Betting on Quantum Objects

Dutch book arguments have been applied to beliefs about the outcomes of measurements of quantum systems, but not to beliefs about quantum objects prior to measurement. In this paper, we prove a quantum version of the probabilists' Dutch book theorem that applies to both sorts of beliefs: roughly, if ideal beliefs are given by vector states, all and only Born-rule probabilities avoid Dutch books. This theorem and associated results have implications for operational and realist interpretations of the logic of a Hilbert lattice. In the latter case, we show that the defenders of the eigenstate-value orthodoxy face a trilemma. Those who favor vague properties avoid the trilemma, admitting all and only those beliefs about quantum objects that avoid Dutch books.Comment: 26 pages, 3 figures, 1 table; improved operational semantics, results unchange

Two Forms of Inconsistency in Quantum Foundations

Recently, there has been some discussion of how Dutch Book arguments might be used to demonstrate the rational incoherence of certain hidden variable models of quantum theory (Feintzeig and Fletcher 2017). In this paper, we argue that the 'form of inconsistency' underlying this alleged irrationality is deeply and comprehensively related to the more familiar 'inconsistency' phenomenon of contextuality. Our main result is that the hierarchy of contextuality due to Abramsky and Brandenburger (2011) corresponds to a hierarchy of additivity/convexity-violations which yields formal Dutch Books of different strengths. We then use this result to provide a partial assessment of whether these formal Dutch Books can be interpreted normatively.Comment: 26 pages, 5 figure

Probabilism for stochastic theories

I defend an analog of probabilism that characterizes rationally coherent estimates for chances. Specifically, I demonstrate the following accuracy-dominance result for stochastic theories in the C*-algebraic framework: supposing an assignment of chance values is possible if and only if it is given by a pure state on a given algebra, your estimates for chances avoid accuracy-dominance if and only if they are given by a state on that algebra. When your estimates avoid accuracy-dominance (roughly: when you cannot guarantee that other estimates would be more accurate), I say that they are sufficiently coherent. In formal epistemology and quantum foundations, the notion of rational coherence that gets more attention requires that you never allow for a sure loss (or “Dutch book”) in a given sort of betting game; I call this notion full coherence. I characterize when these two notions of rational coherence align, and I show that there is a quantum state giving estimates that are sufficiently coherent, but not fully coherent

Extensions of bundles of C*-algebras

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ℏ→0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ℏ→0 case) on the fiber C*-algebras

Is the classical limit “singular”?

We argue against claims that the classical ℏ→0 limit is "singular" in a way that frustrates an eliminative reduction of classical to quantum physics. We show one precise sense in which quantum mechanics and scaling behavior can be used to recover classical mechanics exactly, without making prior reference to the classical theory. To do so, we use the tools of strict deformation quantization, which provides a rigorous way to capture the ℏ→0 limit. We then use the tools of category theory to demonstrate one way that this reduction is explanatory: it illustrates a sense in which the structure of quantum mechanics determines that of classical mechanics

The First Three Rungs of the Cosmological Distance Ladder

It is straightforward to determine the size of the Earth and the distance to the Moon without making use of a telescope. The methods have been known since the 3rd century BC. However, few amateur or professional astronomers have worked this out from data they themselves have taken. Here we use a gnomon to determine the latitude and longitude of South Bend, Indiana, and College Station, Texas, and determine a value of the radius of the Earth of 6290 km, only 1.4 percent smaller than the true value. We use the method of Aristarchus and the size of the Earth's shadow during the lunar eclipse of 2011 June 15 to derive an estimate of the distance to the Moon (62.3 R_Earth), some 3.3 percent greater than the true mean value. We use measurements of the angular motion of the Moon against the background stars over the course of two nights, using a simple cross staff device, to estimate the Moon's distance at perigee and apogee. Finally, we use simultaneous CCD observations of asteroid 1996 HW1 obtained with small telescopes in Socorro, New Mexico, and Ojai, California, to derive a value of the Astronomical Unit of (1.59 +/- 0.19) X 10^8 km, about 6 percent too large. The data and methods presented here can easily become part of a beginning astronomy lab class.Comment: 34 pages, 11 figures, accepted for publication in American Journal of Physic