On the Kolmogorov--Wiener--Masani spectrum of a multi-mode weakly stationary quantum process

We introduce the notion of a $k$-mode weakly stationary quantum process $\varrho$ based on the canonical Schr\"odinger pairs of position and momentum observables in copies of $L^2(\mathbb{R}^k)$, indexed by an additive abelian group $D$ of countable cardinality. Such observables admit an autocovariance map $\widetilde{K}$ from $D$ into the space of real $2k \times 2k$ matrices. The map $\widetilde{K}$ on the discrete group $D$ admits a spectral representation as the Fourier transform of a $2k \times 2k$ complex Hermitain matrix-valued totally finite measure $\Phi$ on the compact character group $\widehat{D}$, called the Kolmogorov-Wiener-Masani (KWM) spectrum of the process $\varrho$. Necessary and sufficient conditions on a $2k \times 2k$ complex Hermitian matrix-valued measure $\Phi$ on $\widehat{D}$ to be the KWM spectrum of a process $\varrho$ are obtained. This enables the construction of examples. Our theorem reveals the dramatic influence of the uncertainty relations among the position and momentum observables on the KWM spectrum of the process $\varrho$. In particular, KWM spectrum cannot admit a gap of positive Haar measure in $\widehat{D}$. The relationship between the number of photons in a particular mode at any site of the process and its KWM spectrum needs further investigation.