On the physical interpretation of Heywood and Redhead's algebraic impossibility theorem

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.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=_ia_i P _i andB=_jb_j P_j be spectral resolutions ofA andB. Then <A,B> is the observable associated with maximal operatorR=_ijf_ij P _iP_j, where f_ij=F(a_i,b_j), 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(d_i,e_j) 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 µ_R the composite state of all the apparatuses needed to measure R. So µ_R 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 , P_(t)(Qf(r), R=r)=0.