This paper uses the sample probability-space description to review, under general conditions, the measurement process whereby an empirical expectation value is obtained for comparison either with other values or with values calculated from a theory. The emphasis here is on some of the conditions which must be satisfied by general sequences of single measurements, about which an observer may have relatively little knowledge, in order that such sequences yield suitable expectation values. In particular, sequences are considered for which the requirements that the single measurements of the sequence be independent and made of the same physical quantity on ensembles of identically prepared systems are not necessarily satisfied. The differences between sequences which satisfy these requirements and those which do not are discussed in terms of the implications or meaning of the resultant expectation value as a point of contact between theory and experiment.
This paper uses the sample probability-space description to review, under general conditions, the measurement process whereby an empirical expectation value is obtained for comparison either with other values or with values calculated from a theory. The emphasis here is on some of the conditions which must be satisfied by general sequences of single measurements, about which an observer may have relatively little knowledge, in order that such sequences yield suitable expectation values. In particular, sequences are considered for which the requirements that the single measurements of the sequence be independent and made of the same physical quantity on ensembles of identically prepared systems are not necessarily satisfied. The differences between sequences which satisfy these requirements and those which do not are discussed in terms of the implications or meaning of the resultant expectation value as a point of contact between theory and experiment.
Abstract Here, some difficulties resulting from the application of any empirical acceptability conditions on sequences of single measurements are investigated. In particular, the often used acceptability requirement that each single measurement be made under the "same conditions" is discussed. In quantum mechanics, this means that each single measurement is made of the same physical quantity on a system in a ensemble of identically prepared systems. One of the resultant difficulties is that such an application leads to an infinite regression of sequences of single measurements. That is, it does not account for the fact that an observer must start the process of measurement or knowledge acquisition. Furthermore, it is seen that there are some basic sequences of single measurements for which an observer can not possibly know at the outset that the "same condition" requirements are satisfied. These include those measurements by which the homogeneity of space-time is tested. The possible relevance of these difficulties to physics is shown by first considering two possibilities of avoiding these difficulties. One is that the "same condition" requirements can be given the weaker interpretation that there be no physical principle forbidding an observer from knowing in terms of limit empirical means, that they are satisfied at the outset of any sequence. This gets rid of the infinite regression problem as it does not mean that an observer must know in fact that these requirements are satisfied. The other possibility is that if physics does not forbid one in principle from measuring an expectation value in an arbitrarily small time interval then both the basic sequence as well as those by which one knows the "same" requirements are satisfied can be relegated to arbitrarily small time intervals. As far as physics is concerned, then the epistemological difficulties while existing in these small intervals, do not exist for other times, or almost all time. It is then shown that quantum mechanics, as distinct from classical mechanics, and the special relativity require that an infinite time interval is necessary to measure, as a limit mean, any expectation value. Thus physics denies both the above possibilities as it forbids an observer from knowing even in principle, by any finite time that the "same" requirements are satisfied. Also, physics forbids the relegation of the epistemological problems to arbitrarily small time intervals.
Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state η where η does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [Qsτ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement ∀Qsτ [U(Qsτ) defined⇒E(H,U(Qsτ),ψQsτ)],where U(Qsτ) is the probability measure U associates with the infinite instruction string Qsτ and ψQsτ is the outcome sequence obtained by carrying out Qsτ. E(H, U(Qsτ),ψQsτ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(Qsτ) are true for ψQsτ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(Qsτ) corresponds to a nontrivial product probability measure and U H-agrees with experiment, then the outcome sequence ψQsτ is H-random. H-randomness is defined here in terms of the statement E(H, P, φ). Another property of a valid physical theory, which is defined here, is that, for some Qsτ, U(Qsτ) must be determinable on much of AH from ψQsτ. Sufficient conditions for this property to hold are given. AH is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, φ) are given. Among other things, it is proved that, if E(H, P, φ) holds and P is a nontrivial probability measure on A, then φ is not definable in H.
In this work, a definition of agreement between a physical theory and experiment, proposed in earlier work, is extended to be relative to τ where τ is Zermelo-Fraenkel set theory. The main aim of this work is to show that this definition, unlike that of earlier work, is sufficiently powerful to include relations between limit properties of empirical outcome sequences and expectation values obtained from the physical theory. We also extend, to the more powerful τ, some earlier results on randomness and the empirical determinability of the probability measure which a physical theory assigns to the outcome set of an infinite sequence of experiments.
In the standard, textbook treatment of quantum mechanics, the contact with experiment is described in terms of probabilities for obtaining this, that, or another result, when identical experiments are performed on a huge number ( ensemble ) of identical objects. Such an ensemble description was adequate during the first half century of quantum theory, because the technology of that era was incapable of making a measurement without destroying or severely changing the measured object (photon, atom, …). In recent years, however, advances in technology and technique have made possible repetitive quantum measurements on a single quantum object, with each measurement influencing the object only minimally. Such measurements cannot be analyzed using solely the ensemble theory. Additional theoretical concepts are needed. The purpose of this book is to describe the methods and theory of such measurements (as well as the more elementary theory of ensembles of measurements). This book is written for two types of readers: those who have had only a little previous contact with quantum mechanics, and those who have had much. Readers of the first type are presumed to understand atomic physics at a qualitative level. The unstarred sections of this book offer such readers an overview of the present state-of-the-art of quantum measurements on single objects and prospects for the future, as well as an elementary overview of the theory of such measurements.
The informal aspects, arising in the interpretation of physical experiments, of the theory of probability and mathematical statistics are discussed. The conditions that verifying experiments must satisfy are presented and the role of heuristic (extralogical) assertions is analyzed using the example of mathematical expectation. The principal hypotheses implicit in experiments are enumerated: the principle of reproducibility (the past will be repeated in the future); the principle of reasonable sufficiency; and, the statistical principle (better to predict something rather than nothing). Considerable attention is devoted to Fisher and multisample confidence intervals. It is noted that Fisher confidence intervals are inconsistent. The arguments for introducing contrivances into practical calculations of probabilities are enumerated: incompleteness of any system of hypotheses; subjective estimates of probabilities; adjoining of statistical ensembles; nonstationariness and instability; rare phenomena; and, the use of classical probabilities and the law of large numbers. It is concluded that the relative frequency of appearance (empirical probability) is a normal physical quantity in the sense that it admits physical measurement. Its abnormality is manifested in the fact that it is burdened, more than other physical quantities, with conventions and hypotheses which must be specially checked (verified).
Measurements play a crucial role in doing physics: Their results provide the basis on which we adopt or reject physical theories. In this note, we examine the effect of subjecting measurements themselves to our experience. We require that our contact with the world is empirically warranted. Therefore, we study theories that satisfy the following assumption: Interactions are accounted for so that they are empirically traceable, and observations necessarily go with such an interaction with the observed system. Examining, with regard to these assumptions, an abstract representation of measurements with tools from quantum logic leads us to contextual theories. Contextuality becomes a means to render interactions, thus also measurements, empirically tangible. The measurement becomes problematic---also beyond quantum mechanics---if one tries to commensurate the assumption of tangible interactions with the notion of a spectator theory, i.e., with the idea that measurement results are read off without effect. The problem, thus, presents itself as the collision of different epistemological stances with repercussions beyond quantum mechanics.