We study the use of methods based on the real symplectic groups $Sp(2n,mathcal{R})$ in the analysis of the Arthurs-Kelly model of proposed simultaneous measurements of position and momentum in quantum mechanics. Consistent with the fact that such measurements are in fact not possible, we show that the observable consequences of the Arthurs-Kelly interaction term are contained in the symplectic transformation law connecting the system plus apparatus variance matrices at an initial and a final time. The individual variance matrices are made up of averages and spreads or uncertainties for single hermitian observables one at a time, which are quantum mechanically well defined. The consequences of the multimode symplectic covariant Uncertainty Principle in the Arthurs-Kelly context are examined.
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