I. Brief History.When I first learned of this conference, and saw the wide range of interests represented by the participants, I was certain that finally a common definition of that most intriguing concept-the soliton-would emerge.And I was anxious that this definition be broad enough to encompass solitons in both their classical and quantum versions.For recent work had convinced me that the general technique known as Bethe's ansatz for solving diverse one-dimensional quantum problems in fact was nothing more nor less than the quantum soliton.However, from conversations with conference participants, I discovered that many were not aware of this accumulated work on exactly soluble quantum systems.I even gained the impression that some feel quantum mechanics to be much harder than classical mechanics-an unnecessary complication.My own feeling, on the other hand, is that in many cases quantum mechanics clarifies matters.It forces one to immediately face problems that would eventually have to be faced in the corresponding classical case, problems such as the counting of states, to determine how important solitons really are.Anyway, the history of the quantum soliton is all to the greater glory of the soliton concept.And quite a respectable history it is.I would date the beginning at 1931 with a paper of H. Bethe [1] on magnetism; that is a total span of 45 years.Quantum mechanics itself has not been around all that much longer!It now seems my hopes for a consensus on a definition of the soliton were premature, and things are still fermenting.Nevertheless, here is my contribution to the conference in the form of a brief history for general interest, and a more original work on the quantization of the Toda lattice.Perhaps next time there will be some consolidation, and we can then decide just how particular and how general a concept is the soliton.1.We begin this historical account with the classic paper of H. Bethe [1], which introduced first the many-body wave function crucial
I. Brief History.When I first learned of this conference, and saw the wide range of interests represented by the participants, I was certain that finally a common definition of that most intriguing concept-the soliton-would emerge.And I was anxious that this definition be broad enough to encompass solitons in both their classical and quantum versions.For recent work had convinced me that the general technique known as Bethe's ansatz for solving diverse one-dimensional quantum problems in fact was nothing more nor less than the quantum soliton.However, from conversations with conference participants, I discovered that many were not aware of this accumulated work on exactly soluble quantum systems.I even gained the impression that some feel quantum mechanics to be much harder than classical mechanics-an unnecessary complication.My own feeling, on the other hand, is that in many cases quantum mechanics clarifies matters.It forces one to immediately face problems that would eventually have to be faced in the corresponding classical case, problems such as the counting of states, to determine how important solitons really are.Anyway, the history of the quantum soliton is all to the greater glory of the soliton concept.And quite a respectable history it is.I would date the beginning at 1931 with a paper of H. Bethe [1] on magnetism; that is a total span of 45 years.Quantum mechanics itself has not been around all that much longer!It now seems my hopes for a consensus on a definition of the soliton were premature, and things are still fermenting.Nevertheless, here is my contribution to the conference in the form of a brief history for general interest, and a more original work on the quantization of the Toda lattice.Perhaps next time there will be some consolidation, and we can then decide just how particular and how general a concept is the soliton.1.We begin this historical account with the classic paper of H. Bethe [1], which introduced first the many-body wave function crucial
Spin, still a concept or property not fully exploited, has already impacted the world in many ways with its many applications. The history of how spin came to light and how it was interpreted and developed as investigations were conducted is of particular interest with the many odd outcomes of experiments concerning this intrinsic property. This article will sort out the historical outline of how spin was first indicated in theories and how it was eventually embraced as a fundamental property of quantum particles in an attempt to holistically review the process of recognition of this remarkable concept. Through this review, it becomes apparent how the recognition and development of a concept involves the intellect of a broad range of physicists who are willing to view things from both classical and quantum perspectives particularly when it comes to new concepts as spin. How elements like luck and chance play a role will also be revealed. As it turns out, our intuitions about the world around us may not be the ultimate truth, especially when it comes to Quantum Mechanics. Space Quantization was once thought of as no more than a theoretical outcome, however, experiments anyway impose this reality on physicists requiring them to abandon stubborn biases. Admittedly, not every speculation was correct, the analogy of “Spin” was an unfortunate one. But this article shows that this is just how understanding the mysteries of the quantum world really unfolds.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Unlike special relativity, quantum mechanics was not the inspired product of one mind. It developed, in fits and starts, over a period of 25 years (1900–1925), and many cooks stirred the broth. Even now, a century later, quantum mechanics raises profound conceptual questions. Every competent physicist can “do” quantum mechanics, and the predictions it makes are in spectacularly good agreement with experimental results; there can be little doubt that quantum mechanics is “right.” On the other hand, as Richard Feynman once remarked, “nobody understands quantum mechanics.” I'll tell the story in two parts. First the history (how the essential pieces of the puzzle were assembled), and then the implications (what the finished picture has to say about our world). Photons Like relativity, quantum mechanics began with the study of light. In both cases, this historical association is misleading. Relativity, really, has nothing to do with light – to be sure, it involves that magic speed limit, c , and of course the most familiar thing that travels at speed c is light. So light is a convenient vehicle for introducing the subject. But it is perfectly possible to imagine a universe in which there is no such thing as light, and yet relativity might still be valid. Quantum mechanics, too, starts with light – in this case the “quantum” of light (the photon). But, as it turns out, this is just one rather specialized example of much more general principles – in fact, a rather bad example, because photons are by their nature relativistic (they travel at the speed of light), and it would be more natural to begin with the quantum mechanics of slow-moving objects, in analogy with Newtonian mechanics.
In applied physics and in engineering intuitive understanding is a boost for creativity and almost a necessity for efficient product improvement. The existence of soliton solutions to the quantum equations in the presence of self-interactions allows us to draw an intuitive picture of quantum mechanics. The purpose of this work is to compile a collection of models, some of which are simple, some are obvious, some have already appeared in papers and even in textbooks, and some are new. However this is the first time they are presented (to our knowledge) in a common place attempting to provide an intuitive physical description for one of the basic particles in nature: the electron. The soliton model applied to the electrons can be extended to the electromagnetic field providing an unambiguous description for the photon. In so doing a clear image of quantum mechanics emerges, including quantum optics, QED and field quantization. To our belief this description is of highly pedagogical value. We start with a traditional description of the old quantum mechanics, first quantization and second quantization. We describe an intuitive description of first and second quantization based on the concept of solitons, pioneered by the double solution of de Broglie in 1927. Many aspects of first and second quantization are clarified and visualized intuitively, including the so called vacuum fluctuations.
In this paper, we pick some less well-known contributions of great minds to the history of ideas from the proceedings of the Solvay Conferences. We hope to show there was nothing inevitable about the new physics winning out. In fact, we suggest modern-day physicists may usefully go back to some of the old ideas – most notably the idea that elementary particles do have some shape and size – and that they should, perhaps, try somewhat harder to explain intrinsic properties of these particles, such as angular momentum and their magnetic moment, in terms of classical physics. The contributions which we discuss are those of Ernest Rutherford, Joseph Larmor, Hendrik Antoon Lorentz and Louis de Broglie.
Two themes are considered in this paper. First of all in the Introduction a comment is made about the Tzitzéica equation which occurred in the author's work as reported in [1]. Much recent work has referred to this equation and a representative bibliography is given in this paper with brief comments. The second theme of this paper is concerned with work surrounding Quantum Solitons and the author's talk at the ISLAND 2 meeting under the title ‘Quantum solitons’ is briefly summarised. An extended review of this subject matter has now appeared in [2] and as there the quantum soliton of the quantum attractive NLS model is seen as a ‘qubit’ for quantum information purposes. It is hoped that this summary and/or the reference [2] can help to stimulate further interest in this ‘quantum’ aspect of our subject in the solitons community. The actual observation of a quantum soliton is also reported and a proper theoretical description of it given. Reference is made to -boson lattices first as a simple example of the quantum inverse method in the Introduction and then as a subject matter in its own right at the end of the paper.