The 10th International Workshop on Maximum Entropy and Bayesian Methods, MaxEnt 90, was held in Laramie, Wyoming from 30 July to 3 August 1990. This volume contains the scientific presentations given at that meeting. This series of workshops originated in Laramie in 1981, where the first three of what were to become annual workshops were held. The fourth meeting was held in Calgary. the fifth in Laramie, the sixth and seventh in Seattle, the eighth in Cambridge, England, and the ninth at Hanover, New Hampshire. It is most appropriate that the tenth workshop, occurring in the centennial year of Wyoming's statehood, was once again held in Laramie. The original purpose of these workshops was twofold. The first was to bring together workers from diverse fields of scientific research who individually had been using either some form of the maximum entropy method for treating ill-posed problems or the more general Bayesian analysis, but who, because of the narrow focus that intra-disciplinary work tends to impose upon most of us, might be unaware of progress being made by others using these same techniques in other areas. The second was to introduce to those who were somewhat aware of maximum entropy and Bayesian analysis and wanted to learn more, the foundations, the gestalt, and the power of these analyses. To further the first of these ends, presenters at these workshops have included workers from area. s as varied as astronomy, economics, environmenta.
In a certain sense this book has been twenty-five years in the writing, since I first struggled with the foundations of the subject as a graduate student. It has taken that long to develop a deep appreciation of what Gibbs was attempting to convey to us near the end of his life and to understand fully the same ideas as resurrected by E.T. Jaynes much later. Many classes of students were destined to help me sharpen these thoughts before I finally felt confident that, for me at least, the foundations of the subject had been clarified sufficiently. More than anything, this work strives to address the following questions: What is statistical mechanics? Why is this approach so extraordinarily effective in describing bulk matter in terms of its constituents? The response given here is in the form of a very definite point of view-the principle of maximum entropy (PME). There have been earlier attempts to approach the subject in this way, to be sure, reflected in the books by Tribus [Thermostat ics and Thermodynamics, Van Nostrand, 1961], Baierlein [Atoms and Information Theory, Freeman, 1971], and Hobson [Concepts in Statistical Mechanics, Gordon and Breach, 1971].
The material contained in this work concerns relativistic quantum mechanics, and as such pertains to classical fields. On the one hand it is meant to serve as a text on the subject, a desire stemming from the author's fruitless searches for an adequate, up-to-date reference when lecturing on these topics. At times the supplementary material was found to exceed by far that in the assigned text. On the other hand, there is some flavor of a monograph to what follows, most particularly in the later chapters, for a major goal is to demonstrate just how far we can advance our understanding of the behavior of stable particles and their interactions without introducing quantized fields. Those wishing to describe the world in this way may view the result as a point of departure, despite the fact that their wish remains unfulfilled. Confirmed quantum-field theorists, however, will doubtless view it as a summary of just why they feel compelled to quantize the fields. Approximately half the book is devoted to the single-particle Dirac equation and its solutions. A great deal of detail is provided in this respect, and the discus sion is reasonably comprehensive. The Dirac equation is extraordinarily important in its own right, particularly as a basis for quantum electrodynamics (QED), and is thus worthy of extensive study.
In this volume we continue the logical development of the work begun in Volume I, and the equilibrium theory now becomes a very special case of the exposition presented here. Once a departure is made from equilibrium, however, the problems become deeper and more subtle-and unlike the equilibrium theory, many aspects of nonequilibrium phenomena remain poorly understood. For over a century a great deal of effort has been expended on the attempt to develop a comprehensive and sensible description of nonequilibrium phenomena and irreversible processes. What has emerged is a hodgepodge of ad hoc constructs that do little to provide either a firm foundation, or a systematic means for proceeding to higher levels of understanding with respect to ever more complicated examples of nonequilibria. Although one should rightfully consider this situation shameful, the amount of effort invested testifies to the degree of difficulty of the problems. In Volume I it was emphasized strongly that the traditional exposition of equilibrium theory lacked a certain cogency which tended to impede progress with extending those considerations to more complex nonequilibrium problems. The reasons for this were adduced to be an unfortunate reliance on ergodicity and the notions of kinetic theory, but in the long run little harm was done regarding the treatment of equilibrium problems. On the nonequilibrium level the potential for disaster increases enormously, as becomes evident already in Chapter 1.
In a certain sense this book has been twenty-five years in the writing, since I first struggled with the foundations of the subject as a graduate student. It has taken that long to develop a deep appreciation of what Gibbs was attempting to convey to us near the end of his life and to understand fully the same ideas as resurrected by E.T. Jaynes much later. Many classes of students were destined to help me sharpen these thoughts before I finally felt confident that, for me at least, the foundations of the subject had been clarified sufficiently. More than anything, this work strives to address the following questions: What is statistical mechanics? Why is this approach so extraordinarily effective in describing bulk matter in terms of its constituents? The response given here is in the form of a very definite point of view-the principle of maximum entropy (PME). There have been earlier attempts to approach the subject in this way, to be sure, reflected in the books by Tribus [Thermostat ics and Thermodynamics, Van Nostrand, 1961], Baierlein [Atoms and Information Theory, Freeman, 1971], and Hobson [Concepts in Statistical Mechanics, Gordon and Breach, 1971].
The material contained in this work concerns relativistic quantum mechanics, and as such pertains to classical fields. On the one hand it is meant to serve as a text on the subject, a desire stemming from the author's fruitless searches for an adequate, up-to-date reference when lecturing on these topics. At times the supplementary material was found to exceed by far that in the assigned text. On the other hand, there is some flavor of a monograph to what follows, most particularly in the later chapters, for a major goal is to demonstrate just how far we can advance our understanding of the behavior of stable particles and their interactions without introducing quantized fields. Those wishing to describe the world in this way may view the result as a point of departure, despite the fact that their wish remains unfulfilled. Confirmed quantum-field theorists, however, will doubtless view it as a summary of just why they feel compelled to quantize the fields. Approximately half the book is devoted to the single-particle Dirac equation and its solutions. A great deal of detail is provided in this respect, and the discus sion is reasonably comprehensive. The Dirac equation is extraordinarily important in its own right, particularly as a basis for quantum electrodynamics (QED), and is thus worthy of extensive study.
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