This chapter first discusses statistical techniques and concepts — including probability, probability distribution, average and expectation value, and uncertainty — that are relevant to quantum mechanics. The physical interpretation of probability in quantum mechanics is then discussed. It is argued that by adopting the statistical interpretation, one can think clearly about quantum mechanics without committing to any of the various physical interpretations, and without tackling the conceptual hurdles that they present. The history and substance of interpretational issues in quantum mechanics is then briefly discussed, the point being to introduce these profound, unresolved questions without taking sides in ongoing scientific debates. The chapter closes with a personal reflection on Albert Einstein, the most revered and fascinating figure in physics, and the one most famously at odds with the conventional interpretation of quantum mechanics.
This chapter first discusses statistical techniques and concepts — including probability, probability distribution, average and expectation value, and uncertainty — that are relevant to quantum mechanics. The physical interpretation of probability in quantum mechanics is then discussed. It is argued that by adopting the statistical interpretation, one can think clearly about quantum mechanics without committing to any of the various physical interpretations, and without tackling the conceptual hurdles that they present. The history and substance of interpretational issues in quantum mechanics is then briefly discussed, the point being to introduce these profound, unresolved questions without taking sides in ongoing scientific debates. The chapter closes with a personal reflection on Albert Einstein, the most revered and fascinating figure in physics, and the one most famously at odds with the conventional interpretation of quantum mechanics.
Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian conception. The paper includes a comparison of the orthodox and Bayesian theories of statistical inference. It concludes with a few remarks concerning the implications for the concept of physical reality.
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.
The objective of Chapter 4 is to give an overview of the basic concepts of quantum mechanics and their interpretation. Thus this chapter functions as a theoretical preparation for a better understanding of the quantum world and of the knowledge measurement process. First, the terminology of the word “quantum” and its impact on the derivation of the term “knowledge quantum” is explained in Chapter 4.1. Then, the discussion moves to the historical development of quantum mechanics. The key issues and current research field for quantum theory will be given in an overview. The following chapters show some insights into key quantum mechanical concepts such as slit-experiments, wave-particle duality, the Copenhagener interpretation, and quantum probability. Chapter 4 closes with an explanation of the Uncertainty Principle of Werner Heisenberg, especially since the Uncertainty Principle is used to derive the knowledge measurement process through the method of analogical reasoning. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This chapter, along with the next two chapters, sets the basic context for the readers to go into the topic of this book. Physicists, mathematicians, and philosophers struggle to clarify the foundations of quantum mechanics and debate on the interpretations of quantum mechanics. What is the foundation of quantum mechanics and how to interpret the formalism of quantum mechanics are the core issues of concern in the philosophy of physics. These two questions also involve fundamental issues in the general philosophy of science and even metaphysics and epistemology. This chapter introduces the original thoughts, historical developments, and current status of the interpretation of quantum mechanics, as well as its connection with the philosophy of science and metaphysics; it also analyzes the classifications by Bell, Leifer, and Cabello on the spectrum of interpretations of quantum mechanics, which can be regarded as a sort of “navigation map”. On this basis, the main object of discussion in this book is introduced – the quantum histories interpretation.
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.
Quantum mechanics, one of the most successful theories in the history of science, was created to account for physical systems not describable by classical physics. Though it is consistent with all experiments conducted thus far, many of its core concepts (amplitudes, global phases, etc.) can not be directly accessed and its interpretation is still the subject of intense debate, more than 100 years since it was introduced. So, a fundamental question is why this particular mathematical model is the one that nature chooses, if indeed it is the correct model. In the past two decades there has been a renewed effort to determine what physical or informational principles define quantum mechanics. In this paper, recent attempts at establishing reasonable physical principles are reviewed and their degree of success is tabulated. An alternative approach using joint quasi-probability distributions is shown to provide a common basis of representing most of the proposed principles. It is argued that having a common representation of the principles can provide intuition and guidance to relate current principles or advance new principles. The current state of affairs, along with some alternative views are discussed.