A striking characteristic of contemporary science is its emphasis upon probability. This is especially notable in quantum mechanics. There is a respect in which probability is the same for all scientific theories. Verifiability requires that any theory predict certain numbers which can be compared with the numbers gained by actual operations of measuring. In actual practice these numbers, which we shall term theoretical measurables and operative measurables respectively, never correspond. It becomes necessary, therefore, for the scientist to specify when the deviation between them is such that verification occurs. These specifications are defined by the theory of errors in which the concept of probability has an essential place and a specific meaning. But the theory of errors is the same for all applications; hence, probability as defined by the theory of errors is the same for all scientific theories.
A striking characteristic of contemporary science is its emphasis upon probability. This is especially notable in quantum mechanics. There is a respect in which probability is the same for all scientific theories. Verifiability requires that any theory predict certain numbers which can be compared with the numbers gained by actual operations of measuring. In actual practice these numbers, which we shall term theoretical measurables and operative measurables respectively, never correspond. It becomes necessary, therefore, for the scientist to specify when the deviation between them is such that verification occurs. These specifications are defined by the theory of errors in which the concept of probability has an essential place and a specific meaning. But the theory of errors is the same for all applications; hence, probability as defined by the theory of errors is the same for all scientific theories.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
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.
Philosophical reflections on physics and its theories have been shaping the agenda of general philosophy of science, including issues such as the nature of scientific theories, the meaning of scientific terms, and scientific modeling. But physics is also well worth being considered a special science of its own, whose methods and tools raised specific questions for philosophers of science: this is the approach followed in this chapter, which has as its main thread the role of mathematics in physical theories. It tackles classical issues concerning measurement and determinism and long-standing controversies in the philosophy of statistical mechanics (how may the reversible laws of mechanics account for the irreversible principles of thermodynamics?) and in the philosophy of quantum physics (does quantum mechanics provide us a with a complete description of quantum phenomena?). It also addresses emerging issues in the field, such as computer simulations and their role between theory and observation.
The abstract theory of probability and its interpretation are briefly reviewed, and it is explicitly demonstrated that the formalism of quantum mechanics satisfies the axioms of probability theory. This refutes the suggestions which have occasionally been made that ‘‘classical’’ probability theory does not apply to quantum mechanics. Several erroneous applications of probability theory to quantum mechanics are examined, and the nature of the errors are exposed. It is urged that more attention be given to probability theory in the physics curriculum.
Concept of has had many transformations throughout the history of physics, and it has been so influential in this field. However, with the growth of quantum physics, became a scientific dilemma and has created many challenges in various contemporary sciences. If this concept is understood correctly, many misconceptions corresponding to it in natural sciences, especially physics can be removed. The aim of this paper is to investigate this concept analytically to solve the existing problems. To do this, first we consider the terminology of including characteristic, necessary connection, and exact prediction. Then, William James definition of determinism, Laplacian determinism, Popper definition of determinism, causal determinism, logical determinism, epistemic determinism, metaphysical determinism, and theological are introduced. After which, we pay attention to different kinds of including global/local domain, complete/incomplete form, and the main factor in the formation of events. In the next step, it is discussed that although had been vague and causal in ancient Greece, it has become global and theological in middle Ages. In classical physics, it took a complete and universal form. Laplace provided the most complete picture of determinism. Because of some philosophical ideas, changed into a different concept in quantum mechanics. Copenhagen school physicists, especially Bohr, regarded determinism as a problem and an undesirable element. It is shown that is a complex subject with a number of aspects and characteristics which cannot be set aside from the science. The final result is that cannot be denied in the science; in particular, the quantum mechanics is a deterministic theory but in an incomplete or local manner.