Our next order of business is to incorporate a compatibility criterion.

To understand why such a criterion is needed, remember the experiment of Englert, Scully, and Walther. The two measurements to which the photon may be subjected answer the respective questions:

These two measurements are incompatible (in the sense that they cannot be simultaneously made), for the first experiment requires that the shutters remain closed, whereas the second requires that they be opened. This incompatibility is not due to a lack of skill on the part of the experimenters. It has a logical basis, inasmuch as the corresponding questions involve contradictory assumptions. The first question takes for granted that the atom went through a single slit and asks: which? The second question takes for granted that the atom went through both slits and asks: how? If the first question makes sense, the second clearly doesn't, and vice versa.

The existence of incompatible measurements is another consequence of the existence of stable objects. For this, it is not enough that the internal relative positions of composite objects are fuzzy. They must also remain fuzzy. To see what this entails, consider an object that is composed of one positive and one negative charge, like atomic hydrogen. In order that its single internal relative position remains as fuzzy as it is, the electrostatic attraction between the two charges (which by itself would cause this relative position to become less fuzzy) has to be counterbalanced by some kind of repulsion (which by itself would cause this relative position to become more fuzzy). What could give rise to this repulsion?

Nature uses a most ingeneous method to produce this repulsion: simply let the relative momentum be fuzzy, too.

A fuzzy position stays as fuzzy as it is if it is subject to no force and if the corresponding momentum is sharp. If both the position and the corresponding momentum are fuzzy, the position changes by a fuzzy distance (in any given time span); as a result, it gets fuzzier.

But even that is insufficient. Suppose that the fuzziness of the relative position decreases. Since, as a consequence, the electrostatic attraction between the charges increases, the repulsion, too, must increase. In other words, a decrease in the fuzziness of the relative position r must be accompanied by an increase in the fuzziness of the relative momentum p — and vice versa. The mathematical formulation of this conclusion is the "uncertainty" relation

Δr·Δp ≥ ħ/2.

The root-mean-square deviations from the mean, Δr and Δp, are the respective measures of fuzziness. ħ is the (reduced) Planck constant.

It follows that the incompatibility of a relative position with the corresponding relative momentum — the impossibility of measuring the two observables simultaneously with arbitrary precision — is a precondition (condition of possibility) of the existence of stable material objects.

Which is why we need a formal criterion of compatibility. It is readily shown that this takes the following form: the outcomes of compatible measurements correspond to commuting projection operators.

Finally we assume non-contextuality. This means: if the interval C is the union of two disjoint intervals A and B, then the probability of finding the value of an observable O in C is the sum of two probabilities, the probability of finding it in A and the probability of finding it in B.

This apparently innocent assumption actually holds only for probabilities that are assigned to individual systems on the basis of past or future measurement outcomes. Probabilities that are assigned to composite systems or on the basis of past and future outcomes are generally contextual. (The experiment of Greenberger, Horne, and Zeilinger provides an example of contextuality for composite systems. The experiment discussed here illustrates contextuality for time-symmetric probability assignments.)

At this point we have everything that is needed to prove Gleason's theorem, according to which the probability of an outcome represented by the projection operator P is the trace of the operator product WP, where W is the system's density operator. It is readily shown how this depends (i) on actual measurement outcomes and (ii) on the times of measurements.

Upshot: the formalism of (non-relativistic) quantum physics is a direct consequence of the existence of stable objects that (i) "occupy space" and (ii) are composed of finite numbers of objects that don't.