The field of quantum mechanics has proved extraordinarily successful at predicting all of the observed behaviours of electrons under the experimental circumstances just described. Indeed, it has proved extraordinarily successful at predicting all of the observed behaviours of all physical systems under all circumstances. Since its development in the late 1920s and early ’30s, it has served as the framework within which virtually the whole of theoretical physics is carried out.

The mathematical object with which quantum mechanics represents the states of physical systems is called a wave function. It is a cardinal rule of quantum mechanics that such representations are complete: absolutely everything there is to say about any given physical system at any given moment is contained in its wave function.

In the extremely simple case of the single-particle system considered above, the wave function of the particle takes the form of a straightforward function of position (among other things). The wave function of a particle that is located in some region A, for example, has a nonzero value in A and the value zero everywhere in space except in A. Likewise, the wave function of a particle that is located in some region B has a nonzero value in B and the value zero everywhere in space except in B. The wave function of a particle that is in a superposition of being in region A and in region B—for example, an electron of x-spin = +1 that has just passed through a y-box—has nonzero values in A and B and the value zero everywhere else.

As formulated in quantum mechanics, the laws of physics are solely concerned with how the wave functions of physical systems evolve through time. It is an extraordinary peculiarity of standard versions of quantum mechanics, however, that there are two very different categories of physical laws: one that applies when the physical system in question is not being directly observed and one that applies when it is.

The laws in the first category usually take the form of linear differential equations of motion. They are designed to entail, for example, that an electron with x-spin = +1 that is fed into a y-box will emerge from that box, just as it actually does, in a superposition of being in the y-spin = +1 path and being in the y-spin = −1 path. All of the experimental evidence currently available suggests that these laws govern the evolutions of the wave functions of all isolated microscopic physical systems, in all circumstances.

Yet there are good reasons for doubting that these laws constitute the true equations of motion for the entire physical universe. First, they are completely deterministic, whereas there seems to be an inevitable element of chance (as discussed above) in the outcome of a measurement of the position of a particle that is in a superposition with respect to two regions. Second, what the linear differential equations of motion predict regarding the process of measuring the position of such a particle is that the measuring device itself, with certainty, will be in a superposition of indicating that the particle is in region A and indicating that it is in region B. In other words, the equations predict that there will be no matter of fact regarding whether the measuring device indicates region A or region B.

This analysis can be extended to include a human observer whose role is to look at the measuring device to ascertain how the measurement comes out. What emerges is that the observer himself will be in a superposition of believing that the device indicates region A and believing that the device indicates region B. Equivalently, the observer will be in a physical state (or brain state) such that there is no matter of fact about what region he believes the device to be indicating. Obviously, this is not what happens in actual cases of measurement by human observers.

How then is it possible to account for the fact that superposition states are never actually observed? According to the standard interpretation of quantum mechanics, when a physical system is being observed, a second category of explicitly probabilistic laws applies exclusively. These laws do not determine a precise position for a given particle but determine only a probability that it will have one position or another. Thus, the laws as applied to a particle in a superposition of regions A and B would predict not that “the particle exists in A and the particle exists in B” but that “there is a 50 percent chance of finding the particle in A and there is a 50 percent chance of finding the particle in B.” That is, there is a 50 percent chance that the measurement alters the particle’s wave function to one whose value is zero everywhere except in A and a 50 percent chance that it alters the particle’s wave function to one whose value is zero everywhere except in B.

As to the distinction between the circumstances in which each category of laws applies, the standard interpretation is surprisingly vague. The difference, it has been said, is that between “measurement” and “ordinary physical processes” or between what does the observing and what is observed or between what lies (as it were) in front of measuring devices and what lies behind them or between “subject” and “object.” Many physicists and philosophers consider it a profoundly unsatisfactory state of affairs that the best formulation of the most fundamental laws of nature should depend on distinctions as imprecise and elusive as these.

Assuming that the existence of two ill-defined categories of fundamental physical laws is rejected, there remains the problem of accounting for the absence of superposition states in measurements of quantum mechanical phenomena. Since the 1970s this so-called “measurement problem” has gradually emerged as the most important challenge in quantum mechanics.