While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible.In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given measurement context specified by a state and a preferred observable.Halvorson and Clifton characterized the algebraic structure of beables as a von Neumann subalgebra, called a beable subalgebra, of the full observable algebra such that the probability distribution of every observable affiliated therewith admits the ignorance interpretation.On the other hand, we have shown that for every von Neumann algebra there is a unique set theoretical universe such that the internal "real numbers" bijectively correspond to the observables affiliated with the given von Neumann algebra.Here, we show that a set theoretical universe is associated with a beable subalgebra if and only if it is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with probability equal to unity.Moreover, we show that there is a unique maximal ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament and others, by the given measurement context.The set theoretical language for the ZFC-satisfiable universe, characterized by the present work, rigorously reconstructs Bohr's notion of the "classical language" to describe the beables in a given measurement context.
While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible.In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given measurement context specified by a state and a preferred observable.Halvorson and Clifton characterized the algebraic structure of beables as a von Neumann subalgebra, called a beable subalgebra, of the full observable algebra such that the probability distribution of every observable affiliated therewith admits the ignorance interpretation.On the other hand, we have shown that for every von Neumann algebra there is a unique set theoretical universe such that the internal "real numbers" bijectively correspond to the observables affiliated with the given von Neumann algebra.Here, we show that a set theoretical universe is associated with a beable subalgebra if and only if it is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with probability equal to unity.Moreover, we show that there is a unique maximal ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament and others, by the given measurement context.The set theoretical language for the ZFC-satisfiable universe, characterized by the present work, rigorously reconstructs Bohr's notion of the "classical language" to describe the beables in a given measurement context.
John Bell once argued that one ought to select, out of the 'observables' of quantum theory, some subset of 'beables' that can be consistently ascribed determinate values. Moreover, this subset should be selected so as to guarantee (among other things) that we can dispense with the orthodox interpretation's loose talk about 'measurement values': "...the probability of a beable being a particular value would be calculated just as was formerly calculated the probability of observing that value". Working in the framework of C*-algebras (in particular, Segal algebras), I propose an algebraic characterization of those subsets of bounded observables of a quantum system that can have beable status with respect to any (fixed) state of the system. It turns out that observables with beable status in a state need not all commute (a possibility Bell himself does not consider), but they must at least form a certain kind of 'quasicommutative' subalgebra determined by the state. A virtue of the analysis is that it applies to beables with continuous spectra, usually neglected in discussions of the no-hidden-variables theorems. In the (very) special case where the algebra of observables for a system is representable on a finite-dimensional Hilbert space, I give a complete characterization of the maximal beable subalgebras determined by any state of the system; the infinite-dimensional case remains open. These results are discussed in relation to previous results of a similar nature, to 'no-collapse' interpretations of quantum mechanics, and to algebraic relativistic quantum field theory.
Abstract In chapter 12 we give an overview of the historical development of hidden-variable theories aiming to attribute ‘hidden’ definite outcomes to any potential measurement on a quantum system. We start with a discussion of von Neumann’s assumption on hidden-variable models, which was later realized not to hold by Hermann, Kochen, Specker, and Bell, going into detail on Bell’s refutation of von Neumann’s arguments. We then turn to the notion of contextuality, which severely restricts hidden-variable models: we discuss Gleason’s theorem before stating and proving the famous Kochen-Specker theorem, as well as the simpler versions of Peres’ nonet and Mermin’s pentagram. These results show that quantum mechanics is not compatible with the assignment of definite measurement outcomes to all possible measurements, and such assignments can hence only be made once a restricted set of observables, a context, is selected
Noncontextual hidden variables theories, assigning simultaneous values to all quantum mechanical observables, are inconsistent by theorems of Gleason and others. These theorems do not exclude contextual hidden variables theories, in which a complete state assigns values to physical quantities only relative to contexts. However, any contextual theory obeying a certain factorisability conditions implies one of Bell's Inequalities, thereby precluding complete agreement with quantum mechanical predictions. The present paper distinguishes two kinds of contextual theories, ‘algebraic’ and ‘environmental’, and investigates when factorisability is reasonable. Some statements by Fine about the philosophical significance of Bell's Inequalities are then assessed.
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truth-functional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "state-induced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are non-Boolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided. Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the pro-positional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partial-Boolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are non-Boolean are discriminated. And two notions of truth-functionality are distinguished: truth-functionality [symbols omitted] applicable to the PQM lattices; and truth-functionality [symbol omitted] applicable to both the lattices and partial-Boolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truth-functional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and Kochen-Specker proofs of the impossibility of hidden-variables for quantum mechanics show the impossibility of a bivalent, truth-functional [symbol omitted] semantics for three-or-higher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum state-induced expectation-function on a PQM truth-functionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectation-function is the quantum analog of a classical semantic mapping, even though the domain where each expectation-function is bivalent and truth-functional is usually a non-Boolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hidden-variable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partial-Boolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.
Noncontextual hidden variables theories, assigning simultaneous values to all quantum mechanical observables, are inconsistent by theorems of Gleason and others. These theorems do not exclude contextual hidden variables theories, in which a complete state assigns values to physical quantities only relative to contexts. However, any contextual theory obeying a certain factorisability condition implies one of Bell's Inequalities, thereby precluding complete agreement with quantum mechanical predictions. The present paper distinguishes two kinds of contextual theories, ‘algebraic’ and ‘environmental’, and investigates when factorisability is reasonable. Some statements by Fine about the philosophical significance of Bell's Inequalities are then assessed. INTRODUCTION In most of the discussions of Bell's Inequalities little notice is taken of the historical circumstances of their discovery. J. S. Bell was told by J. Jauch of the work of A. Gleason [1957] which showed that simultaneous values cannot be assigned to all observables of a quantum mechanical system in a way that respects their algebraic structure (except in the simple case in which the relevant Hilbert space has dimension less than three). Gleason's theorem thus precludes a type of hidden variables theory that has come to be called ‘noncontextual’. The question of agreement between the statistical predictions of quantum mechanics and those of a noncontextual hidden variables theory does not even arise, since such a theory cannot be consistently formulated.
Abstract A difficulty in quantum logic is the well-known arbitrariness in choosing a binary operation for conditional among three principal candidates called the Sasaki, the contrapositive Sasaki, and the relevance conditional, mainly chosen from syntactical grounds. A fundamental problem remains to clarify their semantical differences manifest in operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory, developing models of quantum set theory built upon quantum logics with those three conditionals, each of which defines different quantum logical truth-value assignment for set theoretical statements. We show that each of them satisfies the transfer principle to determine the truth values of theorems of the ZFC set theory and defines the internal reals bijectively corresponding to the observables of the quantum system under consideration. Then, the truth values of their equality relations are identical irrespective of the chosen conditionals. Interestingly, however, their order relations exhibit a strong dependence on the specific conditional employed, while the order relation attains full truth value if and only if Olson’s spectral order relation holds. We further characterize the order relation in terms of experimentally accessible relations for outcomes of successive projective measurements of the corresponding observables, showing that each choice has its own operational meaning with symmetry between the Sasaki and the contrapositive Sasaki conditionals, in contrast to the majority view that favors the Sasaki conditional. Our findings reveal that quantum set theory yields empirically testable predictions concerning state-dependent binary relations between quantum observables, thereby extending Born’s probabilistic interpretation from propositions to relations.