Probability and Randomness

The following sections are included:

• Dedication
• Preface
• Contents

We start with the remark that, in contrast with, e.g., geometry, axiomatic probability theory was created not so long ago. Soviet mathematician Andrei Nikolaevich Kolmogorov presented the modern axiomatics of probability theory only in 1933 in his book [177]. The book was originally published in German. The English translation [178] was published only in 1952 (and the complete Russian translation [179] of the German version [177] only in 1974). Absence of an English translation soon (when the German language lost its international dimension) led to the following problem. The majority of the probability theory community did not have a possibility to read Kolmogorov. Their picture of the Kolmogorov model was based on its representations in English (and Russian) language textbooks. Unfortunately, in such representations a few basic ideas of Kolmogorov disappeared, since they were considered as philosophical remarks with no direct relevance to mathematics. This is partially correct, but probability theory is not just mathematics. It is a physical theory and, as any physical theory, its mathematical formalism has to be endowed with some interpretation. In Kolmogorov's book the interpretation question was discussed in very detail. However, in the majority of mathematical representations of Kolmogorov's approach, the interpretation issue is not enlightened at all. From my viewpoint, one of the main negative consequences of this ignorance was oblivion of contextuality of Kolmogorov's theory of probability. Kolmogorov designed his probability theory as a mathematical formalization of random experiments (see also discussion below). For him, each experimental context C generates its own probability space (endowed with its own probability measure). It is practically impossible to find a probability theory textbook mentioning this key interpretational issue of Kolmogorov's theory. We remark that contextuality of classical probability theory plays very important role when classical and quantum probability theories are being compared, see Chapter 5…

In this chapter we shall briey present basics of various mathematical models of classical randomness:

• randomness as unpredictability (von Mises-Wald-Church),
• randomness as typicality (Laplace-Ville-Martin-Löf),
• randomness as high algorithmic complexity (Kolmogorov, Solomonoff, Chaitin, Schnorr)

Each approach by itself is well formalized mathematically. The starting points of these approaches are very different, but surprisingly it happens that the last two viewpoints on randomness are equivalent, see Theorem 6 (Schnorr [227]), section 2.3. Moreover, such random sequences are also random in the sense of the first approach: they are von Mises-Wald-Church collectives, Corollary 1, section 2.3. However, inter-relation of the first approach with the other two is sufficiently complicated…

The following sections are included:

• Extension of Probability Measure
  • Lebesgue measure on the real line
  • Outer and inner probabilities, Lebesgue measurability
• Complete Probability
• Von Mises Views
  • Problem of verification
  • Jordan measurability
• Role of the Axiom of Choice in the Measure Theory
• Possible Generalizations of Probability Theory
  • Negative probabilities
  • On generalizations of the frequency theory of probability
  • p-adic probability
• Quantum Theory: No Statistical Stabilization for Hidden Variables?

In this chapter we present a very brief introduction to the quantum formalism. On one hand, the chapter can serve as the first step towards understanding quantum theory. We present sufficient material to understand essentials of quantum theory and especially its probability structure. On the other hand, the chapter may be of some interest for physicists who know quantum theory very well, since this is quite an unusual introduction to the quantum formalism. To escape mathematical difficulties, we restrict our considerations to the case of finite dimensional state spaces. However, the basic definitions and constructions are applicable even for infinite-dimensional state spaces. Interpretational issues will be considered in Chapter 6.

We begin with the following terminological remark. In this chapter, the term “classical probability” is used for probability represented mathematically by the measure-theoretic model of Kolmogorov (so to say, Kolmogorovean probability); the term “quantum probability” is used for theory of probability based on the complex Hilbert representation and Born's rule connecting states with probabilities. We recall that in Chapter 1 we also considered another classical probability model developed by von Mises. In consideration of the present chapter we shall proceed solely with the Kolmogorov probability model, see [133], [156] for the corresponding presentation based on the von Mises probability model…

The modern situation in QM is characterized by the diversity of its interpretations. Moreover, as was remarked in my book [167] (and section 5.2.2) each basic interpretation has a variety of individual flavors. For example, suppose that somebody claims to be an adherent of the Copenhagen interpretation of QM. When you ask for details, most probably you will meet with a very specific view on what this interpretation is about. I have the impression that there are so many versions of the Copenhagen interpretation of QM as there are people knowing about it. The same can be said about, e.g., the statistical or Bohmian interpretations of QM.

The following sections are included:

• Irreducible Quantum Randomness
• Lawless Universe? Digital Philosophy?
• Unpredictability and Indeterminism

By starting to study some scientific theory it is very useful to know philosophic views of its creators. Regarding Bell's inequality project we have to be aware that J. Bell was one of those using the realist interpretation of QM, as well as, e.g. Einstein, Podolsky, and Rosen, De Broglie, early Schrödinger, …, D. Bohm, L. Ballentine, the author of this book,… . In contrast to Einstein, Bell (following Bohm) did not consider nonlocality as “an absurd alternative to incompleteness of QM”. As well as for Bohm, the universe was the indivisible entity, cf. with the book of D. Bohm and B. Hiley [36] about “undivided universe.” This philosophy of wholeness played the determining role in Bell's research on quantum foundations. Roughly speaking his main aim (at least at the beginning) was to justify Bohmian mechanics. Thus by studying Bell's argument it is important to be aware that Bell was Bohmian…

Recent years were characterized by the tremendous extension of the domain of applications of the quantum probability (QP). The operator representation of observables and states is fruitful not only in quantum physics but also for applications outside of physics, the keywords are non-Kolmogorovness and simplicity…

The following sections are included:

• Appendix A Växjö Interpretation-2002
  • A.1 Contextual Statistical Realist Interpretation of Physical Theories
  • A.2 Citation with Comments
  • A.3 On Romantic Interpretation of Quantum Mechanics
• Appendix B Analogy between non-Kolmogorovian Probability and non-Euclidean Geometry

The following sections are included:

• Bibliography
• Index