In order to deal with Kochen and Specker's no-go result, van Fraassen invoked 'splitting' of non-maximal observable depending on 'context'. In his contextual approach there are different physical quantities corresponding to a non-maximal operator. Such physical quantities are different, so those values are not required to coincide. However, Heywood and Redhead showed the impossibility of local truth-value assignment in contextual approach. Their argument is restricted to finite dimensional and non-relativistic case. In this paper we shall show a similar no-go result in more general framework (von Neumann algebra). Also, our result will be applied to Algebraic quantum field theory.
In order to deal with Kochen and Specker's no-go result, van Fraassen invoked 'splitting' of non-maximal observable depending on 'context'. In his contextual approach there are different physical quantities corresponding to a non-maximal operator. Such physical quantities are different, so those values are not required to coincide. However, Heywood and Redhead showed the impossibility of local truth-value assignment in contextual approach. Their argument is restricted to finite dimensional and non-relativistic case. In this paper we shall show a similar no-go result in more general framework (von Neumann algebra). Also, our result will be applied to Algebraic quantum field theory.
The semantics underlying the propositional structure of Hilbert space quantum mechanics involves an inherent ambiguity concerning the impossibility of assigning definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. Although the preceding Kochen-Specker result forbids a global, absolute assignment of truth values to quantum mechanical propositions, it does not exclude ones that are contextual. In this respect, the Bub-Clifton “uniqueness theorem” is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a suitable “preferred” observable. It is suggested that the most natural choice of the latter, especially for confronting the problem of truth valuation in quantum mechanics, results in the determinateness of the observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The resulting account of truth is compatible with a realist conception of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature.
What if Einstein, Podolsky, and Rosen were right that quantum mechanics is incomplete, and there do exist ‘‘elements-of-reality’’ corresponding to all observables whether or not they are compatible? The implications are surprising. Not only must the values of these elements-of-reality depend upon what measurements are actually performed on the system, but also upon measurements performed at spacelike separation from it! The former dependence has been dubbed ‘‘contextualism,’’ and the latter ‘‘nonlocality.’’ It is important for students to understand the necessity of these two features, if only to reinforce the point that a straightforward classical recasting of quantum theory in terms of pre-existing measurement independent elements-of-reality is not feasible. Unfortunately the argument for contextualism, in particular, is daunting due to its reliance on an elaborate geometrical argument. I aim here to minimize the geometry needed to argue for contextualism by using some ideas inspired by the original arguments of Bell and of Kochen and Specker. Using similar methods, I simplify the closely related geometrical argument for nonlocality due to Heywood and Redhead and to Stairs.
Abstract The logic of a physical theory reflects the structure of the propositions referring to the behaviour of a physical system in the domain of the relevant theory. It is argued in relation to classical mechanics that the propositional structure of the theory allows truth-value assignment in conformity with the traditional conception of a correspondence theory of truth. Every proposition in classical mechanics is assigned a definite truth value, either ‘true ’ or ‘false’, describing what is actually the case at a certain moment of time. Truth-value assignment in quantum mechanics, however, differs; it is known, by means of a variety of ‘no go ’ theorems, that it is not possible to assign definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. In this respect, the Bub-Clifton ‘uniqueness theorem ’ is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a particular observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The conceptual implications of the resulting account are traced down and analyzed at length. In this light, the traditional conception of correspondence truth may be viewed as a species or as a limit case of the more generic proposed scheme of contextual correspondence when the non-explicit specification of a context of discourse poses no further consequences.
There is no such thing as “The Quantum Logical Interpretation of Quantum Mechanics”. Rather, there is a cluster of interpretations, all of which can be described as “quantum logical”. Here I provide a general framework for discussing interpretations of this kind, and then locate various suggestions within it. The presentation owes much to van Fraassen (see, in particular, van Fraassen 1974 ); his “modal interpretation” is one of those I discuss, along with those of Jauch, Putnam and Kochen. I begin by rehearsing some orthodox quantum theory. Within quantum mechanics we deal with a set 0 of measurable quantities, orobservables(position, momentum, components of spin and so on). Experiment can determine the value of an observable for a given system: the values so determined will be real numbers. A maximal amount of information about what the result would be for a given system, whatever experiment we chose to perform on it, is available once we know thestateof the system.
The Kochen–Specker theorem shows that it is impossible to assign sharp values to all dynamical variables in quantum mechanics in such a way that the algebraic relations among the values of dynamical variables whose self-adjoint operators commute are the same as those among the operators themselves. We point out that, for quantum theories obtained by quantizing some classical theory, this condition – Kochen–Specker non-contextuality – is implausible from the start because quantization usually changes algebraic relations. We explain why this is so, using the formalism of deformation quantization and its conception of star products , and we illustrate the relevance of this point using various examples of dynamical variables quantized via Weyl quantization and coherent state quantization. Our observations suggest that the relevance of the Kochen–Specker theorem to the question of whether one can assign sharp values to all dynamical variables is rather limited.
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: