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The British Journal for the Philosophy of Science

Independently Motivating the Kochen—Dieks Modal Interpretation of Quantum Mechanics

Rob Clifton

Published Mar 1, 1995• 83 citations
Publisher

Abstract

The distinguishing feature of ‘modal’ interpretations of quantum mechanics is their abandonment of the orthodox eigenstate–eigenvalue rule, which says that an observable possesses a definite value if and only if the system is in an eigenstate of that observable. Kochen's and Dieks' new biorthogonal decomposition rule for picking out which observables have definite values is designed specifically to overcome the chief problem generated by orthodoxy's rule, the measurement problem, while avoiding the no-hidden-variable theorems. Otherwise, their new rule seems completely ad hoc.The ad hoc charge can only be laid to rest if there is some way to give Kochen's and Dieks' rule for picking out which observables have definite values some independent motivation. And there is, or so I will argue here. Specifically, I shall show that theirs is the only rule able to save Schrödinger's cat from a fate worse than death, and sidestep the Bell–Kochen–Specker no-hidden-variables theorem, once we impose four independently natural conditions on such rules

DOI
10.1093/bjps/46.1.33

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The distinguishing feature of ‘modal’ interpretations of quantum mechanics is their abandonment of the orthodox eigenstate–eigenvalue rule, which says that an observable possesses a definite value if and only if the system is in an eigenstate of that observable. Kochen's and Dieks' new biorthogonal decomposition rule for picking out which observables have definite values is designed specifically to overcome the chief problem generated by orthodoxy's rule, the measurement problem, while avoiding the no-hidden-variable theorems. Otherwise, their new rule seems completely ad hoc.The ad hoc charge can only be laid to rest if there is some way to give Kochen's and Dieks' rule for picking out which observables have definite values some independent motivation. And there is, or so I will argue here. Specifically, I shall show that theirs is the only rule able to save Schrödinger's cat from a fate worse than death, and sidestep the Bell–Kochen–Specker no-hidden-variables theorem, once we impose four independently natural conditions on such rules
Orthodox quantum mechanics includes the principle that an observable of a system possesses a well-defined value if and only if the presence of that value in the system is certain to be confirmed on measurement. Modal interpretations reject the controversial ‘only if’ half of this principle to secure definite outcomes for quantum measurements that leave the apparatus entangled with the object it has measured. However, using a result that turns on the construction of a Kochen–Specker contradiction, I argue that modal interpretations cannot deliver a metaphysically tenable conception of properties in quantum mechanics unless they also abandon the less controversial ‘if’ half of the orthodox principle.
One of the statements one would hear pretty early in any quantum mechanics course is that for quantum particles there are properties which do not have definite values. For example, if we measure |+〉 in standard basis, the best we can do to describe the outcome is to say, that with probability 1/2 we will get 0 and with probability 1/2 we will get 1. Yet one could wonder, is this really the way things are or is it just because our description of this quantum particle is incomplete, but in fact there are some hidden variables inside every quantum system that contain enough information to tell the outcome of measurement (see Fig. 1). However, many people disdain the idea of hidden variables, so there has been rather long history of theorems that forbid theories involving hidden variables. Usually if one wants to prove some no hidden variables theorem he has to assume something about the hidden variables theories he is going to forbid. One of such assumptions is non-contextuality, which means that the outcome we get after measuring with an observable is the same regardless of what other observables we choose to measure at the same time. The first theorem forbidding hidden variables theories is due to von Neumann in 1932. Then for a third of a century everyone cited von Neumann’s work carelessly, till Bell in 1966 indicated that there was an unreasonable assumption in von Neumann’s proof and urged for further examination of the problem (see [6] and [1] for more historical details). Bell also constructed a hidden variables model for one qubit that could assign the outcome of any observable in a way consistent with quantum mechanics. He also proved two theorems that forbid non-contextual hidden variables theories using counterexamples with larger quantum systems. Yet surprisingly he himself thought that hidden variables do exists and considered these theorems only as characterizations of the hidden variables theory we should look for. In this essay we will discuss several versions of one of the two Bell’s theorems. The theorem we will consider was also independently proven by Kochen and Specker in 1967 (see [5]), hence it is called Bell-Kochen-Specker theorem. For shortness and in order not to mix this theorem with another Bell’s theorem from now on we will refer to it as Kochen-Specker theorem. We will be interested in proving several versions of this theorem and the following lemma will turn out to be of great use:
Abstract All interpretations of quantum mechanics still face the issue vigorously debated by Einstein and Bohr in the 1930s: do the theory’s mere probabilistic predictions for measurement outcomes indicate that observables lack definite values prior to measurement? Or is it just that the theory is lacking, making it currently not possible to know with certainty what the true values are until they are measured? Kochen (1985) and Dieks (1989), or ‘KD’ for short, embrace the For countless useful comments and often penetrating criticism, I would like to thank Guido Bacciagaluppi, John L. Bell, Harvey Brown, Jeffrey Bub, Bill Demopoulos, Michael Dickson, Dennis Dieks, Arthur Fine, Martin Jones, Simon Kochen, Richard Healey, Meir Hemmo, David Malament, Hamar Pitowsky, Laura Ruetsche, Abner Shimony, and Howard Stein. (This is not a list of ‘endorsements’.) And I wish to express gratitude to the Social Sciences and Humanities Research Council of Canada for their continued generous support.
It is well-known that developments of quantum mechanical states according to the Schrödinger equation during a measurement seem to prevent measurements from having definite results. For, the usually assumed (idealized) Schrödinger development of the measured object and the measuring apparatus during a measurement typically results in a state of the entire system which is a superposition of the eigenstates of the measured observable and measuring observable. And the most common interpretations of quantum mechanics state that an observable does not have a definite value if the quantum mechanical state is a superposition of the eigenstates of the observable. Many ways out of this problem have been suggested. I want to discuss an interpretation suggested by S. Kochen (Kochen 1985), and suggest similar interpretations, each of which accept the universal validity of the Schrödinger equation, and yet yield definite results for measurements.
The decoherence approach to the measurement problem invokes dissipative interactions between a measuring apparatus and its environment to explain, within the context of ‘pure’ quantum mechanics (QM), why such devices appear to possess definite pointer readings. By ‘pure’ QM, I mean Schrödinger evolution with no wavefunction collapse. Several classes of interpretations of pure QM can rely on decoherence. One class is the ‘modal’ interpretations of van Fraassen (1979), Dieks (1989), Healey (1989), and others. Another class is the relative-state and many-world interpretations. I will argue that decoherence cannot help these interpretations address the general metaphysical challenges raised against them. But decoherence can help pick out a ‘special’ basis that determines which observables receive definite values. I'll explore to what extent decoherence rescues the modal (biorthogonal) basis-selection rule, and Zurek's (environmental interaction) basis-selection rule, from the basis degeneracy problem and the imperfect measurement problem.