Graphical representations of the
q-creation and the
q-annihilation operators and set partition statistics (Infinite Dimensional Analysis and Quantum Probability Theory)
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)⟩ ,
Mathematical Theory of Quantum Particles Interacting with a Quantum Field (A Arai) H-P Quantum Stochastic Differential Equations (F Fagnola) Quantum White Noise Calculus (U C Ji & N Obata) Can Quantumness Be an Origin of Dissipation? (T Arimitsu) What is Stochastic Independence? (U Franz) Creation-Annihilation Processes on Cellar Complecies (Y Hashimoto) Fock Space and Representation of Some Infinite-Dimensional Groups (T Matsui & Y Shimada) Free Product Actions and Their Applications (Y Ueda) Remarks on the s-Free Convolution (H Yoshida) and other papers.