On \(q\)-deformed quantum stochastic calculus. (Q2772077)

Let \(\mathcal H\) be a Hilbert space. For \(q\in(-1,1)\), a Fock space \(\Gamma\) is introduced using the \(q\)-deformed symmetrization operator \(P^{(n)}(\psi_1\otimes\cdots\otimes\psi_n)=\sum_{\sigma\in{}S_n}q^{l(\sigma)} \psi_{\sigma(1)}\otimes\cdots\otimes\psi_{\sigma(n)}\). Standard \(q\)-deformed annihilation and creation operators \(a(\phi)\) and \(a^*(\phi)\) are defined on \(\Gamma\) for each \(\phi\in{\mathcal H}\). For a complex parameter \(\mu\) (\(| \mu| <1\)), a deformed analogue of the number of particles operator is introduced; for a bounded operator \(T:{\mathcal H}\rightarrow{\mathcal H}\), \(\lambda_n(T)\) is defined by \(\lambda_n(T)(\psi_1\otimes\cdots\otimes\psi_n)= \sum_i\mu^n\psi_1\otimes\cdots\otimes\psi_{i-1}\otimes{} T\psi_i\otimes\psi_{i+1}\otimes\cdots\otimes\psi_n\) and a deformed identity operator \(\gamma_\mu\) is defined as acting as \(\mu^n\) on \({\mathcal H}^{\otimes{}n}\). These operators satisfy deformed commutation relations, and, being bounded, the author avoids some of the problems encountered in other non-deformed theories [see \textit{R. L. Hudson} and \textit{K. R. Parthasarathy}, Commun. Math. Phys. 93, 301--323 (1984; Zbl 0546.60058)].NEWLINENEWLINEThe paper investigates the \(q\)-deformed quantum stochastic calculus based on these operators, defining suitable stochastic integrals, iterated integrals and finally proving the appropriate Itô's formula.