Foundational issues in statistical mechanics and the more general question of how probability is to be understood in the context of physical theories are both areas that have been neglected by philosophers of physics. This book fills an important gap in the literature by providing a most systematic study of how to interpret probabilistic assertions in the context of statistical mechanics. The book explores both subjectivist and objectivist accounts of probability, and takes full measure of work in the foundations of probability theory, in statistical mechanics, and in mathematical theory. It will be of particular interest to philosophers of science, physicists and mathematicians interested in foundational issues, and also to historians of science.
Foundational issues in statistical mechanics and the more general question of how probability is to be understood in the context of physical theories are both areas that have been neglected by philosophers of physics. This book fills an important gap in the literature by providing a most systematic study of how to interpret probabilistic assertions in the context of statistical mechanics. The book explores both subjectivist and objectivist accounts of probability, and takes full measure of work in the foundations of probability theory, in statistical mechanics, and in mathematical theory. It will be of particular interest to philosophers of science, physicists and mathematicians interested in foundational issues, and also to historians of science.
Abstract This chapter states the aims of this volume and provides an introduction to the material to follow. In the first section, various interpretations of probabilities are surveyed and discussed with an eye to physics and its probabilities. It is stressed that the axioms of the probability calculus do not suffice to understand probabilities from physics. Objectivist interpretations take it that probabilistic statements reflect mind-independent facts. However, as a discussion of frequentism, propensity views, and other objectivist interpretations shows, it is difficult to say what exactly these facts are. According to subjectivist views, probabilities are used to express degrees of belief. These views have problems to explain why physicists reasonably agree on the values of many probabilities. The second section of this introduction summarizes the chapters of this volume.
Abstract Many theories and models from physics are probabilistic. This observation raises several philosophical questions: What are probabilities in physics? Do they reflect objective chances which exist independently of the human mind? Or do they only express subjective credences and thus capture our own uncertainty about the world? Finally, which metaphysical lessons, if at all, can one draw from the largely probabilistic character of physics? The chapters collected in this volume address these questions and provide a detailed philosophical appraisal of the status of probabilities in all of physics. Particular emphasis is laid upon statistical physics and quantum mechanics. Many chapters reflect a desire to understand probabilities from physics as objective chances. These chances are characterized, e.g., as time-averages, as probabilities from a best system in the terms of David Lewis, or using the Boltzmannian typicality approach. Other chapters are sympathetic to a Bayesian view of probabilities in physics. The chapters about quantum mechanics elucidate the peculiar characteristics of quantum correlations and discuss strategies to justify the Born Rule. Finally, the chapters of this volume demonstrate how closely interpretive issues about probabilities are entangled with other foundational problems of physics such as the Reversibility Paradox, the ontology of the quantum world and the question whether the world is deterministic.
There are several (general) philosophy of physics textbooks on the market.1 An
obvious question to ask at the outset then is: Why another one? In his “Some
Philosophical Aspects of Particle Physics”, Michael Redhead bemoans the common practice of philosophers of science of dealing with examples “which are no
longer of current research interest in science” ((1980), p. 279). This practice is
still common even among many philosophers of physics, and, for the most part,
the available textbooks continue to engage in this practice. Generally, one finds a
little spacetime, a little statistical mechanics, and a little quantum theory. Now,
these are of course the ‘pillars’ of modern physics, so it is what we should expect
to be covered in any textbook on the philosophy of physics worth its salt (and,
indeed, they are well represented in this book). However, the issues that are
dealt with are usually very old fashioned and very limited in scope: ‘spacetime’
means ‘the twins paradox’ (and possibly conventionalism); ‘statistical mechanics’
means ‘time asymmetry’; and ‘quantum theory’ means ‘the measurement problem’. Redhead is surely right that this leads physicists to “regard philosophy of
science as somewhat irrelevant”. Rightly so, too, if this were in fact representative of much of what actually goes on in philosophy of physics. However, the
state of play as represented in philosophy of physics and (some) philosophy of
science journals shows a very different level of engagement, with philosophers
of physics investigating the frontiers of scientific research. This disparity (or the
impression of such) is the raison d’eˆtre for the book you are now reading. The result is an introductory textbook covering those portions of philosophy of physics
research that other textbooks fail to cover, or cover only very briefly or very
simplistically. The three pillars of modern physics-relativity, quantum theory
and statistical mechanics-are still represented, then, but from a more advanced
and contemporary perspective.
The mathematical formalism of quantum theory in terms of vectors and operators in infinite-dimensional complex vector spaces is very abstract. The definitions of many mathematical quantities used do not seem to have an intuitive meaning, which makes it difficult to appreciate the mathematical formalism and understand quantum mechanics. This book provides intuition and motivation to the mathematics of quantum theory, introducing the mathematics in its simplest and familiar form, for instance, with three-dimensional vectors and operators, which can be readily understood. Feeling confident about and comfortable with the mathematics used helps readers appreciate and understand the concepts and formalism of quantum mechanics. This book is divided into four parts. Part I is a brief review of the general properties of classical and quantum systems. A general discussion of probability theory is also included which aims to help in understanding the probability theories relevant to quantum mechanics. Part II is a detailed study of the mathematics for quantum mechanics. Part III presents quantum mechanics in a series of postulates. Six groups of postulates are presented to describe orthodox quantum systems. Each statement of a postulate is supplemented with a detailed discussion. To make them easier to understand, the postulates for discrete observables are presented before those for continuous observables. Part IV presents several illustrative applications, which include harmonic and isotropic oscillators, charged particle in external magnetic fields and the Aharonov–Bohm effect. For easy reference, definitions, theorems, examples, comments, properties and results are labelled with section numbers. Various symbols and notations are adopted to distinguish different quantities explicitly and to avoid misrepresentation. Self-contained both mathematically and physically, the book is accessible to a wide readership, including astrophysicists, mathematicians and philosophers of science who are interested in the foundations of quantum mechanics.
The Statistics of Non-Boolean Event Structures.- Possibility and Probability.- Some Remarks on Hamiltonian Systems and Quantum Mechanics.- The Possibility Structure of Physical Systems.- Quantum Mechanical Physical Quantities as Random Variables.- On the Interference of Probabilities.- Classical and Quantum Probability and Set Theory.- Discussion.- A Generalized Measure and Probability Theory for the Physical Sciences.- Discussion.- Quantum Logic, Convexity, and a Necker-Cube Experiment.- On the Applicability of the Probability Concept to Quantum Theory.- Discussion.- A Mathematical Setting for Inductive Reasoning.- Discussion.- Classical Statistical Mechanics Versus Quantal Statistical Thermodynamics: A Study in Contrasts.- Discussion.- A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory.
In this paper I propose an interpretation of classical statistical mechanics that centers on taking seriously the idea that probability measures represent complete states of statistical mechanical systems. I show how this leads naturally to the idea that the stochasticity of statistical mechanics is associated directly with the observables of the theory rather than with the microstates (as traditional accounts would have it). The usual assumption that microstates are representationally significant in the theory is therefore dispensable, a consequence which suggests interesting possibilities for developing non-equilibrium statistical mechanics and investigating inter-theoretic answers to the foundational questions of statistical mechanics.