  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.
