In his last memoir on mathematical physics, Henri Poincaré presented one of the most profound and compelling proofs of the hypothesis of quanta. This highly original proof, which is actually three separate proofs, is based on first principles and is full of physical insight, mathematical rigor, and elegant simplicity. The memoir is refreshingly uncluttered by some of the conventional, and more abstract concepts, such as temperature and entropy, that Planck and others relied on in their work. Poincaré’s analysis is based on an ingenious physical model consisting of long-period resonators interacting with short-period resonators. A unique formulation of statistical mechanics, based on the calculus of probabilities, Fourier’s integral, and complex analysis, logically unfolds throughout the memoir. Poincaré invents an ‘‘inverse statistical mechanics’’ that allows him to prove a crucial result that no one had proved before: The hypothesis of quanta is both a sufficient and a necessary condition to account for Planck’s law of radiation. In a separate, more universal proof, Poincaré proves that the existence of a discontinuity in the motion of a resonator is necessary to explain any observed law of radiation. Given the significant impact of Poincaré’s memoir on quantum theory and statistical physics, it is surprising that most physicists are not aware of its valuable mathematical and physical ideas. Poincaré’s tour de force proofs are presented here in a form suitable for use in a standard course in quantum mechanics, statistical mechanics, or mathematical physics.
In his last memoir on mathematical physics, Henri Poincaré presented one of the most profound and compelling proofs of the hypothesis of quanta. This highly original proof, which is actually three separate proofs, is based on first principles and is full of physical insight, mathematical rigor, and elegant simplicity. The memoir is refreshingly uncluttered by some of the conventional, and more abstract concepts, such as temperature and entropy, that Planck and others relied on in their work. Poincaré’s analysis is based on an ingenious physical model consisting of long-period resonators interacting with short-period resonators. A unique formulation of statistical mechanics, based on the calculus of probabilities, Fourier’s integral, and complex analysis, logically unfolds throughout the memoir. Poincaré invents an ‘‘inverse statistical mechanics’’ that allows him to prove a crucial result that no one had proved before: The hypothesis of quanta is both a sufficient and a necessary condition to account for Planck’s law of radiation. In a separate, more universal proof, Poincaré proves that the existence of a discontinuity in the motion of a resonator is necessary to explain any observed law of radiation. Given the significant impact of Poincaré’s memoir on quantum theory and statistical physics, it is surprising that most physicists are not aware of its valuable mathematical and physical ideas. Poincaré’s tour de force proofs are presented here in a form suitable for use in a standard course in quantum mechanics, statistical mechanics, or mathematical physics.
Quantum Mechanics and its Interpretations Peter E. Hodgson (bio) Quantum Mechanics is an elegant and indeed beautiful formalism, and enables the results of many physical measurements to be calculated with great accuracy. But 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. It should therefore be evident that the most that can be done here is to try to show where the real difficul-ties 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. The Discovery of the Quantum The quantum nature of the physical world was discovered by Max Planck in the early years of the twentieth century. He was studying thermodynamics and his attention was drawn to the frequency distribution of black-body radiation. This radiation is emitted from a small hole in a heated enclosure with rough walls, which ensures [End Page 62] that the radiation reaches statistical equilibrium before emission. The frequency distribution is found to be independent of the material lining the walls of the oven, so Planck realized that it must be a fundamental characteristic of all matter. This frequency distribution had been measured by several experimentalists, and Planck resolved to try to see how it can be derived theoretically. Already Rayleigh and Jeans had used classical radiation theory to derive an expression that fitted the data well at low frequencies but tended to infinity at high energies, which was of course unacceptable. The physicist Wien had derived a formula that fitted the data at high frequencies but failed at low frequencies. Planck used an ingenious argument based on entropy to interpolate between these two formulae and obtained a formula, now known as the Planck distribution, that fitted the data perfectly. His next task was to understand the physics behind his formula. He used the familiar technique of assuming that the energy is emitted in small finite amounts, and then let their size tend to zero, so as to represent a continuous distribution. To his great surprise, the intermediate formula with discrete amounts of energy, called quanta, gave the Planck distribution, but the final result, with infinitely small quanta, did not. All his instincts as a physicist rebelled against this conclusion. Surely energy must be emitted continuously. He tried hard to find a way to avoid this unwelcome conclusion, but without success. He had to admit that it is a feature of the world. Soon it was confirmed by Einstein’s interpretation of the photoelectric effect. This story shows clearly that physicists do not impose their ideas on the world, but recognize and accept what they find, whether or not it agrees with their original ideas. The world is made by God in ways that often we cannot imagine. [End Page 63] The Old Quantum Theory A few years later Rutherford discovered that the atom has a small central nucleus containing most of its mass surrounded by a number of electrons. Bohr used the idea of the quantum to build his model of the atom. He assumed that the electrons go around the nucleus in discrete orbits, and that radiation is emitted when an electron falls from one orbit to a lower one. With this model, he was able to calculate to high accuracy the frequencies of the radiation emitted from an excited hydrogen atom. This was a great achievement, but to get his result he had to assume that the electrons stayed in their orbits, whereas classical theory predicted that they would continuously lose energy and rapidly spiral into the nucleus. Quantum Mechanics and its Interpretations These problems were solved by the development of quantum mechanics in the 1920s by Heisenberg, Schrodinger, Dirac, and others. This enabled what is called the wave function to be calculated for any physical system, and from the wave function the probabilities of occurrence of measurable quantities can be found. It is a beautiful theory, and is the indispensable means of calculation for atomic, nuclear...
The origin of the relativity theories was closely bound up with the development of electromagnetic concepts, a development that approached a coherent fieldtheoretical formulation, according to which all actions may vary in a continuous manner. In contrast, quantum theory arose out of the development of atomic concepts, a development that was characterized by the acknowledgment of a fundamental limitation to classical physical ideas when applied to atomic phenomena. This restriction was expressed in the so-called quantum postulate, which attributed to any atomic process an essential discontinuity that was symbolized by Planck's quantum of action. Quantum field theory (QFT) is a later phase of the conceptual developments of quantum theory, and has the old quantum theory and non-relativistic quantum mechanics, essentially the preliminary analyses of the interactions between atoms and radiation, as its predecessors. This chapter will review some features of quantum physics that are relevant to the rise of QFT. The quantization of motion In solving the problem of the equilibrium between matter and radiation, Max Planck (1900) showed that the laws of heat radiation demanded an element of discontinuity in the description of atomic processes. In the statistical behavior of atoms represented, in Planck's description, by linear resonators in their interactions with radiation, only states of vibration should be taken into account whose energy was an integral multiple of a quantum, hv , where h is Planck's constant and v is the frequency of the resonator. Planck himself believed that the discontinuity of energy was only a property of atoms, and was not ready to apply the idea of energy quantization to radiation itself.
Doubts and misconceptions about the origin and justification of a physical theory\'s most fundamental concept are bound to spill over into doubts and misconceptions about the origin and foundation of the theory as a whole: such is the case, in point of fact, of the concept of quantum discontinuity of orthodox quantum mechanics. This work, which comprises three chapters, each of which is an independent paper, is a comprehensive investigation of the origin and nature of the concept of quantum discontinuity. The first paper, entitled \"Rewriting the Quantum \'Revolution\',\" gives a novel account of the origin of quantum theory. It refutes, by direct counter-example, Thomas Kuhn\'s well-known thesis that revolutions in science do not happen on the basis of theory and experiment alone; the quantum revolution happened exactly in this way. The second paper, \"On the Lawfulness of Quanta,\" is a sequel to \"Rewriting the Quantum \'Revolution\'.\" It evinces that and how the existence of the energy quanta is a principled result; it refutes the traditional viewthat the existence of the energy quantawas originally introduced in physics as an ad hoc hypothesis in a self-contradictory argument by showing that it is instead a necessary consequence of the laws of thermodynamics of black-bodies. This result affects our understanding of what is actually involved in the classical-to-quantum paradigm-shift, of scientific method, and of the specific nature of the quantum hypothesis which was carried over into contemporary physics in the course of the quantum revolution. The third paper, \"The Dual Dynamical Foundation of Orthodox Quantum Mechanics,\" was written in collaboration with the mathematical physicist Dr. Ricardo Correa da Silva. It is a detailed critical study of the dynamical foundations of orthodox quantum mechanics; it contains several original results of mathematical, historical, and theoretical interest. The most important of them is the discovery of a dual action functional from which both of the dynamical postulates of von Neumann\'s quantum mechanics, process 1 and process 2, necessarily follow. It also contains the following results that have interest on their own: (i) a derivation of the canonical commutation relations from Matrix Mechanics; (ii) an analysis of Schrödinger\'s first derivation of the wave equation that improves on the understanding of it that is now current in the literature; (iii) a demonstration of how the canonical commutation relations follow fromWave Mechanics; (iv) a discussion, that improves on F. A. Muller\'s work, about the theoretical equivalence of Matrix andWave Mechanics; (v) a revaluation of von Neumann\'s statement of the measurement problem. Broadly speaking, the three papers, taken together, drive home the lesson that the concept of quantum discontinuity stands on a more solid foundation than the scholarship on the foundations of quantum mechanics has hitherto acknowledged
Abstract With the failure of classical physics to explain the experimentally observed black body radiation spectrum, the photoelectric effect and the existence of narrow spectral lines in the light emitted by excited atoms, physicists were forced to reach out for new ideas. Following Planck’s initial “drastic hypothesis” of light “quanta”, Einstein led the way in exploring some of the myriad implications of this idea. In particular, the study of fluctuations led to a new understanding of the interaction of electromagnetic radiation with atoms. Einstein invented the photon by showing, through fluctuation studies, that light emitted by atoms must be in the form of plane waves. These ideas led to the first notions regarding the quantization of matter and its dynamics.
This text presents an intuitive and robust mathematical image of fundamental particle physics based on a novel approach to quantum field theory, which is guided by four carefully motivated metaphysical postulates. In particular, the book explores a dissipative approach to quantum field theory, which is illustrated for scalar field theory and quantum electrodynamics, and proposes an attractive explanation of the Planck scale in quantum gravity. Offering a radically new perspective on this topic, the book focuses on the conceptual foundations of quantum field theory and ontological questions. It also suggests a new stochastic simulation technique in quantum field theory which is complementary to existing ones. Encouraging rigor in a field containing many mathematical subtleties and pitfalls this text is a helpful companion for students of physics and philosophers interested in quantum field theory, and it allows readers to gain an intuitive rather than a formal understanding.