We investigate epistemologically the meaning of probability as implied in quantum physics in connection with a proposed direct interpretation of the state function and of the related quantum theoretical quantities in terms of physical systems having physical properties, through an extension of meaning of the notion of physical quantity to complex mathematical expressions not reductible to simple numerical values. We show how the changes occurred in the implication of probabilities in quantum physics, from a penetration tool (probability as statistics and frequencies of occurencies of events) to a theoretical concept (probability given by a physically aimed at «amplitude of probability») actually make this view a somewhat «natural» one.
We investigate epistemologically the meaning of probability as implied in quantum physics in connection with a proposed direct interpretation of the state function and of the related quantum theoretical quantities in terms of physical systems having physical properties, through an extension of meaning of the notion of physical quantity to complex mathematical expressions not reductible to simple numerical values. We show how the changes occurred in the implication of probabilities in quantum physics, from a penetration tool (probability as statistics and frequencies of occurencies of events) to a theoretical concept (probability given by a physically aimed at «amplitude of probability») actually make this view a somewhat «natural» one.
Despite its power as the conceptual basis for a huge range of physical phenomena in atomic and subatomic physics, quantum mechanics still suffers from a lack of clarity regarding the physical meaning of its fundamental theoretical concepts, such as those of quantum state and of quantum theoretical quantities or variables , dealt with by the known mathematical-theoretical rules. These concepts have generally been considered as not giving a direct description of physical systems, for they do not correspond to what is usually understood by `physical states' or `physical properties', notably characterized by definite numerical values such as those obtained from measurement. The situation has been tentatively expressed in terms of various `interpretations', conceived not only with regard to the physical meanings of mathematical quantities, but also in connection with philosophical statements about `physical reality'. The question of whether or not quantum theoretical quantities describe definite physical systems existing in nature has therefore generally been considered as beyond the possibilities of physics, because of the definitions that are commonly taken for `physical state' and `physical quantity'. We sketch the main ideas on this problem and propose a possible way out of the puzzle, in terms of an extension of the meaning given to the concepts of the physical state and physical quantity of a system, which would allow one, without any theoretical change in quantum mechanics, to speak consistently of real quantum systems as having definite physical properties .
In this expository paper we discuss the role of probability in statistical physics: it should be that of a general tool guiding induction and must not be restricted only to the empirical interpretation coming from its evaluation by an observed frequency. It is true that in many cases the value of a probability can be sensibly expressed by a suitable frequency, but this does not entails that the latter can be taken as definition of probability. Moreover, in each situation in which one thinks that probabilities are physically real things and believes that something which is only “very probable” can be regarded as being “practically certain”, some “small” probabilities are actually ignored: this attitude of mind may render illegitimate also the probabilistic interpretation of the relevant physical laws. A “law” which is based on a probabilistic interpretation cannot aim at explaining why a fact that we forecast will occur, but rather at explaining why we forecast that this fact will occur. Finally, we emphasize the role of conditional probability, and we show that looking at it from the right perspective allows to avoids also classic paradoxical aspects of quantum mechanics, without the need to resort to the interpretation based on the dualism “particle-wave”.
The article discusses approaches to understanding the nature of probability in mathematics and physics with the consequences of these approaches, as well as the difficulties that arise. The ontological status of probability is discussed. It is shown that probability theory, contrary to R. Mises, should be classified as a mathematical science, using the definition formulated by A.N. Kolmogorov. The application of this definition in practice is provided within the framework of the classical approach, provided that the connection with physical reality is defined through the concept of equal opportunity. To do this, the concepts of simple and complex events are introduced, and the method of decomposing complex events into simple ones is explained. The point of view is substantiated, according to which a mathematical probability determined and related to physical reality in this way can be considered an objective probability, that is, existing independently of any tests. Various interpretations of quantum mechanics and approaches to understanding probability are also revealed. It is shown that in physics today the existing approaches to understanding the nature of probability contradict each other. In the Bohm interpretation, probability is an analogue of the classical approach – statistical objective probability, which has a mathematical formalization, and in quantum Bayesianism, probability quantifies the degree of reliability and depends on the subject conducting the experiment. The Copenhagen interpretation of quantum mechanics tries to eliminate the difficulties associated with combining the wave and corpuscular properties of light and electron beams into a single picture by giving the ontological status of probability, which leads to the problem of understanding physical reality. The article argues for the need for a new connection between the mathematical description of probability and physical reality. The empirical (frequency) approach of R. Mises is analyzed, which causes a lot of criticism, as it does not have a clear consistent formalization. The article argues that this approach is inconsistent with quantum mechanics. The probability given by M. Born in the Copenhagen interpretation of quantum mechanics is also considered, which formulates a new definition of probability, which does not exclude the difficulties associated with the physical interpretation of probability. Born’s approach is based on the classical approach and is reduced to a formal axiomatic definition, which assumes the interpretation of probability as a mathematical construct. The connection of probability with physical reality and the consistency of the statistical method of finding probability values remain unresolved problems, which creates a dualism of understanding probability: in mathematics – one thing, in physics – another. If in the mathematical approach, the solution of these problems is carried out through the connection of probability with physical properties, then in quantum mechanics, in the Copenhagen interpretation, problems persist, which is caused by giving the wave function an ontological status.
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.
The chance of a physical event is the objective, single-case probability that it will occur. In probabilistic physical theories like quantum mechanics, the chances of physical events play the formal role that the values of physical quantities play in classical (deterministic) physics, and there is a temptation to regard them on the model of the latter as describing intrinsic properties of the systems to which they are assigned.I argue that this understanding of chances in quantum mechanics, despite being a part of the orthodox interpretation of the theory and the most prevalent view in the physical community, is incompatible with a very wide range of metaphysical views about the nature of chance. The options that remain are unlikely to be attractive to scientists and scientifically minded philosophers.
We have seen that there is an intimate relationship between probability and information. The values we assign to probabilities depend on the information available, and information is a function of probabilities. This connection makes it inevitable that information will be an important concept in any statistical theory, including thermodynamics and, of course, quantum physics. The probabilistic interpretation of quantum theory has probability amplitudes rather than probabilities as the fundamental quantities. This feature, together with the associated superposition principle, is responsible for intrinsically quantum phenomena and gives quantum information theory its distinctive flavour. We shall see that the quantum rules for dynamical evolution and measurement, together with the existence of entangled states, have important implications for quantum information. They also make it possible to perform tasks which are either impractical or impossible within the classical domain. In describing these we shall make extensive use of simple but fundamental ideas in quantum theory. This chapter introduces the mathematical description of quantum physics and the concepts which will be employed in our study of quantum information.