An Indeterminacy Relation for Several Observables and Its Classical Interpretation

An examination of the limitations placed by the conceptual structure of quantum mechanics on the simultaneous measurement of two or more observables leads, for the case of an even number $m=2k$ of observables, to a quantitative estimate, depending only on the mean values of their commutators in the state in question, of the least value which the product of their $m$ uncertanties may assume. This indeterminacy relation is interpreted classically to mean that, by measurement of these observables alone, a point ($p, q$) in phase space can be determined only to within a certain channel whose normal cross section, as measured by the value of the $k\mathrm{th}$ integral invariant taken over it, must be at least of order $h^k$.