Michael E. Miller

We articulate a collection of desiderata for an account of the dynamical quantities of a physical theory, and we present a theory that meets these desiderata in the case of quantum mechanics. Our theory retains a distinction between the values of dynamical quantities and the truth values of sentences asserting that a system has a particular value of a particular quantity. This allows our theory to incorporate the phenomenon of quantum indeterminacy as a pattern in the properties instantiated by a system in a particular quantum state, while also retaining the semantics of classical logic. We contrast our theory with quantum logic, which flattens the distinction between quantity values and truth values, and we address a series of objections that have been raised to previous attempts to reconcile quantum indeterminacy with classical logic.

The representational theory of measurement provides a collection of results that specify the conditions under which an attribute admits of numerical representation. The original architects of the theory interpreted the formalism operationally and explicitly acknowledged that some aspects of their representations are conventional. There have been a number of recent efforts to reinterpret the formalism to arrive at a more metaphysically robust account of physical quantities. In this paper we argue that the conventional elements of the representations afforded by the representational theory of measurement require careful scrutiny as one moves toward such an interpretation. To illustrate why, we show that there is a sense in which the very number system in which one represents a physical quantity such as mass or length is conventional. We argue that this result does not undermine the project of reinterpreting the representational theory of measurement for metaphysical purposes in general, but it does undermine a certain class of inferences about the nature of physical quantities that some have been tempted to draw.

Del Santo and Gisin have recently argued that classical mechanics exhibits indeterminacy and that by treating the observables of classical mechanics with real number precision we introduce hidden variables that restore determinacy. In this article we introduce the conceptual machinery required to critically evaluate these claims. We present a characterization of indeterminacy which can capture both quantum indeterminacy and the classical indeterminacy Del Santo and Gisin propose. This allows us to show that there is an important difference between the two: their classical indeterminacy can be resolved with hidden variables in a manner which is not possible for quantum indeterminacy.

In the framework of quantum field theory, one finds multiple load-bearing locality and causality conditions. One of the most important is the cluster decomposition principle, which requires that scattering experiments conducted at large spatial separation have statistically independent results. The principle grounds a number of features of quantum field theory, especially the structure of scattering theory. However, the statistical independence required by cluster decomposition is in tension with the long-range correlations characteristic of entangled states. In this paper, we argue that cluster decomposition is best stated as a condition on the dynamics of a quantum field theory, not directly as a statistical independence condition. This redefinition avoids the tension with entanglement while better capturing the physical significance of cluster decomposition and the role it plays in the structure of quantum field theory.

Del Santo and Gisin have recently argued that classical mechanics exhibits a form of indeterminacy and that by treating the observables of classical mechanics with real number precision we introduce hidden variables that restore determinacy. In this article we introduce the conceptual machinery required to critically evaluate these claims. We present a characterization of indeterminacy which can capture both quantum indeterminacy and the classical indeterminacy of Del Santo and Gisin. This allows us to show that there is an important difference in kind between the two: their classical indeterminacy can be resolved with hidden variables in a manner which is not possible for quantum indeterminacy.

Approaches to the interpretation of physical theories provide accounts of how physical meaning accrues to the mathematical structure of a theory. According to many standard approaches to interpretation, meaning relations are captured by maps from the mathematical structure of the theory to statements expressing its empirical content. In this article I argue that while such accounts adequately address meaning relations when exact models are available or perturbation theory converges, they do not fare as well for models that give rise to divergent perturbative expansions. Since truncations of divergent perturbative expansions often play a critical role in establishing the empirical adequacy of a theory, this is a serious deficiency. I show how to augment state-space semantics, a view developed by Beth and van Fraassen, to capture perturbatively evaluated observables even in cases where perturbation theory is divergent. This new semantics establishes a sense in which the calculations that underwrite the empirical adequacy of a theory are both meaningful and true, but requires departure from the assumption that physical meaning is captured entirely by the exact models of a theory.

Physical theories often characterize their observables with real number precision. Many non-fundamental theories do so needlessly: they are more precise than they need to be to capture the physical matters of fact about their observables. A natural expectation is that a truly fundamental theory will require its full precision in order to exhaustively capture all of the fundamental physical matters of fact. I argue against this expectation and I show that we do not have good reason to expect that the standard of precision set by successful theories, or even by a truly fundamental theory, will match the granularity of the physical facts.

Haag's theorem has been interpreted as establishing that quantum field theory cannot consistently represent interacting fields. Earman and Fraser have clarified how it is possible to give mathematically consistent calculations in scattering theory despite the theorem. However, their analysis does not fully address the worry raised by the result. In particular, I argue that their approach fails to be a complete explanation of why Haag's theorem does not undermine claims about the empirical adequacy of particular quantum field theories. I then show that such empirical adequacy claims are protected from Haag's result by the techniques that are required to obtain theoretical predictions for realistic experimental observables. I conclude by showing how Haag's theorem is illustrative of a general tension between the foundational significance of results that can be obtained in perturbation theory and non-perturbative characterizations of the content of quantum field theory.

This paper places Julian Schwinger's development of the Euclidean Green's function formalism for quantum field theory in historical context. It traces the techniques employed in the formalism back to Schwinger's work on waveguides during World War II, and his subsequent formulation of the Minkowski space Green's function formalism for quantum field theory in 1951. Particular attention is dedicated to understanding Schwinger's physical motivation for pursuing the Euclidean extension of this formalism in 1958.