Scaled Boolean Algebras

Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but rather as degrees of belief in uncertain propositions. This paper separates the study of these objects from that not-entirely-mathematical problem that motivated them. That motivating problem is explicated in the first section, and the application of scaled Boolean algebras to it is explained in the last section. The intermediate sections deal only with the mathematics. It is hoped that this isolation of the mathematics from the motivating problem makes the mathematics clearer.Comment: 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press, To appear in Advances in Applied Mathematic

The extension problem for partial Boolean structures in Quantum Mechanics

Alternative partial Boolean structures, implicit in the discussion of classical representability of sets of quantum mechanical predictions, are characterized, with definite general conclusions on the equivalence of the approaches going back to Bell and Kochen-Specker. An algebraic approach is presented, allowing for a discussion of partial classical extension, amounting to reduction of the number of contexts, classical representability arising as a special case. As a result, known techniques are generalized and some of the associated computational difficulties overcome. The implications on the discussion of Boole-Bell inequalities are indicated.Comment: A number of misprints have been corrected and some terminology changed in order to avoid possible ambiguitie

Uncertainty relations and possible experience

The uncertainty principle can be understood as a condition of joint indeterminacy of classes of properties in quantum theory. The mathematical expressions most closely associated with this principle have been the uncertainty relations, various inequalities exemplified by the well known expression regarding position and momentum introduced by Heisenberg. Here, recent work involving a new sort of “logical” indeterminacy principle and associated relations introduced by Pitowsky, expressable directly in terms of probabilities of outcomes of measurements of sharp quantum observables, is reviewed and its quantum nature is discussed. These novel relations are derivable from Boolean “conditions of possible experience” of the quantum realm and have been considered both as fundamentally logical and as fundamentally geometrical. This work focuses on the relationship of indeterminacy to the propositions regarding the values of discrete, sharp observables of quantum systems. Here, reasons for favoring each of these two positions are considered. Finally, with an eye toward future research related to indeterminacy relations, further novel approaches grounded in category theory and intended to capture and reconceptualize the complementarity characteristics of quantum propositions are discussed in relation to the former

Effects and Propositions

The quantum logical and quantum information-theoretic traditions have exerted an especially powerful influence on Bub's thinking about the conceptual foundations of quantum mechanics. This paper discusses both the quantum logical and information-theoretic traditions from the point of view of their representational frameworks. I argue that it is at this level, at the level of its framework, that the quantum logical tradition has retained its centrality to Bub's thought. It is further argued that there is implicit in the quantum information-theoretic tradition a set of ideas that mark a genuinely new alternative to the framework of quantum logic. These ideas are of considerable interest for the philosophy of quantum mechanics, a claim which I defend with an extended discussion of their application to our understanding of the philosophical significance of the no hidden variable theorem of Kochen and Specker.Comment: Presented to the 2007 conference, New Directions in the Foundations of Physic

Measure Recognition Problem

This is an article in mathematics, specifically in set theory. On the example of the Measure Recognition Problem (MRP) the article highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra \algB and a property $\Phi$ of measures one can recognize by purely combinatorial means if \algB supports a strictly positive measure with property $\Phi$. The most famous instance of this problem is MRP(countable additivity), and in the first part of the article we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP(nonatomic) and MRP(separable). Then we show how our recent work D\v zamonja and Plebanek (2006) gives an easy solution to the former of these problems, and gives some partial information about the latter. The long term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem which gives such a structure theorem for measure algebras

Betting on the Outcomes of Measurements: A Bayesian Theory of Quantum Probability

We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the other measurements is returned to the agent. We show how the rules of rational betting imply all the interesting features of quantum probability, even in such finite gambles. These include the uncertainty principle and the violation of Bell's inequality among others. Quantum gambles are closely related to quantum logic and provide a new semantics to it. We conclude with a philosophical discussion on the interpretation of quantum mechanics.Comment: 21 pages, 2 figure

Kochen-Specker theorem for von Neumann algebras

The Kochen-Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type $I_{n}$ factor as algebra of observables, including $I_{\infty}$. Afterwards, we give a proof of the Kochen-Specker theorem for an arbitrary von Neumann algebra $\mathcal{R}$ without summands of types $I_{1}$ and $I_{2}$, using a known result on two-valued measures on the projection lattice $\mathcal{P(R)}$. Some connections with presheaf formulations as proposed by Isham and Butterfield are made.Comment: 22 pages, no figure