A compatible statistical interpretation of a wave packet is proposed. De Broglian probabilities which unite wave and particle features of quantons are evaluated for free wave packets and Jor a superposition of wave packets. The obtained expressions provide a very plausible and physically appealing explanation of coherence in apparently incoherent beams and of the characteristic modulation of the momentum distribution, found recently in neutron interferometry combined with spectral filtering. Certain conclusions about dualism and objectivity in quantum domain are (...) also derived. (shrink)

The statistical interpretation introduced by Born in mid-1926 is not the interpretation now associated with his name. Born's own understanding of that interpretation is revealed by looking at some of its roots in Born's earlier work with Franck on collisions, his collaboration with Jordan on that topic, his contributions to matrix mechanics, his attempt in collaboration with Wiener at an operator formulation of quantum mechanics, and at the exposition of the interpretation in Born's first papers (...) on a wave mechanical treatment of collisions. (shrink)

The ontological models framework distinguishes ψ-ontic from ψ-epistemic wave- functions. It is, in general, quite straightforward to categorize the wave-function of a certain quantum theory. Nevertheless, there has been a debate about the ontological status of the wave-function in the statistical interpretation of quantum mechanics: is it ψ-epistemic and incomplete or ψ-ontic and complete? I will argue that the wave- function in this interpretation is best regarded as ψ-ontic and incomplete.

In his paper titled ‘Against “measurement” ’ [Physics World 3(8), 33–40 [1990]], Bell criticised arguments that use the concept of measurement to justify the statistical interpretation of quantum theory. Among these was the text of Gottfried [Quantum Mechanics (Benjamin, New York, [1966])]. Gottfried has replied to this criticism, claiming to show that, for systems with both continuous and discrete degrees of freedom, the statistical interpretation for the discrete variables is implied by requiring that the continuous variables (...) are described classically. In the present paper, Gottfried’s argument is criticised. It is suggested that he takes over aspects of classical physics which are in conflict with the classical limit of the Schrödinger equation. He incorrectly assumes that, in the output from a Stern-Gerlach apparatus, the wave-function of any ion is restricted to one or another of the beams. (shrink)

It is argued that the so-called minimal statistical interpretation of quantum mechanics does not completely resolve the measurement problem in that this view is unable to show that quantjum mechanics can dispense with classical physics when it comes to a treatment of the measuring interaction. It is suggested that the view that quantum mechanics applies to individual systems should not be too hastily abandoned, in that this view gives perhaps the best hope of leading to a version of (...) quantum mechanics which does provide a complete solution to the measurement problem. (shrink)

Population-level theories of evolution—the stock and trade of population genetics—are statistical theories par excellence. But what accounts for the statistical character of population-level phenomena? One view is that the population-level statistics are a product of, are generated by, probabilities that attach to the individuals in the population. On this conception, population-level phenomena are explained by individual-level probabilities and their population-level combinations. Another view, which arguably goes back to Fisher but has been defended recently, is that the population-level statistics (...) are sui generis, that they somehow emerge from the underlying deterministic behavior of the individuals composing the population. Walsh et al. label this the statistical interpretation. We are not willing to give them that term, since everyone will admit that the population-level theories of evolution are statistical, so we will call this the emergentist statistical interpretation. Our goals are to show that: This interpretation is based on gross factual errors concerning the practice of evolutionary biology, concerning both what is done and what can be done; its adoption would entail giving up on most of the explanatory and predictive projects of evolutionary biology; and finally a rival interpretation, which we will label the propensity statistical interpretation succeeds exactly where the emergentist interpretation fails. (shrink)

Starting with the Born interpretation of quantum mechanics, we show that the quantum theory of measurement, supplemented by the strong law of large numbers, leads to a measurement statistics interpretation of quantum mechanics. A probabilistic characterization of the spectrum of a physical quantity is given, and an analysis of the notions of possible values and possible measurement results is carried out.

The Born’s rule to interpret the square of wave function as the probability to get a specific value in measurement has been accepted as a postulate in foundations of quantum mechanics. Although there have been so many attempts at deriving this rule theoretically using different approaches such as frequency operator approach, many-world theory, Bayesian probability and envariance, literature shows that arguments in each of these methods are circular. In view of absence of a convincing theoretical proof, recently some researchers have (...) carried out experiments to validate the rule up-to maximum possible accuracy using multi-order interference (Sinha et al, Science, 329, 418 [2010]). But, a convincing analytical proof of Born’s rule will make us understand the basic process responsible for exact square dependency of probability on wave function. In this paper, by generalizing the method of calculating probability in common experience into quantum mechanics, we prove the Born’s rule for statistical interpretation of wave function. (shrink)

What is natural selection? I address this question by exploring the relation between two debates: Is natural selection a mechanism? Is natural selection a causal or a statistical theory? I argue that the first can be assessed only relative to a model and that, following the second, there are two fundamentally different and independent kinds of models, Modern-Synthesis and Darwinian models. MS-models, I argue, are not mechanistic even if they are causal. D-models, in contrast, are mechanistic. A causal-mechanistic (...) class='Hi'>interpretation of D-models is thus compatible with a statistical interpretation of MS-models. Natural selection, I conclude, lacks a single, unifying nature. (shrink)

I attempt to develop further the statistical interpretation of quantum mechanics proposed by Einstein and developed by Popper, Ballentine, etc. Two ideas are proposed in the present paper. One is to interpret momentum as a property of an ensemble of similarly prepared systems which is not satisfied by any one member of the ensemble of systems. Momentum is regarded as a statistical parameter like temperature in statistical mechanics. The other is the holistic assumption that a probability (...) distribution is determined as a whole as most likely to be realized. This is the same as the chief assumption in statistical mechanics, and maximum likelihood in classical statistics. These ideas enable us to understand statistically (1) the formalism of quantum mechanics, (2) Heisenberg's uncertainty relations, and (3) the origin of quantum equations. They also explain violation of Bell's inequality and the interference of probabilities. (shrink)

The conspicuous similarities between interpretive strategies in classical statistical mechanics and in quantum mechanics may be grounded on their employment of common implementations of probability. The objective probabilities which represent the underlying stochasticity of these theories can be naturally associated with three of their common formal features: initial conditions, dynamics, and observables. Various well-known interpretations of the two theories line up with particular choices among these three ways of implementing probability. This perspective has significant application to debates on primitive (...) ontology and to the quantum measurement problem. (shrink)

To complete our ontological interpretation of quantum theory we have to conclude a treatment of quantum statistical mechanics. The basic concepts in the ontological approach are the particle and the wave function. The density matrix cannot play a fundamental role here. Therefore quantum statistical mechanics will require a further statistical distribution over wave functions in addition to the distribution of particles that have a specified wave function. Ultimately the wave function of the universe will he required, (...) but we show that if the universe in not in thermodynamic equilibrium then it can he treated in terms of weakly interacting large scale constituents that are very nearly independent of each other. In this way we obtain the same results as those of the usual approach within the framework of the ontological interpretation. (shrink)

Philosophical accounts of quantum theory commonly suppose that the observables of a quantum system form a Type-I factor von Neumann algebra. Such algebras always have atoms, which are minimal projection operators in the case of quantum mechanics. However, relativistic quantum field theory and the thermodynamic limit of quantum statistical mechanics make extensive use of von Neumann algebras of more general types. This chapter addresses the question whether interpretations of quantum probability devised in the usual manner continue to apply in (...) the more general setting. Features of non-type I factor von Neumann algebras are cataloged. It is shown that these novel features do not cause the familiar formalism of quantum probability to falter, since Gleason's Theorem and the Lüders Rule can be generalized. However, the features render the problem of the interpretation of quantum probability more intricate. (shrink)

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. (shrink)

The statistical aspects of quantum explanation are intrinsic to quantum physics; individual quantum events are created in the interactions associated with observation and are not describable by predictive theory. The superposition principle shows the essential difference between quantum and non-quantum physics, and the principle is exemplified in the classic single-photon two-slit interference experiment. Recently Mandel and Pfleegor have done an experiment somewhat similar to the optical single-photon experiment but with two independently operated lasers; interference is obtained even with beam (...) intensity so small that only one photon is in the apparatus at a time. The result can be understood in terms of the superposition of states; or, in terms of the Uncertainty Principle, which is found to forbid the determination of which of the two lasers is the source of a given photon (if conditions for interference are to obtain). The Mandel-Pfleegor experiment gives a physical argument against the continuous localization of a photon that is assumed in the hidden variable theories and therefore gives further support for the generally accepted view that an observed entity (observed state) is created in the observation event. This aspect of quantum physics implies a subjectivism on the level of individual quantum-level occurrences, since there is in quantum theory no basis for asserting the existence of the event independently of observation of it. Extension of this subjectivism to large scale, non-quantum phenomena falls within the principles of quantum theory; counter considerations that argue against such an extension are noted. (shrink)

It has been claimed that if statistical power and p-values are both used to measure the strength of our evidence for the null-hypothesis when the results of our tests are not significant, then they can also be used to derive inconsistent epistemic judgements as we compare two different experiments. Those problematic derivations are known as power approach paradoxes. The consensus is that we can avoid them if we abandon the idea that statistical power can measure the strength of (...) our evidence. In this paper however, I put forward a different solution. I argue that every power approach paradox rests on an equivocation on "strong evidence". The main idea is that we need to make a careful distinction between the evidence provided by the quality of the test and the evidence provided by the outcome of the test. Both provide different types of evidence and their respective strength are to be evaluated differently. (shrink)

Although philosophers have considered some of the implications of the nature of quantum statistics of many-particle systems for the interpretation problem, e.g., Reichenbach, they have not produced a complete analysis of the relationship between aspects of quantum statistics and complications and/or possible solutions of the interpretation problem. While the present work by no means provides a complete account, it does explore some heretofore uncharted regions. One of the latter is an analysis of a situation that I call 'The (...) Paradox of Identical Particles ,' which arises in Bose-Einstein statistics. PIP refers to the fact that Elementary Quantum Mechanics seems to be committed to the thesis that systems of particles of the same species, "identical" particles, exhibit behavior differing from that of systems of particles of different species, simply in virtue of the fact that the former contain entities of the same species while the latter do not. Taken at face value this is tantamount to the Empedoclean "axiom" that like attracts like, while like and unlike repel, a somewhat embarassingly naive proposition for modern physicists to embrace, and one which, furthermore, poses a host of perplexing problems, e.g., how do the particles "know" whether they are of the same species or not? A result proven in the present work is that any "pluralistic" interpretation of quantum theory is unable to provide a satisfactory way of ridding quantum physicists of this appeal to what I call "species sensitivity." Arguing that Quantum Field Theory is most naturally and plausibly interpreted as a "monistic" theory--which means, inter alia, that for a system of n identical particles to exist is not for n distinct objects of any kind to exist, but rather for one unified underlying entity, a quantum field, to be in a certain kind of state or condition--I show that from this point of view PIP can be dissolved. On the basis of this and other considerations, I argue that the interpretation problem is more amenable to solution if we take the QFT point of view as the fundamental expression of quantum theory. (shrink)

A statistical mechanical treatment is given of thermal contact between two systems. Reciprocal temperature emerges from this as the relative change in the number of microscopic states a macroscopic system at equilibrium ranges over, at constant volume and chemical composition, with change in internal energy. The significance of this is discussed in detail with reference to a monatomic gas and an Einstein solid.

The attempt to derive (rather than assume) the statistical postulate of quantum theory from the many-universes interpretation of Everett and De Witt is analyzed The many-universes interpretation is found to be neither necessary nor sufficient for the task.

The traditional use of ergodic theory in the foundations of equilibrium statistical mechanics is that it provides a link between thermodynamic observables and microcanonical probabilities. First of all, the ergodic theorem demonstrates the equality of microcanonical phase averages and infinite time averages (albeit for a special class of systems, and up to a measure zero set of exceptions). Secondly, one argues that actual measurements of thermodynamic quantities yield time averaged quantities, since measurements take a long time. The combination of (...) these two points is held to be an explanation why calculating microcanonical phase averages is a successful algorithm for predicting the values of thermodynamic observables. It is also well-known that this account is problematic.

This survey intends to show that ergodic theory nevertheless may have important roles to play, and it explores three other uses of ergodic theory. Particular attention is paid, firstly, to the relevance of specific interpretations of probability, and secondly, to the way in which the concern with systems in thermal equilibrium is translated into probabilistic language. With respect to the latter point, it is argued that equilibrium should not be represented as a stationary probability distribution as is standardly done; instead, a weaker definition is presented. (shrink)

In following paper an attempt will be made to analyse the statistical relationships between variables as the functions of causal relations existing between them. Our basic assumption here is that statistical relationships between traits, events, or characteristics of objects, may be logically derived from the pattern of their mutual causal connections, if this pattern is described by appropriate concepts and with sufficient precision. The first part of the paper presents basic concepts, which according to author's view may serve (...) for the description of different patterns of causal relations in such a way, that statistical relationships corresponding to each pattern may be derived. It gives also examples of such a derivation for some less complicated cases. The second part of the paper is an attempt of application of proposed method to the understanding and critical consideration of some standard techniques of statistical analysis, especially those mostly used in social sciences. (shrink)

In The Point of View for my Work as an Author, Kierkegaard declares that the works discussed therein move from the aesthetic to the religious, that the Postscript represents the turning point in this movement, etc. In this brief and preliminary study we use a ?change?point? version of the chi?square test on the frequencies of selected sets of ?aesthetic? and ?religious? words to determine the degree of statistical evidence for these and other related claims. Briefly, these tests show that (...) there is strong statistical evidence for all but one of these various claims and that there are therefore good statistical grounds for taking Kierkegaard's ?aesthetic/religious? interpretation seriously. (shrink)

In their influential paper “Ceteris Paribus, There is No Problem of Provisos”, Earman and Roberts (Synthese 118:439–478, 1999) propose to interpret the non-strict generalizations of the special sciences as statistical generalizations about correlations. I call this view the “statistical account”. Earman and Roberts claim that statistical generalizations are not qualified by “non-lazy” ceteris paribus conditions. The statistical account is an attractive view, since it looks exactly like what everybody wants: it is a simple and intelligible theory (...) of special science laws without the need for mysterious ceteris paribus conditions. I present two challenges to the statistical account. According to the first challenge, the statistical account does not get rid of so-called “non-lazy” ceteris paribus conditions. This result undermines one of the alleged and central advantages of the statistical account. The second challenge is that the statistical account, qua general theory of special science laws, is weakened by the fact that idealized law statements resist a purely statistical interpretation. (shrink)

This paper attempts to analyze the concept of quantum statistical determinism. This is done after we have clarified the epistemic difference between causality and determinism and discussed the content of classical forms of determinism—mechanical and dynamical. Quantum statistical determinism transcends the classical forms, for it expresses the multiple potentialities of quantum systems. The whole argument is consistent with a statistical interpretation of quantum mechanics.

Many philosophers have pointed out that statistical evidence, or at least some forms of it, lack desirable epistemic or non-epistemic properties, and that this should make us wary of litigations in which the case against the defendant rests in whole or in part on statistical evidence. Others have responded that such broad reservations about statistical evidence are overly restrictive since appellate courts have expressed nuanced views about statistical evidence. In an effort to clarify and reconcile, I (...) put forward an interpretive analysis of why statistical evidence should raise concerns in some cases but not others. I argue that when there is a mismatch between the specificity of the evidence and the expected specificity of the accusation, statistical evidence—as any other kind of evidence—should be considered insufficient to sustain a conviction. I rely on different stylized court cases to illustrate the explanatory power of this analysis. (shrink)

I briefly sketch Bohm's causal interpretation (BCI) and its solution to the measurement problem. Crucial to BCI's no-collapse account of both ideal and non-ideal measurement is the existence of particles in addition to wavefunctions. The particles in their role as the producers of the observable experimental outcomes render practical considerations, such as what observables can be reasonably measured or how to get rid of interference terms in non-ideal measurements, secondary to BCI's account of measurement. I then explain why it (...) is not easy for BCI to justify its statistical postulate. To successfully justify the postulate would be to solve the distribution problem. Two proposed deterministic solutions to this problem are only briefly set out and not discussed in detail. BCI can solve the measurement problem whether or not the distribution problem is solved. However, if the distribution problem is not solved, BCI cannot be shown to be empirically adequate. (shrink)

In the last years, the surprising bosonic behavior that a many-fermion system may acquire has raised interest because of theoretical and practical reasons. This trans-statistical behavior is usually considered to be the result of approximation modeling methods generally employed by physicists when faced with complexity. In this paper, we take a tensor product structure and an ontology of properties approach and provide two versions of a toy model in order to argue that trans-statistical behavior allows for a realistic (...) interpretation. (shrink)

Over the past fifteen years there has been a considerable amount of debate concerning what theoretical population dynamic models tell us about the nature of natural selection and drift. On the causal interpretation, these models describe the causes of population change. On the statistical interpretation, the models of population dynamics models specify statistical parameters that explain, predict, and quantify changes in population structure, without identifying the causes of those changes. Selection and drift are part of a (...) statistical description of population change; they are not discrete, apportionable causes. Our objective here is to provide a definitive statement of the statistical position, so as to allay some confusions in the current literature. We outline four commitments that are central to statisticalism. They are: 1. Natural Selection is a higher order effect; 2. Trait fitness is primitive; 3. Modern Synthesis (MS)-models are substrate neutral; 4. MS-selection and drift are model-relative. (shrink)

The major competing statistical paradigms share a common remarkable but unremarked thread: in many of their inferential applications, different probability interpretations are combined. How this plays out in different theories of inference depends on the type of question asked. We distinguish four question types: confirmation, evidence, decision, and prediction. We show that Bayesian confirmation theory mixes what are intuitively “subjective” and “objective” interpretations of probability, whereas the likelihood-based account of evidence melds three conceptions of what constitutes an “objective” probability.

Statistical evidence is crucial throughout disparate impact’s three-stage analysis: during (1) the plaintiff’s prima facie demonstration of a policy’s disparate impact; (2) the defendant’s job-related business necessity defense of the discriminatory policy; and (3) the plaintiff’s demonstration of an alternative policy without the same discriminatory impact. The circuit courts are split on a vital question about the “practical significance” of statistics at Stage 1: Are “small” impacts legally insignificant? For example, is an employment policy that causes a one percent (...) disparate impact an appropriate policy for redress through disparate impact litigation? This circuit split calls for a comprehensive analysis of practical significance testing across disparate impact’s stages. Importantly, courts and commentators use “practical significance” ambiguously between two aspects of practical significance: the magnitude of an effect and confidence in statistical evidence. For example, at Stage 1 courts might ask whether statistical evidence supports a disparate impact (a confidence inquiry) and whether such an impact is large enough to be legally relevant (a magnitude inquiry). Disparate impact’s texts, purposes, and controlling interpretations are consistent with confidence inquires at all three stages, but not magnitude inquiries. Specifically, magnitude inquiries are inappropriate at Stages 1 and 3—there is no discriminatory impact or reduction too small or subtle for the purposes of the disparate impact analysis. Magnitude inquiries are appropriate at Stage 2, when an employer defends a discriminatory policy on the basis of its job-related business necessity. (shrink)

After some general remarks about the interrelation between philosophical and statistical thinking, the discussion centres largely on significance tests. These are defined as the calculation of p-values rather than as formal procedures for ‘acceptance‘ and ‘rejection‘. A number of types of null hypothesis are described and a principle for evidential interpretation set out governing the implications of p- values in the specific circumstances of each application, as contrasted with a long-run interpretation. A number of more complicated situ- (...) ations are discussed in which modification of the simple p-value may be essential. (shrink)

Statistical mechanics is often taken to be the paradigm of a successful inter-theoretic reduction, which explains the high-level phenomena (primarily those described by thermodynamics) by using the fundamental theories of physics together with some auxiliary hypotheses. In my view, the scope of statistical mechanics is wider since it is the type-identity physicalist account of all the special sciences. But in this chapter, I focus on the more traditional and less controversial domain of this theory, namely, that of explaining (...) the thermodynamic phenomena.What are the fundamental theories that are taken to explain the thermodynamic phenomena? The lively research into the foundations of classical statistical mechanics suggests that using classical mechanics to explain the thermodynamic phenomena is fruitful. Strictly speaking, in contemporary physics, classical mechanics is considered to be false. Since classical mechanics preserves certain explanatory and predictive aspects of the true fundamental theories, it can be successfully applied in certain cases. In other circumstances, classical mechanics has to be replaced by quantum mechanics. In this chapter I ask the following two questions: I) How does quantum statistical mechanics differ from classical statistical mechanics? How are the well-known differences between the two fundamental theories reflected in the statistical mechanical account of high-level phenomena? II) How does quantum statistical mechanics differ from quantum mechanics simpliciter? To make our main points I need to only consider non-relativistic quantum mechanics. Most of the ideas described and addressed in this chapter hold irrespective of the choice of a (so-called) interpretation of quantum mechanics, and so I will mention interpretations only when the differences between them are important to the matter discussed. (shrink)

Different realistic attitudes towards wavefunctions and quantum states are as old as quantum theory itself. Recently Pusey, Barret and Rudolph on the one hand, and Auletta and Tarozzi on the other, have proposed new interesting arguments in favor of a broad realistic interpretation of quantum mechanics that can be considered the modern heir to some views held by the fathers of quantum theory. In this paper we give a new and detailed presentation of such arguments, propose a new taxonomy (...) of different realistic positions in the foundations of quantum mechanics and assess the scope, within this new taxonomy, of these realistic arguments. (shrink)

An intricate, long, and occasionally heated debate surrounds Boltzmann’s H-theorem (1872) and his combinatorial interpretation of the second law (1877). After almost a century of devoted and knowledgeable scholarship, there is still no agreement as to whether Boltzmann changed his view of the second law after Loschmidt’s 1876 reversibility argument or whether he had already been holding a probabilistic conception for some years at that point. In this paper, I argue that there was no abrupt statistical turn. In (...) the first part, I discuss the development of Boltzmann’s research from 1868 to the formulation of the H-theorem. This reconstruction shows that Boltzmann adopted a pluralistic strategy based on the interplay between a kinetic and a combinatorial approach. Moreover, it shows that the extensive use of asymptotic conditions allowed Boltzmann to bracket the problem of exceptions. In the second part I suggest that both Loschmidt’s challenge and Boltzmann’s response to it did not concern the H-theorem. The close relation between the theorem and the reversibility argument is a consequence of later investigations on the subject. (shrink)