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PSA Proceedings of the Biennial Meeting of the Philosophy of Science Association

The Logic of Experimental Questions

R. I. G. Hughes

Published Jan 1, 1982• 15 citations
Publisher

Abstract

The experimental procedures of physics assign values to physical quantities; we measure the temperature of a gas, the luminosity of a light source, the momentum of a particle. Any pair, q = (A, Δ), with A an observable quantity, Δ a Borel subset of the reals, we will call an experimental question, though some prefer the term proposition. For example, in lowering my big toe into the bath, I address the experimental question (T,Δ) to the system consisting of my bathwater, where T is the temperature measured in °F, and Δ is (approximately) the interval (110,120). Theory tells us what answers we may expect, given certain initial conditions. A determinist theory like classical mechanics tells us that, under specified conditions, the measurement of a quantity will yield a particular result with certainty, whereas a statistical theory like quantum mechanics tells us what probability attaches to various possible outcomes of a measurement.

DOI
10.1086/psaprocbienmeetp.1982.1.192671

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The experimental procedures of physics assign values to physical quantities; we measure the temperature of a gas, the luminosity of a light source, the momentum of a particle. Any pair, q = (A, Δ), with A an observable quantity, Δ a Borel subset of the reals, we will call an experimental question, though some prefer the term proposition. For example, in lowering my big toe into the bath, I address the experimental question (T,Δ) to the system consisting of my bathwater, where T is the temperature measured in °F, and Δ is (approximately) the interval (110,120). Theory tells us what answers we may expect, given certain initial conditions. A determinist theory like classical mechanics tells us that, under specified conditions, the measurement of a quantity will yield a particular result with certainty, whereas a statistical theory like quantum mechanics tells us what probability attaches to various possible outcomes of a measurement.
This chapter presents the measurement problem from the point of view of the thermal interpretation of quantum physics. Everything physicists measure is measured in a thermal environment, for which statistical thermodynamics is relevant. The thermal interpretation agrees with how one interprets measurements in thermodynamics, the macroscopic part of quantum physics, derived via statistical mechanics. By its very construction, the thermal interpretation naturally matches the classical properties of our quantum world: The thermal interpretation assigns states-and a realistic interpretation for them-to individual quantum systems, in a way that large quantum systems are naturally described by classical observables. Since measurement devices are large quantum systems, this is the key to solving the measurement problem. We discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. Since quantum physics makes many deterministic predictions, for example regarding observed spectra, but also assertions about probabilities, we distinguish deterministic and statistical measurements. To clarify the meaning of the concept of measurement, we postulate (in Section 11.1) a measurement principle that defines what it means in the thermal interpretation to measure a quantity with a specified uncertainty. This turns the measurement problem from a philosophical riddle into a scientific problem in the domain of quantum statistical mechanics, namely how the quantum dynamics correlates macroscopic readings from an instrument with properties of the state of a measured microscopic system. The essence of scientific practice is the reproducibility of measurements, discussed in Section 11.2. Unlike in traditional interpretations, single, nonreproducible observations do not count as measurements, since this would violate the reproducibility of measurements-the essence of scientific practice. As a consequence, the measurement of a Hermitian quantity A is regarded as giving an uncertain value approximating the q-expectation {A} rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of A. This difference is most conspicuous in the interpretation of single discrete microscopic events. Except in very special circumstances, these are not reproducible. Thus, they have no scientific value in themselves and do not constitute measurement results. Scientific value is, however, in ensembles of such observations, which result in approximate measurements of q-probabilities and q-expectations. Sections 11.3 and 11.4 distinguish deterministic and statistical measurements, depending on whether a single observation is reproducible, and discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. How to measure the q-probabilities of events, described in terms of POVMs, is discussed in Section 11.5. Section 11.6 poses the measurement problem as a problem of the statistical mechanics of measurement devices, and discuss recent work by Allahverdyan, Balian, and Nieuwenhuizen on its resolution. Section 11.7 discusses the problem how, in the thermal interpretation, discrete events can arise, although qexpectations are by their very nature continuous. This is resolved by pointing to work by Breuer and Petruccione on piecewise deterministic processes (PDP). Section 11.8 sheds additional light on the origin of discrete events by an analysis of the role dissipative bistability plays in obtaining discreteness in classical physics.
Statistical physical theories are frequently formulated in terms of probabilistic structures founded on a ’’logic of experimentally verifiable propositions.’’ It is argued that to each experimentally verifiable proposition there corresponds an experimental procedure which, in general, alters the state of the system, and is completely characterized by a ’’measurement transformation’’ or ’’operation.’’ An analysis of the relations among these experimental procedures leads us to a ’’logic of operations’’ which is quite different from the ’’lattice theoretic logics’’ that are often considered (albeit inadequate empirical justification), as models for the calculus of experimentally verifiable propositions of quantum theory. It is seen that the quantum probability theory based on the logic of operations provides the proper mathematical framework for discussing the statistics of successive observations in quantum theory. We also indicate how a theory of quantum stochastic processes can be formulated in a way similar to the Kolmogorov formulation of the classical theory.
The measurement problem of quantum mechanics was probably born in 1926: ‘Thus Schrödinger’s quantum mechanics gives a very definite answer to the question of the outcome of a collision; however, this does not involve any causal relationship. One obtains no answer to the question “what is the state after the collision,” but only to the question “how probable is a specific outcome of the collision” (in which the quantum-mechanical law of [conservation of] energy must of course be satisfied). This raises the entire problem of determinism. From the standpoint of our quantum mechanics, there is no quantity that could causally establish the outcome of a collision in each individual case; however, so far we are not aware of any experimental clue to the effect that there are internal properties of atoms that enforce some particular outcome. Should we hope to discover such properties that determine individual outcomes later (perhaps phases of the internal atomic motions)? Or should we believe that the agreement between theory and experiment concerning our inability to give conditions for a causal course of events is some pre-established harmony that is based on the non-existence of such conditions? I myself tend to relinquish determinism in the atomic world. But this is [also] a philosophical question, for which physical arguments alone are not decisive.’ (Born, 1926a, p. 866; translation by the author) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
How can quantum mechanics be (i) the fundamental theoretical framework of contemporary physics and (ii) a probability calculus that presupposes the events to which, and on the basis of which, it assigns probabilities? The question is answered without invoking knowledge or observers, by interpreting the necessary distinction between two kinds of physical quantities - unconditionally definite quantities and quantities that have values only if they are measured - as a distinction between the manifested world and its manifestation.
In the paper, the question whether truth values can be assigned to the propositions about properties of a state of a physical system before the measurement is discussed. To answer this question, a notion that a propositionally noncontextual theory can provide a map linking each element of a bounded lattice to a truth value so as to explain the outcomes of experimental propositions associated with the state of the system is introduced. The paper demonstrates that no model based on the propositionally noncontextual theory can be consistent with the occurrence of a non-vanishing two-path quantum interference term and the quantum collapse postulate.
One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).