Abstract
Heywood and Redhead's 1983 algebraic (Kochen-Specker type) impossibility proof, which establishes the inconsistency of a broad class of contextualized local realistic theories, assumes two locality conditions and two auxiliary assumptions. One of those auxiliary conditions, FUNC*, has been called a physically unmotivated,ad hoc formal constraint.
In this paper, we derive Heywood and Redhead's auxiliary conditions from physical assumptions. This allows us to analyze which classes of hidden-variables theories escape the Heywood-Redhead contradiction. By doing so, we hope to clarify the physical and philosophical ramifications of the Heywood-Redhead proof. Most current hidden-variables theories, it turns out, violate Heywood and Redhead's auxiliary conditions.
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1. See Redhead [1], pp. 133–136, for a complete discussion.
2. Arthur Fine first pointed out the implicit reliance on FUNC*, and proved FUNC* to be both consistent with and independent of the Value Rule.
3. LetA=∑iai P i andB=∑jbj P′j be spectral resolutions ofA andB. Then <A,B> is the observable associated with maximal operatorR=∑ijfij P i⊗P′j, where fij=F(ai,bj), and where function F is 1:1.
4. Heywood and Redhead's versions of these conditions employ equivalence-class notation to specify the ontological context. {<D,E>}={R} refers to the equivalence class of all possible <D,E> formed by using different F functions (cf. Footnote 3). Clearly, such notation assumes that ifR andR′ are two distinct commuting maximal operators formed as described in Fn. 3 fromD andE using two different F(di,ej) functions, then [Q]t (R)(R)=[Q]t (R′)(R), so that [Q]t {R}(R) is uniquely defined.
Heywood and Redhead never rely upon this assumption in their proof, however. It is easily checked that a Heywood-Redhead contradiction follows from my non-equivalence class versions of OLOC, ELOC, VR, and FUNC*. Therefore, I will not use equivalence class notation.
5. Here I denote by µR the composite state of all the apparatuses needed to measure R. So µR may represent the state of more than one device.
6 This is because in a hidden-variables framework, quantum mechanical probabilities are a weighted average of the underlying hidden-variables probabilities.
7. This argument resembles a proof given by Fine [8].
8. Recall from theorem 1 that ifQ=f(R), then for all quantum states φ, Pφ(t)(Q≠f(r), R=r)=0.
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Elby, A. On the physical interpretation of Heywood and Redhead's algebraic impossibility theorem. Found Phys Lett 3, 239–247 (1990). https://doi.org/10.1007/BF00666014
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DOI: https://doi.org/10.1007/BF00666014