Recursion relation for Wick products of the CCR algebra (Q841662)

Let \({\mathcal H}\) be a complex Hilbert space and \({\mathcal W}({\mathcal H})\) the corresponding CCR algebra with the canonical linear map \(a:{\mathcal H}\to{\mathcal W}({\mathcal H})\). For every \(f\in{\mathcal H}\) define \(\varphi(f)=a(f)^*+a(f)\), referred to as the Bose field. The main result of the paper is the formula \[ [{:}\varphi(f_1)\cdots\varphi(f_n){:},\varphi(f)] = \sum_{j=1}^n[\varphi(f_j),\varphi(f)]\, {:}\varphi(f_1)\cdots\widehat{\varphi(f_j)}\cdots\varphi(f_n){:} \] for all \(f,f_1,\dots,f_n\in{\mathcal H}\), \(n\in\mathbb N\), where \(\widehat{\varphi(f_j)}\) means that \(\varphi(f_j)\) is removed, and \({:}\varphi(f_1)\cdots\varphi(f_n){:}\) stands for the Wick product of \(\varphi(f_1),\dots,\varphi(f_n)\). The authors have developed a similar formula for \(\mathcal C(\mathcal H)\), the corresponding CAR algebra of \(\mathcal H\), in [J.~Math.\ Anal.\ Appl.\ 318, No.\,1, 380--386 (2006; Zbl 1101.46037)].