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

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