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.
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.
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.
In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics. The present Part I defines the concepts of observables, states and ensembles, clarifies the logical relations and operations for them, and shows how they give rise to dynamics and probabilities. States are identified with maximal consistent sets of weak equalities in the algebra of observables (instead of, as usual, with the rays in a Hilbert space). This leads to a concise foundation of quantum mechanics, free of undefined terms, separating in a clear way the deterministic and the stochastic features of quantum physics. The traditional postulates of quantum mechanics are derived from well-motivated axiomatic assumptions. No special quantum logic is needed to handle the peculiarities of quantum mechanics. Foundational problems associated with the measurement process, such as the reduction of the state vector, disappear. The new interpretation of quantum mechanics contains `elements of physical reality' without the need to introduce a classical framework with hidden variables. In particular, one may talk about the state of the universe without the need of an external observer and without the need to assume the existence of multiple universes.
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum Σ(A) in T(A) , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on Σ , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from Σ to Scott’s interval domain. Noting that open subsets of Σ(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A) .These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
In this chapter we shall mainly present the basic formalism that was initially developed by Heisenberg, also known as matrix mechanics (see Subsec. 1.5.7). We will first introduce in Sec. 2.1 the concept of quantum observables. Then, the problem of discrete and continuous spectra will be discussed and the basic non-commutability of quantum-mechanical observables will be deduced. While in Sec. 2.1 we discuss observables on a general formal level, in Sec. 2.2 some basic quantum-mechanical observables will be defined, and then different representations discussed and commutation relations derived. In Sec. 2.3 a basic uncertainty relation is derived. In the same section the relationship between uncertainty, superposition, and complementarity will be discussed. Finally, in Sec. 2.4 complete subsets of commuting observables will be shown to be Boolean subalgebras pertaining to a quantum algebra which is not Boolean. BASIC FEATURES OF QUANTUM OBSERVABLES This section is devoted to a general and formal exposition of quantum observables. In Subsec. 2.1.1 we shall learn how one can mathematically represent quantum observables as Hermitian operators. In Subsec. 2.1.2 we shall see how to change a basis, while in Subsec. 2.1.3 we shall find the relationship between eigenvalues of the observables and probabilities and learn how to calculate mean values. In Subsec. 2.1.4 we shall deal with an operator diagonalization. Finally, in Subsec. 2.1.5, the basic non-commutability of quantum observables will be presented by means of an example.
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.
It has been experimentally demonstrated that quantum coherence can persist in macroscopic phenomena [J.R. Friedman et al.,Nature, 406 (2000) 43]. To face the challenge of this new fact, in this article QM in its standard form is assumed to be extended by one beable (hidden variable), i. e., a quantum observable with always definite values in nature (but usually only statistically given in the quantum state). Localization is taken as the most plausible beable. The paradoxical aspects of conventional QM take now a different form. Suitably defining the notion of "subject" fully within the QM formalism, proving the quantum conditional subsystem-state theorem, and choosing the relative-decoherence interpretation of QM, the paradoxes formally disappear, leaving one with decoherence relative to the definite values of the beable; thus being only appearance, not absolute reality in QM. Relative to a different subject one has perseverance of coherence. Hence, in this approach it is claimed that decoherence and coherence, both exist in reality, but are not "seen" by the same subject, and "subject" is, in this interpretation, indispensable. The two mentioned, apparently contradictory phenomena, are in this way decoupled from each other, contradiction is avoided, and any one of the two can be treated in the way that is usual in QM.