I have proposed OST (Observable System Theory) as a foundational theory of system theory. Roughly speaking, OST is a theory that unifies statistics and quantum mechanics. In comparison with ordinary statistics, its main advantage is that it possesses an explicit axiomatic structure. In this sense, I have claimed the following two points:(i): OST will, in the future, become the standard understanding of statistics.(ii): OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics.Statistical mechanics, however, involves microscopic states, probability distributions, Gibbs states, coarse graining, time evolution, thermodynamic limits, and so on. For this reason, a complete axiomatization of statistical mechanics is generally regarded as difficult. In the present paper, we propose a different strategy: the system-theoretical part of statistical mechanics is formulated by means of the axiomatic structure of OST, while the remaining parts are treated in the conventional way, withoutforcing them into a single axiomatic system.The aim of this paper is to understand statistical mechanics in the language of OST. In other words, the parts of statistical mechanics in which statistics has traditionally been used are here reformulated by means of OST. Thus, statistical mechanics is examined from the viewpoint of the three-term structure: state, observable, measured value. Although this is only a partial axiomatization, it nevertheless allows us to make the following claim: (iii): OST-based statistical mechanics will, in the future, become the standard understanding of statistical mechanics.This should be regarded as a corollary of (i) above. At least from the foundational viewpoint of how statistical mechanics should be read as a descriptive specification, the author believes that the OST-based understanding of statistical mechanics is more natural than the conventional one.
I have proposed OST (Observable System Theory) as a foundational theory of system theory. Roughly speaking, OST is a theory that unifies statistics and quantum mechanics. In comparison with ordinary statistics, its main advantage is that it possesses an explicit axiomatic structure. In this sense, I have claimed the following two points:(i): OST will, in the future, become the standard understanding of statistics.(ii): OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics.Statistical mechanics, however, involves microscopic states, probability distributions, Gibbs states, coarse graining, time evolution, thermodynamic limits, and so on. For this reason, a complete axiomatization of statistical mechanics is generally regarded as difficult. In the present paper, we propose a different strategy: the system-theoretical part of statistical mechanics is formulated by means of the axiomatic structure of OST, while the remaining parts are treated in the conventional way, withoutforcing them into a single axiomatic system.The aim of this paper is to understand statistical mechanics in the language of OST. In other words, the parts of statistical mechanics in which statistics has traditionally been used are here reformulated by means of OST. Thus, statistical mechanics is examined from the viewpoint of the three-term structure: state, observable, measured value. Although this is only a partial axiomatization, it nevertheless allows us to make the following claim: (iii): OST-based statistical mechanics will, in the future, become the standard understanding of statistical mechanics.This should be regarded as a corollary of (i) above. At least from the foundational viewpoint of how statistical mechanics should be read as a descriptive specification, the author believes that the OST-based understanding of statistical mechanics is more natural than the conventional one.
I have proposed OST (Observable System Theory) as a foundational theory of system theory.3 This theory adopts two principles, namely, the Born principle and the Causal principle, and furthermore explicitly uses the one-shot measurement principle as a rule of use. OST is one of the most fundamental scientific theories, alongside mathematics and physics, and it can provide a unified framework for discussing quantum mechanics (information), statistics (system theory), practical logic, epistemology (philosophy), statistical mechanics, and so on.In this paper, I clarify that Fisherian statistics, Bayesian statistics, and quantum mechanics share a common mathematical foundation, namely OST. In statistics, an OST-based understanding of the Kalman filter is presented. Moreover, the so-called “interpretation problem” of quantum mechanics after measurement is unnecessary, and the “Wigner’s friend paradox” is resolved within OST. In addition, the FR paradox (= Frauchiger–Renner paradox) is understood as a paradox arising from the violation of the one-shot measurement principle, and thus it suggests the validity of OST. Through these discussions, I claim the following two points:(i) OST-based statistics will, in the future, become the standard understanding of statistics.(ii) OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
Roughly a quarter of a century ago, the author proposed "Observable System Theory" (OST) as a foundational theory of systems theory. OST describes both statistics and quantum mechanics within a single framework of state–observable–measured value, governed by two explicitly formulated principles: the Born principle and the causal principle. This paper examines Bayesian networks (BNs) from the standpoint of OST, with particular attention to the following fundamental question: what, ultimately, do DAG calculations compute? The answer proposed here is that DAG calculations ultimately compute the probabilistic causality of measured values. Accordingly, OST admits two distinct notions of causality. The first is the causal evolution of states—equivalently, the causal evolution of observables in the Heisenberg picture—governed by OST's causal principle. The second is the probabilistic dependence structure among measured values, represented by a directed acyclic graph (DAG) together with local conditional probability distributions. The main result of this paper is the following fundamental theorem: for every finite Bayesian network specified by a DAG and local conditional probability distributions, there exists a corresponding measurement in OST whose outcome distribution coincides exactly with the joint probability distribution generated by the Bayesian network. We also aim to dispel the notion that theories involving multiple, distinct kinds of causal relation—such as Aristotle's doctrine of the four causes—are inherently untenable. Keywords: Observable System Theory; Bayesian network; directed acyclic graph; measurement; measured value; measurement probability.
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
An investigation is made into how the foundations of statistical mechanics are affected once we treat classical mechanics as an approximation to quantum mechanics in certain domains rather than as a theory in its own right; this is necessary if we are to understand statistical-mechanical systems in our own world. Relevant structural and dynamical differences are identified between classical and quantum mechanics (partly through analysis of technical work on quantum chaos by other authors). These imply that quantum mechanics significantly affects a number of foundational questions, including the nature of statistical probability and the direction of time.
According to the Kolmogorovian Censorship Hypothesis, everything that quantum theory says about the world in the language of the quantum mechanical Hilbert space formalism is actually about relationships between ordinary relative frequencies expressible in operational terms using classical Kolmogorovian probability theory. In other words, a quantum theoretical description of a system should in principle be translatable into a purely operational-probabilistic description. However, our goal in this paper is different; we do not want to deal with the problem how to translate the known theory of quantum mechanics into operational terms, or to reconstruct the theory from postulates which can be interpreted in operational terms. Our aim is somewhat broader and points in the opposite direction. We start with a general scheme for the operational description of an arbitrary physical system. The description is based solely on the notion of observable events (measurement operations and measurement results) and on general, empirically established simple laws concerning their relative frequency. These laws are so simple and fundamental that they apply equally to any physical system. In the first part of the paper, we outline the basic elements of such an operational-probabilistic theory. In the second part, we discuss how this operational-probabilistic description compares to the quantum mechanical description and to what extent the standard Hilbert space quantum mechanics can be regarded as a reformulation of the general operational-probabilistic theory.