themselves; but this is already obvious from the properties of the spectrum of operators. Only the values of functions of [*commuting*]{} observables (and, in particular, of functions of a single observable) can be predicted with certainty from the values of the observables themselves.

However, as was shown for the case of linear combinations of the Pauli spin matrices, his argument does not demonstrate that one cannot assign natural values to all observables that lead to natural and consistent classical probabilities.

The nature of quantum observations may put severe limits on what is observable through experiment and how these are combined to estimate the values of other observables, but it does not seem to put restrictive limits on the joint values of noncommuting observables. Indeed, [von Neumann]{} [@vNeu] shows in the same section mentioned above how to get estimates of joint probabilities of noncommuting quantities if sufficiently large ensembles are available.

Particle paths {#path}
==============

It is widely believed that it is impossible to ascribe definite paths to moving quantum particles, except at special points where the position has been measured. The particle paths seen in a cloud chamber, say, can still be explained as an illusion created by correlation patterns 