seen the subject evolve from a different point of view, through the search for quantum central limit theorems and non-commuting analogues of Brownian motion. The marriage of these two viewpoints turned out to be a wonderful thing, leading as it did to the creation of quantum stochastic calculus.