Nodal Domains I by Eric J. Heller.

The existence of such objects is made possible by the fuzziness of their internal relative positions and momenta — the relative positions and momenta of what we are used to calling their "components." But the proper (mathematically rigorous and philosophically sound) way to define and quantify a fuzzy observable is to assign probabilities to the possible outcomes of a measurement of this observable. What we are looking for, therefore, is a mathematical tool for calculating the probabilities of measurement outcomes.

The exact form of this probability algorithm can be pinned down in a few easy steps. The details are available in this article.

We begin by observing that the classical probability algorithm is represented by a point P in a set called phase space, and that the measurement outcomes to which this algorithm assigns probabilities are represented by the subsets of this set. If P is contained in a given subset, the probability of obtaining the outcome represented by this subset is 1; otherwise it is 0.

Because this algorithm only assigns trivial probabilities (0 or 1), it is possible (and customary) to think of P as describing the state of the system in the classical sense of "state" — a collection of properties that are possessed regardless of measurements.

To be able to handle fuzzy observables, we need a probability algorithm that can accommodate nontrivial probabilities — probabilities in the entire range between 0 and 1. The most straightforward way of upgrading from the classical algorithm to what we need is

If L is contained in a given subspace S, it assigns probability 1 to the outcome represented by S. If L is orthogonal to S, it assigns probability 0 to the outcome represented by S. And if L is neither contained in nor orthogonal to S, it assigns a probability greater than 0 and less than 1 to this outcome.

The reason why a complex vector space is needed hinges again on the stability of matter. Stable objects require stable (non-decaying) particles, and stable particles require the use of complex numbers.

Let us note, in passing, that because this algorithm, unlike its classical counterpart, assigns nontrivial probabilities, it cannot be re-interpreted as a classical state (in the sense of a collection of possessed properties).