Coherent state representation and unitarity condition in white noise calculus (Q2718433)

Let \({\mathcal S}_C'({\mathbf R})\) be the space of complex tempered distributions and \(\nu\) be the white noise measure on \({\mathcal S}_C'({\mathbf R})\) that is \(\int_{{\mathcal S}_C'({\mathbf R})} e^{\langle z,\xi\rangle}d\nu(z)=e^{-\|\xi\|^2/2}\), where \(\xi\in{\mathcal S}_C\) and \(\|\cdot \|\) is the Hermitian norm. Let us identify \(L^2({\mathcal S}_C'({\mathbf R}),\nu)\) with the Fock space NEWLINE\[NEWLINE\Gamma(H)=\Bigl\{\varphi=(f_n):f_n\in H^{{\widehat \otimes}n},\;\|\varphi\|^2=\sum n!\|f_n\|^2<\infty\Bigr\}, \quad H=L^2({\mathbf R}).NEWLINE\]NEWLINE Define NEWLINE\[NEWLINE{\mathcal W}= \overline{ \{\varphi_\xi=(1,\xi,\frac {\xi^{\otimes 2}}{2!},\dots, \frac{\xi^{\otimes n}}{n!},\ldots)},\;\xi\in{\mathcal S}_C\}\quad(\text{linear span}),NEWLINE\]NEWLINE and consider the triplet \({\mathcal W}' \supset \Gamma(H) \supset {\mathcal W}\). The creation and annihilation at time \(t\) are defined as \(a_t = \int_{{\mathcal S}_C'} z(t) Q_z\nu(dz), a_t^*= \int_{{\mathcal S}_C'} \overline{z(t)} Q_z \nu(dz)\), respectively, where \(Q_z \varphi = \langle \varphi_{\overline{z}},\varphi\rangle\varphi_z, \varphi\in{\mathcal W}, z\in {\mathcal S}_C'.\) Any element \(\Phi\in {\mathcal W}^*\) has the following representation, called diagonal coherent state representation, \(\Phi = \int_{{\mathcal S}_C'} {\mathcal W}_\Phi (z)Q_z\nu(dz)\), where \({\mathcal W}_\Phi (z) = \Gamma(\sqrt 2)^*\Phi(\frac{z+\overline{z}}{\sqrt 2})\) and \(\Gamma(\sqrt 2)\) is the quantization of the scalar \(\sqrt 2 I\).