Classical objects have been excluded as subjects of the observed quantum properties, and the related problem of quantum objects nature has been suspended since the early days of Quantum Theory. Recent experiments show that the problem could be reasonably revisited. The outlined model indicates new issues, which could result from following and exploring the canonical idea of Dirac. Topological defects in solids are considered as an example. The aim is helping to grasp the underlying pre-theoretical new intuitions, which should replace the old ones attached to the background of classical physics.
Classical objects have been excluded as subjects of the observed quantum properties, and the related problem of quantum objects nature has been suspended since the early days of Quantum Theory. Recent experiments show that the problem could be reasonably revisited. The outlined model indicates new issues, which could result from following and exploring the canonical idea of Dirac. Topological defects in solids are considered as an example. The aim is helping to grasp the underlying pre-theoretical new intuitions, which should replace the old ones attached to the background of classical physics.
Authoritative appraisals qualified this book as an axiomatic theory. However, being its essential content no more than an analogy, its theoretical organization cannot be an axiomatic one. In fact, in the first edition Dirac declares to avoid an axiomatic presentation. Moreover, I show that the text is aimed at solving a basic problem: How quantum mechanics is similar to classical mechanics. A previous paper analyzed all past theories of physics, chemistry and mathematics, presented by the respective authors in a non-axiomatic way. Four characteristic features of a new model of organizing a theory have been recognized. An accurate inspection of Dirac text proves that it actually applied this kind of organization of a theory. This fact gives a formal reason of what Kronz and Lupher suggested through intuitive categories, pragmatism and rigor, i.e. Dirac formulation of Quantum mechanics represents a distinct theoretical attitude from von Neumann axiomatic attitude. However, since the second edition Dirac changed his attitude; although relying again on an analogy, his theory was referred to the axiomatic method. Some considerations on the odd paths which led to the present formulation of QM are added. They suggest to look for a new foundation of this theory.
The relationship between physical notions and the structures expressing them is an issue of primary importance the history and philosophy of pre-modern natural philosophy and of modern science. Historical evidence suggests that, when investigating the relationship between physics and mathematics, one should pay attention to the material, sensually perceivable forms mediating it: spoken or written words, drawings, artefacts, descriptions of experimental set-ups or, as the case discussed the present paper, symbolic notation. The reason for this is that, historically, physical notions and their expressions have very often been developed as one, and only distinguished from each other at a later stage. This was for example the case with what we today regard as mathematical and classical mechanics, which were developed the course of the eighteenth century as the discipline of the analysis of motion. When considered from a later point of view, both the physical notions and the concepts involved historical developments appear as being in flux, vague, and, as far as the side is concerned, non-rigorous. In such cases, verbal description, mechanical artefacts or symbols do not simply mediate between physical notions and structures, as might be the case at a later stage, but represent both of them one, allowing the historical actors to construct and manipulate physical-mathematical meaning by following, at times, their physical reasoning and, at times, their criteria of rigour. The history of quantum mechanics offers many examples of how new physics and new mathematics can emerge as one, with symbolic notation playing a central role shaping both. Among the many creators of quantum theory, Paul Dirac was particularly sensitive to questions of notation:
T OPOLOGICAL MATTER IS FUNDAMENTALLY QUANTUM mechanical, with no correspondent in the classical world. Therefore, it would be quite natural to begin our journey with a review of topics that are particularly relevant to understanding the nature of topological quantum states, such as Berry phases and anyon physics. Yet, even before discussing those topics, pausing to think about some fundamental aspects of quantum theory would be extremely worthwhile. For several decades after the triumph of quantum mechanics in the 1920s and early 1930s, thinking about the foundations of the theory and challenging its standard interpretation were viewed by a majority of physicists as a waste of time, if not plain heresy. The judgment of the physics community on the value of such an endeavor has drastically changed after the groundbreaking work of John Bell and the subsequent experimental tests during the 1970s and 1980s. Today, studying the foundations of quantum mechanics has turned into a flourishing field intimately linked with the explosive rise of quantum information theory. Yet, the subject still has a rather marginal place in standard quantum mechanics textbooks, which focus on the formalism and on the recipes necessary for doing calculations. This chapter is intended for the reader who is less familiar with the problems brought to the fore by quantum information theory and the work on the foundations of quantum mechanics as a brief introduction to these fields and a summary of the basic terminology. But why should one bother about such things when discussing the physics of topological quantum matter? Basically, because the two areas are intimately connected: on the one hand, quantum computation is a prominent item on the list of possible applications for topological quantum states, on the other hand, key concepts in quantum theory, such as quantum entanglement, play a central role in understanding topological matter. Hence, starting the journey with a thought about the foundations of quantum mechanics should enable us to grasp the delicate root that makes the quantum world utterly strange and beautiful, then follow it as it grows out of the sphere of few particle physics into the realm of many-body systems. 1.1 THE QUANTUM MEASUREMENT PROBLEM There is a certain uneasiness about quantum mechanics that every student of this field has probably experienced in some measure. And it is not just the insecurity that results from losing the firm ground of classical intuition; special relativity also deals with this type of problem and does not generate similar feelings. Nor is it at the center of any controversy similar in scale and depth with, for example, the debate [455] between Einstein and Bohr on the foundations of quantum theory. To better understand the source of the problem, let us recall the “standard” way of presenting quantum theory, i.e., starting from certain principles or postulates [126, 440] — a typical approach that can be found in most of the commonly used textbooks. More specifically, let us focus on the key postulates that define the state vectors and their evolution, as well as the observables and the measurements.
In these continuation papers (VI and VII) we are interested in approach the problem of spin from a classical point of view. In this first paper we will show that the spin is neither basically relativistic nor quantum but reflects just a simmetry property related to the Lie algebra to which it is associated. The classical approach will be paraleled with the usual quantum one to stress their formal similarities and epistemological differences. The important problem of Einstein-Bose condensation for fermions will also be addressed.
Abstract 1. In the general synthesis of classical dynamics with the quantum theory, the Uncertainty Principle plays a most useful part. It is of course only one aspect of the new mechanics, but it is a very helpful one since by its means it becomes easy to see where the old classical ideas broke down. The state of affairs in the quantum theory is not unlike that of the early days of relativity, when most of those who studied the subject felt the need of supporting the formal theory by seeing how the old ideas failed in specific cases. Here the formal theory is very abstract and is not easy to follow intuitively, and the Uncertainty Principle plays much the same rôle as did the examples of clocks and rods in relativity theory. For this reason it is more appropriate for illustrative examples, than for any extreme generality, and though a number of examples have been already given by Heisenberg and Bohr, it may not be amiss to have some more. There are probably some who will have shared my experience that it is often by no means easy to detect how the uncertainty enters into a given experiment, though once detected the arguments are usually very simple. In a recent conversation Professor Bohr criticised some rather careless remarks that I had made, and the present work was undertaken to clear matters up. It may be shortly described as a study of the Uncertainty Principle in connection with electrometers and magnetometers. It may be of interest to point out that the Uncertainty Principle can be regarded from a rather different aspect. The "resolving power" of optical instruments was discussed very fully by Rayleigh on wave principles, but as long as matter was regarded in the classical manner, a mechanical instrument could be considered as capable of measuring quantities with absolute accuracy. Now that we know that matter also has wave properties, there is need of a theory of the resolving power of mechanical instruments, and this is exactly what the Uncertainty Principle supplies.