A generalized probability theory arises naturally from an attempt to formulate a statistical model for quantum mechanics.One soon finds that the probability theory of Kolmogorov is inadequate for the description of a quantum mechanical system mainly because the quantum mechanical events fail to form a er-algebra of subsets of a set.Instead, they have a much less richly endowed algebraic structure which we shall a "logic".We therefore look upon generalized probability as a stochastic theory in which the events form a logic.Besides being useful in the study of quantum mechanics, this theory has a purely mathematical interest since it constructs a stochastic formalism on an algebraic structure and at the same time gives an abstract generalization of Hubert space theory.The first formulation of this theory is due to G. Bodiou [1], [2].More recently, V. S. Varadarajan [8] has emphasized the stochastic properties of an axiomatic formulation of quantum mechanics due to G. W. Mackey [5] and has proved some important results in this theory.The author heartily recommends these references for the background and motivation of this study.In this paper, the main interest centers on the spectral theory of the subject.The spectrum is important physically since it represents the "allowable" values of a quantum mechanical observable.It is also mathematically useful since it introduces topological methods into the formalism.2. Definitions and notation.Let L be a complemented lattice.That is, L is a lattice with first and last elements, 0,1 respectively and a-*a' satisfying the conditions (i) ia'Y = a for all a eL;(ii) a z% b implies b' zi a'; (iii) a V a ' -1 for all a e L.
A generalized probability theory arises naturally from an attempt to formulate a statistical model for quantum mechanics.One soon finds that the probability theory of Kolmogorov is inadequate for the description of a quantum mechanical system mainly because the quantum mechanical events fail to form a er-algebra of subsets of a set.Instead, they have a much less richly endowed algebraic structure which we shall a "logic".We therefore look upon generalized probability as a stochastic theory in which the events form a logic.Besides being useful in the study of quantum mechanics, this theory has a purely mathematical interest since it constructs a stochastic formalism on an algebraic structure and at the same time gives an abstract generalization of Hubert space theory.The first formulation of this theory is due to G. Bodiou [1], [2].More recently, V. S. Varadarajan [8] has emphasized the stochastic properties of an axiomatic formulation of quantum mechanics due to G. W. Mackey [5] and has proved some important results in this theory.The author heartily recommends these references for the background and motivation of this study.In this paper, the main interest centers on the spectral theory of the subject.The spectrum is important physically since it represents the "allowable" values of a quantum mechanical observable.It is also mathematically useful since it introduces topological methods into the formalism.2. Definitions and notation.Let L be a complemented lattice.That is, L is a lattice with first and last elements, 0,1 respectively and a-*a' satisfying the conditions (i) ia'Y = a for all a eL;(ii) a z% b implies b' zi a'; (iii) a V a ' -1 for all a e L.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.
Quantum probability and quantum mechanics The formalism of quantum probability theory Discussions of quantum mechanics are often confused by a lack of clarity about what exactly constitutes ‘quantum mechanics’. It is therefore useful to try at the start to isolate a consistent mathematical core of quantum mechanics, and consider anything that goes beyond this core to be ‘interpretation’. For us, this core is quantum probability theory. Quantum probability is a generalization of classical probability, and therefore I begin with a brief review of the latter. I assume that the reader has some familiarity with the ideas of probability theory. What follows is just to provide a quick review, and to establish some notation and terminology. In modern classical probability theory, probabilities are defined over algebras of events. The motivation is straightforward: we begin with a set of ‘primitive’, or ‘simple’, events (the ‘sample space’), and form an algebra of events by taking all logical combinations of the simple events. For example, let us take the simple events to be the possible results of rolling a six-sided die one time, so that the sample space is the set {1,2,3,4,5,6}. We then form an algebra of events from the sample space by taking all possible logical combinations of the simple events. Logical combinations include, for example, ‘either 3 or 5’ and ‘not 3 and not 2’.
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.
Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.