What chances could not be. (quantum mechanics)

ABSTRACT

The chance of a physical event is the objective, single-case probability that it will occur. in probabilistic physical theories like quantum mechanics, the chances of physical events play the formal role that the values of physical quantities play in classical (deterministic) physics, and there is a temptation to regard them on the model of the latter as describing intrinsic properties of the systems to which they are assigned.

I argue that this understanding of chances in quantum mechanics, despite being a part of the orthodox interpretation of the theory and the most prevalent view in the physical community, is incompatible with a very wide range of metaphysical views about the nature of chance. The options that remain are unlikely to be attractive to scientists and scientifically minded philosophers.

1 Introduction 2 Underminability 3 The prevalence of underminability, 4 Views of chance 5 The chances are not intrinsic if they are underminable 6 Intrinsic properties 7 Chance in quantum mechanics 8 What the options are

1 Introduction

Philosophers of science do not like to think of themselves as metaphysicians. Their work might have something to offer the metaphysicians but proceeds without their input, and is so much the better for doing so. When a philosopher of science asks what the laws of nature are, for example, he is asking a question that can be answered by presenting the equations of the best physical theories, one that does not presuppose a particular answer to the deep debate which divides his more speculative brethren. Or so the story goes. In at least one very important case, I will argue, this assumption of autonomy from metaphysics is mistaken. The Interpretation of physical probability in quantum mechanics cannot proceed without choosing sides (and, on what passes for the standard view, not choosing very well) in the metaphysical debate about the nature of chance.

The chance of a given physical event, for instance the chance that an electron in state [S.sub.i] will be deflected up when passed through a non-uniform magnetic field with gradient along the z@axis, is the objective single-case probability that the event will occur. In quantum mechanics the chances play roughly the formal role that the values of basic quantities play in classical physics, they are represented by real numbers assigned to localized systems and evolve over time. Hence it is natural to construe them on the model of the values of the basic physical quantities as intrinsic properties pertaining to a system at a time.(1) Certain accounts of the nature of objective chance, however, are incompatible with the construal. In particular, any view of chance which allows for underminable statements about chance, i.e. statements about the chances that obtain at some time t which are incompatible with certain post-t histories. I will remain neutral on the question of whether this is bad news for the notion that the chances are intrinsic or bad news for the view that the chances supervene on (non-chance) history, but by arguing against the combination of the two views, and by showing how many views of chance are infected by underminability, I want to emphasize that one cannot -- and ought not try to -- remain innocent of metaphysics while thinking about the interpretation of chance in physics.

2 Underminability

I will follow David Lewis in using the term `theory of chance of w', written [T.sub.w], to refer to the set of conditionals which give the chances of future events at any time on the basis of preceding history, reserving the term `view of chance' for the philosophical analysis which gives the theory of chance for any particular world. I will assume a world to have the structure of a manifold of events, and the complete description of a world to be given by an assignment of values of basic quantities to all points. By `pre-t history', I mean an assignment of basic quantities to points up until and including a time t, and by `post-t history, an assignment of basic quantities to points after t. It is assumed that the chances are themselves not among the basic quantities, so when I talk about pre- and post-t histories, I mean these to be characterized without reference to the chances. Pre-t history does not include, for example, the fact that the chance of a post-t coin toss coming out heads is 1/2.(2)

A statement S about the t-chances is underminable just in case there are possible post-t histories with which it is incompatible. In any such case, the pre-t distribution of chances contains information about post-t history. To see how this works, consider a particularly simple view of chance, the so-called actual frequentist view according to which the probability of an event A in a reference class B is the relative frequency of occurrences of A within B.(3,4) So, for example, the chance that an electron is deflected up when passed through a non-uniform field with a gradient along the z-axis, if we let [N.sup.n](A) = the number of electrons in state Si actually passed through such a field and [N.sup.n] (A.B) = the number of such electrons which are actually deflected upwards, is …