The subject of my talk is the relations between the mathematical theory of extremal problems and the physical theory of quanta.1 The theory of quanta, or quantum mechanics, studies phenomena in the world of bodies of atomic scale. It turned out that the usual notions of motion and the usual methods of description of a mechanical system were inapplicable for such bodies. One needed to find another formulation of the basic laws of nature instead of the old one based on observations of bodies containing a very large number of atoms. The drastic revision of all usual notions of mathematics and physics was required. As a result of this revision and due to the efforts of several scientists such formulation has been found and the theory, which is as elegant and complete as the Newton mechanics is, has been created. This new theory — the new quantum mechanics — contains the Newton mechanics as a limiting case. Although some particular questions and problems remain open, the basic principles and the mathematical technique of the new theory can be considered to be completely established.
The subject of my talk is the relations between the mathematical theory of extremal problems and the physical theory of quanta.1 The theory of quanta, or quantum mechanics, studies phenomena in the world of bodies of atomic scale. It turned out that the usual notions of motion and the usual methods of description of a mechanical system were inapplicable for such bodies. One needed to find another formulation of the basic laws of nature instead of the old one based on observations of bodies containing a very large number of atoms. The drastic revision of all usual notions of mathematics and physics was required. As a result of this revision and due to the efforts of several scientists such formulation has been found and the theory, which is as elegant and complete as the Newton mechanics is, has been created. This new theory — the new quantum mechanics — contains the Newton mechanics as a limiting case. Although some particular questions and problems remain open, the basic principles and the mathematical technique of the new theory can be considered to be completely established.
The modern physics is characterized by deeper and deeper penetration into the laws of the world of atoms and other minute particles of matter. These laws require not only new experimental techniques to be developed, but also new notions to formulate them. To describe atomic phenomena new methods that are completely different from those used to investigate larger objects of our world are required. The new problem of description of atomic objects is of particular practical importance since the arising notions could be applied to other fields of knowledge. Therefore on the ground of the modern physics some questions of the philosophical character arise. We are going to consider them here.
Scope of this book Quantum mechanics is a difficult subject, and this book is intended to help the reader overcome the main difficulties in the way to understanding it. The first part of the book, Chs. 2–16, contains a systematic presentation of the basic principles of quantum theory, along with a number of examples which illustrate how these principles apply to particular quantum systems. The applications are, for the most part, limited to toy models whose simple structure allows one to see what is going on without using complicated mathematics or lengthy formulas. The principles themselves, however, are formulated in such a way that they can be applied to (almost) any nonrelativistic quantum system. In the second part of the book, Chs. 17–25, these principles are applied to quantum measurements and various quantum paradoxes, subjects which give rise to serious conceptual problems when they are not treated in a fully consistent manner. The final chapters are of a somewhat different character. Chapter 26 on decoherence and the classical limit of quantum theory is a very sketchy introduction to these important topics along with some indication as to how the basic principles presented in the first part of the book can be used for understanding them. Chapter 27 on quantum theory and reality belongs to the interface between physics and philosophy and indicates why quantum theory is compatible with a real world whose existence is not dependent on what scientists think and believe, or the experiments they choose to carry out. The Bibliography contains references for those interested in further reading or in tracing the origin of some of the ideas presented in earlier chapters.
This book examines a number of problems of quantum mechanics, most of which are not usually discussed. What is the origin of probabilities in the mechanics of the microworld? What is the nature of Planck's constant h? What is the nature of probability amplitudes? What is the wave function? A system of axioms for quantum theory is formulated. A model is studied according to which a classical oscillator in a thermostat can be interpreted as a quantum one. The measurement problem is discussed in detail.
For advanced undergraduate students, graduate students, and specialists interested in the foundations of quantum theory.
Quantum mechanics was developed in the early 20th century as a computational algorithm to organize and predict the strange results of a wide variety of experiments involving the microscopic world (molecules, atoms, elementary particles), which appeared random but had a predictable probabilistic distribution. Despite its overwhelming success, the conceptual foundations of this algorithm remain unclear and extremely controversial. How can we connect the microscopic world described by quantum mechanics with the macroscopic world of our daily experience? What is the role of observers in this connection? Why don't strange quantum phenomena occur on macroscopic scales? How can we explain the nonlocal nature of quantum phenomena? In this article I will present some attempts to answer these and other basic questions, which constitute different theories with completely different worldviews, commenting on their strengths and weaknesses. We will conclude that quantum theory is still under construction, as there is no consensual formulation. Its development is a fundamental problem in contemporary physics, and in thought in general.
The contents of the article is mirrored in its title. In science the fact is well-known, that often opposite notions about that or diverse phenomenon result in identical end results, that produces bitter controversies of the opponents. The example can be served by century dispute between the supporters of the wave and corpuscular nature of light, dispute between a system of Ptolemy and Copernicus and many other conflictings in the most miscellaneous areas of science. The wave description of physical phenomena cannot be executed without of elastic medium (ether) in which one there is a wave process and is diffused by means of this elastic medium. Therefore description of wave phenomena in a solid, liquid and gas adequately mirrors the test datas and problems here does not arise. There, where the elastic medium misses, it is necessary to invent miscellaneous ruses to distribute the wave description and to these objects: empty world space (world ether in the miscellaneous fancy forms), internal space of atoms (wave functions of probability), fields (short-range interaction of fields by means of transfer of a disturbance from one point to other), electromagnetic radiation of the Maxwell (simultaneous oscillation magnetic and electrical field in an elastic medium). The wave description of world is fine is designed, since above it many generations of the known scientists worked, but main brake here it is necessary to have in stock an elastic medium. There, where it no or there are doubts in its existence, a field for critics there is opened. In the formal-mathematical description of the world the elastic medium is not necessary any more. Record a wave equation and decide it for the diversified cases. Though in mathematical relation such description of the world perfectly, but the physical sense of mathematical manipulations is lost. Specially it notably in a quantum mechanics. As a consequence, the similar description it is easy to subject a severe critic. To be salvaged from of intrusive critics, it is necessary to invent completely fancy notions about a microcosmos, which one do not have conformity in a macroworld. Thus, the unity of the world is upset, that directly contradicts fundamental laws of universe. The absence of physical sense is the main unremovable brake of the formal-mathematical description of the world. The corpuscular description of the world, basically, is a merit of Newton. Because of small number of the supporters of such description «have defeated» wave notions. The main brake for of corpuscular world was established by Newton, by formulating the law even and rectilinear motion of free bodies. This law has become force majeure for explanation of an interference, polarization and some other properties of light. Attempts to mate the wave description with corpuscular (wave-corpuscle dualism of microparticles [1]) are originally doomed on a fall because of a principled difference between a wave and particle. The ether is not necessary for the corpuscular description of the world in any to the form, therefore photon, for example, billions of years not «being spread» can move in space, that is impossible from the point of view of the wave description of its motion. The main lack of the corpuscular description of the world consists in incompletness of official toolkit for such description. Only new corpuscular quantum physics has entered notion about originating of a gravidynamic field at motion of gravitational charges and bound with this fact by screw motion of free bodies. It has allowed to revise many fundamental rules of classic physics and quantum mechanicses and to realise a capability of the corpuscular description of the world to the full. Unfortunately, it has taken place after in heads of the scientists strongly was fixed paradigm, designed Einstein and fathers of a quantum mechanics. This paradigm had time knock home in heads of millions students and even of the homemakers. In outcome new scientific paradigm on the basis of new physics meets aggressive hostility. It, certainly, temporary phenomenon and the victory new paradigm in science is foregone, as was already repeatedly. It is a pity, that this the temporary interval, usually, exceeds temporary existence of the person in the land of the living, and orthodoxes permanently self-reproduction by conservative science.
It was with some trepidation that I agreed to speak today, because of a strong doubt that I could say anything substantial not already to be found in the literature of the subject. I cannot say that this trepidation has been subsequently relieved: all I can claim to offer in this paper is a review of certain basic characteristics or themes in the quantum-mechanical situation (which by now should, I think, be thoroughly understood by everyone engaged with the matter), supplemented by some rather general reflections on our philosophical predicament. In aid of these more general reflections, I shall indulge a proclivity for calling on historical matters—some fairly recent, some older, some ancient—which I hope may serve to place current issues in a useful perspective; and I ask your forgiveness for allowing myself to quote certain previous, but hitherto unpublished, remarks of my own.