The quantum decomposition of random variables without moments (Q2842032)

Let \(\mu\) be an infinitely divisible law on \(\mathbb{R}\), with Lévy-Khintchine exponent \(\psi\) and triple \((\alpha,\sigma= |b|^2,\nu)\). Let \(e_x(t):= e^{itx}\), \(u_k:= bx+ e_2-1\), and denote by \(K\) the closed linear subspace of \(L^2(\nu)\) generated by \(\{e_x-1\mid x\in\mathbb{R}\}\), and by \(U\) the unique unitary isomorphism from the Fock space \(\Gamma(\mathbb{C}\oplus K)\) into \(L^2(\mu)\), such that \(U(e^{\psi(x)}\text{Exp}(U_x))= e_x\), for any real \(x\) and where \(\text{Exp}(U_x):= \sum_{n\geq 0} {u_x\oplus n\over \sqrt{n!}}\in \Gamma(\mathbb{C}\oplus K)\).NEWLINENEWLINE The so-called generalized field operator is \(Q:= U^*qU\), where \(q\) is the portion operator in \(L^2(\mu): (qf)(t):= t\times f(t)\).NEWLINENEWLINE The authors show first that if \(\mu\) has a second moment, then identifying \(\Gamma(\mathbb{C}\otimes L^2(\nu))\) with \(\Gamma(\mathbb{C})\otimes \Gamma(L^2(\nu))\), \(Q\) decomposes as \(Q_G\otimes 1+ 1\otimes Q_{CP}\), with in particular \(Q_{CP}= A^+(q)+ A^-(q)+ \Lambda(q)+ \operatorname{E}(\mu)\cdot 1\), where \(A^+\), \(A^-\), \(\Lambda\) denote the creation, annihilation, preservation operators, respectively, in the Fock representation of \(L^2(\nu)\).NEWLINENEWLINE The authors adress then the question of an analogous (weak) quantum decomposition, when \(\mu\) has no moment.NEWLINENEWLINE Defining distribition-valued versions of the above operators \(A^+\), \(A^-\), \(\Lambda\), and noticing that the term \(\operatorname{E}(\mu)\) in the above decomposition results from the \(e^{\psi(x)}\) contribution in \(Q_{CP}\), they finally establish that the strongly continuous one-parameter subgroup NEWLINE\[NEWLINEx\mapsto e^{A^+(e_x- 1)} e^{\Lambda(ixq)} e^{A^-(e_{-x}- 1)}NEWLINE\]NEWLINE has a weak generator \(A^+(q)+ A^-(q)+ \Lambda(q)\).