Formalism and Interpretation in Quantum Theory

Quantum Mechanics can be viewed as a linear dynamical theory having a familiar mathematical framework but a mysterious probabilistic interpretation, or as a probabilistic theory having a familiar interpretation but a mysterious formal framework. These points of view are usually taken to be somewhat in tension with one another. The first has generated a vast literature aiming at a “realistic” and “collapse-free” interpretation of quantum mechanics that will account for its statistical predictions. The second has generated an at least equally large literature aiming to derive, or at any rate motivate, the formal structure of quantum theory in probabilistically intelligible terms. In this paper I explore, in a preliminary way, the possibility that these two programmes have something to offer one another. In particular, I show that a version of the measurement problem occurs in essentially any non-classical probabilistic theory, and ask to what extent various interpretations of quantum mechanics continue to make sense in such a general setting. I make a start on answering this question in the case of a simplified version of the Everett interpretation. 1 Two Views of Quantum Mechanics Like any physical theory, Quantum Mechanics has both kinematical and dynamical aspects. The former delineate what changes, and the later delineate how it changes. In the particular case of quantum mechanics, this picture is obscured by the fact that the things that change – quantum states and observables – are related to one another probabilistically. To the extent that we view probabilities as attaching themselves to events – that is, to things that happen – and to the extent we think of these happenings as involving a change of state, we seem to be importing a secondary dynamics. To the extent that we think of probabilities rather as averages over static states of affairs, we seem to be committed to hidden variables – which must be both contextual and non-local (the former by Gleason’s Theorem, the latter, by Bell’s). This dilemma frames the so-called measurement problem: to give an account of quantum mechanics that embraces neither hidden variables nor any secondary dynamics, but still preserves probabilistic appearances. There is, however, another way to look at quantum mechanics. It is a remarkable mathematical fact that, given only the barest essentials of its probabilistic apparatus, the rest of the structure of quantum mechanics, including 1For Jeffrey Bub on his 65th Birthday