Generalized Measure Theory Research Articles

QUANTUM MECHANICS AS QUANTUM MEASURE THEORY

The additivity of classical probabilities is only the first in a hierarchy of possible sum rules, each of which implies its successor. The first and most restrictive sum rule of the hierarchy yields measure theory in the Kolmogorov sense, which is appropriate physically for the description of stochastic processes such as Brownian motion. The next weaker sum rule defines a generalized measure theory which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed "as the squares of quantum amplitudes" is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a "realistic" interpretation of quantum mechanics.

A theory of measurement in diagnosis from first principles

Reiter and de Kleer have independently developed a theory of diagnosis from first principles. Reiter's approach to computing all diagnoses for a given faulty system is based upon the computation of all minimal hitting sets for the collection of conflict sets for ( sd,components,obs). Unfortunately, his theory does not include a theory of measurement. De Kleer and Williams have developed GDE—general diagnostic engine. Their procedure computes all minimal conflict sets resulting from a measurement before discriminating the candidate space. However, they do not provide a formal justification for their theory. We propose a general theory of measurement in diagnosis and provide a formal justification for our theory. Several novel contributions make up the central focus of this paper. First, this work provides an efficient incremental method for computing new diagnoses given a new measurement, based on the previous diagnoses predicting the opposite. Second, this work defines the concepts of conflict set resulting from a measurement, equivalence classes and homogeneous diagnoses as the basis of the method. Finally, this work leads to a procedure for computing all diagnoses and discriminating among competing diagnoses resulting from a measurement.

Tensor products in generalized measure theory

Klay, Randall, and Foulis established that the signed weight space of the tensor product of two quasimanuals each having a positive, finite-dimensional state space is isomorphic to the algebraic tensor product of the signed-weight spaces of the factors. We obtain a generalization of this result for arbitrary quasimanuals. A compactness condition due to Cook—here calleddiscreteness—is discussed and shown to be preserved under the formation of tensor products. It is shown that the predual of the signed weight space of a tensor product of discrete manuals is the projective (ordered) tensor product of the preduals of the signed weight spaces of the factors.

A general axiomatization of additive measurement with applications

Necessary and sufficient conditions are specified for a general theory of additive measurement that presumes very little set-theoretic structure. The theory is illustrated for numerical representations in extensive, conjoint, difference, threshold, expected utility, probability, ambiguity, and subset measurement. © 1992 John Wiley & Sons, Inc.

A skeptic's beliefs and disbeliefs

Two kinds of skepticism are distinguished: systematic (or radical) and methodological (or moderate). It is argued that the former is logically impossible, whereas the latter is part of the scientific outlook. When beliefs in the supernatural and the paranormal are subjected to methodical doubt, they turn out to be not only lacking in empirical support but also at variance with certain scientific principles and with certain general philosophical presuppositions of scientific research. It is noted that, when scientists overlook these principles, they run the risk of consuming or even producing some non-scientific ideas. The cases of parapsychology and psychoanalysis are examined in some detail. Next ten other specimens of pseudoscience are briefly analyzed: general measurement theory, quantum theory of measurement, creationist cosmology, genetic instruction hypothesis, selfish gene hypothesis, human sociobiology, applied catastrophe theory, applied game theory, neoclassical microeconomics, and textualism. Finally some of the key tacit principles of the creed of the methodological skeptic are formulated: materialism (an extension of naturalism), realism, rationalism, empiricism, and systemism.

Combining qualitative and quantitative factors—an analytic hierarchy approach

The Analytic Hierarchy Process (AHP) provides a general theory of measurement for expressing both tangible and intangible factors. In this paper, intangible or qualitative factors are looked upon as dimensions which we have not yet learned how to measure very well. Through a redundant paired-comparison process. AHP allows us to translate qualitative preferences into ratio scaled data. In addition, the structuring stage of AHP facilitates problem understanding.

Theory and Optimization of the Excess Bias Measurement

In this paper we explain the general theory of the excess bias measurement as a radar parameter suitable for deriving information of meteorological relevance from the fluctuation of weather echoes. Since the excess bias is the difference between the estimates of the radar reflectivity factor within the radar measurement cell obtained with the logarithmic and simulated receivers, the integral and differential characteristics of this measurement are accurately analyzed. It is shown that, for specific reflectivity models, the excess bias is a monotonic function of the reflectivity variation within the measurement cell, and the experimental data confirm a close correlation between these two parameters. Finally, we assess the influence of the simulated receiver on detecting reflectivity variations through excess bias measurements. The study refers to the class of receiver responses for which the output y is related to the input power P by: y = aPb, where a and b are characteristic of the receiver. The optimum receiver responses varies with the reflectivity models considered. Whenever the reflectivity field exhibits linear variation in a logarithmic scale, the optimum response approaches the response of a quadratic receiver (b = 1), as much as the reflectivity variations increase. For the models which describe reflectivity steps at the edge of precipitation cells, the optimum b is smaller than the corresponding one for the linear model; the greater the fraction of the measurement cell filled with precipitation, the smaller becomes the optimum b.

Irreversible thermodynamics, quantum mechanics and intrinsic time scales

It is often assumed that phenomenology is a rather weak tool for the analysis of natural systems because it lacks generality. However, in a series of papers we have developed a phenomenological calculus based upon a general theory of measurement and mathematical representations (or, equivalently, upon system response as a bilinear form) which has a broad range of application. The present paper illustrates its power and versatility by demonstrating that irreversible thermodynamics and quantum mechanics are homomorphic. This result is, in itself, interesting since it shows that a large class of dissipative, deterministic systems are homomorphic to a large class of ideal, stochastic systems. In both cases, the metrical structure of the phenomenological calculus allows us to define a “proper time” intrinsic to the system dynamics. With this intrinsic time, a dynamics of aging can be defined upon the system's parameter space. In this context, Schrödinger's equation is seen as a dynamics of aging.

A Measure-Theoretic Approach to Urban Fields with Applications to Origin-Constrained Travel

Mathematical formulations of urban structure usually invoke either a strictly continuous or a strictly discrete account of spatial distributions. The arbitrary nature of this dichotomy is illustrated by urban retail and commercial activity, which can be usefully represented as a mixture of discrete and continuous components. This paper exploits concepts from general measure theory to extend the notion of an urban field. The resulting formulation yields discrete and continuous models as special cases. The abstract-field model is interpreted for the case of urban commercial structure, and is applied to develop a generalized extension of the origin-constrained interaction model. The descriptive utility of the formulation is then demonstrated in the case of commercial arterials in Albany, New York.

Continuum calculus and Feynman’s path integrals

An operational calculus is set up with the specific aim of resolving the problem of the integral of functionals in a general complex Banach algebra. The functionals that occur frequently in physics are Feynman’s path integrals for quantum mechanics which usually appear in the form of an exponential of an integral. Through the establishment of two operations, the r differentiation and p integration, we succeed in constructing a formula, in closed form, for the integrals of Feynman type functionals. Applications to known problems (quantum harmonic oscillator, the electron–phonon system) corroborate conventional results. This formula is found to be at once consistent and more general than the method of projection into cylinder functionals of Frederichs type. It does not require an additive Gaussian measure, and it admits integration with finite limit functions. The methodology developed is general and applicable to other branches of mathematics. It is particularly suited to the study of infinitely divisible distributions in probability theory. We rederive, with facility, Lévy’s formulas for continuous sums, and Lévy–Khintchine and Kolmogorov formulas. We also find it applicable to continuum matrix algebra, where the formula for the determinant of matrices of continuous indices is given as a p integral. As to algebraic identities, we give a continuum version of the binary expansion, and retrieve Stirling’s formula of factorials by p integration. The idée-clef lies in the concept of infinitesimal ratio of a function in the same way that differential calculus deals with infinitesimal differences. Then the functional integral appears to be a natural product of the interaction between the conventional integration and the proposed p integration. It also heralds the possibility of a generalized measure theory for integrals where the basic operation between the measure and the integrand is not bilinear.

Steroids and the practical aspects of performing binding studies

Many cellular regulatory processes are described by ligand interactions with macromolecules. Steroid complexing with specific cytoplasmic receptor proteins and the subsequent binding of these complexes to nuclear constituents are two such processes which are fundamental to the understanding of steroid hormone action. These reactions are characterized by the binding parameters k, the equilibrium association constant, and M 0, the binding site molarity. The current methods used to determine these two parameters are discussed in relation to the general theory of binding parameter measurement. A number of practical aspects of binding parameter measurement are considered. These include problems encountered by working at dilute non-uniform protein concentrations, non-specific binding, and the limitations of all such experimental systems. An exact solution to the non-linear Scatchard plot is presented for the case when only two non-interacting classes of binding sites need be considered. Finally, a set of specific conclusions is offered which should enable investigators to obtain the most accurate information possible whenever carrying out steroid hormone or receptor binding studies.

The appropriateness of the expected utility model

There seems to be general agreement that the choice between risky alter natives depends on two factors : what the possible outcomes may be and how possible they are, i.e. how likely they are to occur. A theory of choice typically proceeds by giving numerical measures to these two factors and combining these numbers into a single index, which the agent is supposed to maximalize if he is deciding rationally. Therefore decision theory can conveniently be divided into two parts the first one about the proper way to measure our two factors, thus being closely connected with the general theory of measurement, and the second about the proper way to combine these measures.

Rational choice and polynomial measurement models

Rational choice behavior is defined in a manner that subsumes rational choice theory as part of general measurement theory. The special class of polynomial measurement problems are defined, and their solutions are reduced to solving polynomial inequalities. An algebraic criterion is given for the solvability of arbitrary finite sets of polynomial inequalities, resolving a conjecture of Tversky.

On interactions between dynamical systems

Given a known dynamical system S in interaction with an unknown system T, this paper describes a method whereby a dynamical description of T, together with the mode of interaction or coupling between S and T, can be inferred from the knowledge of S alone. Thus, the analysis to be presented forms part of the general theory of measurement or observation. Some applications of this analysis to biological problems are described, and some of its general theoretical applications are discussed.

Generalized measure theory

It is argued that a reformulation of classical measure theory is necessary if the theory is to accurately describe measurements of physical phenomena. The postulates of a generalized measure theory are given and the fundamentals of this theory are developed, and the reader is introduced to some open questions and possible applications. Specifically, generalized measure spaces and integration theory are considered, the partial order structure is studied, and applications to hidden variables and the logic of quantum mechanics are given.

Toward a General Theory of Measurement. I

A self-consistent measurement theory is set forth in this paper, which can be applied to the problem of measurement in field theory. Following the formulation of Birkhoff and von Neumann, we construct a system of axioms sufficient to ensure that it is possible to find an implicit mechanism within each theory for the experimental procedure corresponding to each observable proposition of that theory. It is shown that any theory of a type similar to quantum mechanics cannot consistently describe a closed system from the point of view of a single observer.

On the general quantum theory of measurement

We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.

Theory of measurement of blood flow by dye dilution technique

The general mathematical theory of measurement of blood flow by dye dilution techniques is discussed. The mathematical procedure for handling the recirculation problem is described, and it is shown that, in general, sampling of concentration at a single point is not sufficient to uniquely determine both the volume and flow in a vascular bed. The more basic problem of synthesizing a mathematical model of the circulation is analyzed. A general probabilistic scheme for handling transport problems of this kind is outlined, and results obtained for exchange of labeled material between the vascular and extravascular fluid compartment are described.