On the analytical approach to infinite-mode Boson-Gaussian states

We develop an analytical approach to quantum Gaussian states in infinite-mode representation of the Canonical Commutation Relations (CCR's), using Yosida approximations to define integrability of possibly unbounded observables with respect to a state $\rho$ ($\rho$-integrability). It turns out that all elements of the commutative $*$-algebra generated by a possibly unbounded $\rho$-integrable observable $A$, denoted by $\langle A\rangle$, are normal and $\rho \,$-integrable. Besides, $\langle A\rangle$ can be endowed with the well-defined norm $\|\cdot\|_\rho:= {\rm tr}\,(\rho |\cdot| )$. Our approach allows us to rigorously establish fundamental properties and derive key formulae for the mean value vector and the covariance operator. We additionally show that the covariance operator $S$ of any Gaussian state is real, bounded, positive, and invertible, with the property that $S-iJ\geq 0$, being $J$ the multiplication operator by $-i$ on $\ell_2({\mathbb N})$.