Representing Spence's integral by elementary functions

A few years ago, when some of us were working to introduce particle physics into A-level courses, I exchanged letters with a professor of particle physics who expressed the view that we would not succeed in our aims, on the grounds that there were no textbooks that treated particle physics at an appropriate level. Our opinion at the time was that the professor had got his causality back to front, as history has indeed shown. The establishment of A-level particle physics options has prompted authors and publishers to produce several texts for sixth-formers, of which this is one. This book is a companion to Patrick Fullick's Physics in the Heinemann Advanced Science series, and it has the same general design features: full colour, many photographs, attractive layout, boxes containing detailed information and extension material, questions, summaries. These features make it more attractive than some of its rivals. It also covers the current syllabuses comprehensively, with chapters on High energy physics (i.e. accelerating particles), Detectors, Leptons, Antimatter, The nucleus, Hadrons and quarks, The forces of nature, Particle physics and cosmology, and Nuclear power. A short Introduction `In search of perfect symmetry' sets the tone for the rest of the book and illustrates what are, in my view, its main strengths and weaknesses. Following a discussion of reflection and rotation, there are accounts of Noether's Theorem (which shows that momentum and energy conservation arise from symmetry in the mathematical description of the universe), CPT invariance (illustrated with Feynman diagrams), and supersymmetry and superstrings. There are appeals to symmetry throughout the book, notably in a discussion of Dirac's prediction of antimatter. This is both a strength and a weakness. Symmetry offers insights not normally found in A-level work, but I suspect that the majority of sixth-form physicists will find it hard to engage with. The Feynman diagrams in the Introduction illustrate a recurrent problem, namely that some unfamiliar and quite difficult ideas are used before they are explained. Feynman diagrams are `done' in chapter 4; terms such as hadron and strong force are used in chapters 2 and 3 but not really explained until chapter 6. (In a variation on this theme, some ideas, e.g. energy-time uncertainty, are introduced from scratch in two different chapters.) Also, while there are boxes summarizing some key areas of physics (charged particles in electric and magnetic fields, for example), a lot of general A-level physics knowledge is assumed without comment (e.g. projection of vectors, frames of reference, the Planck equation). The book is thus most likely to be useful in a nuclear or particle physics option taught quite late in an A-level course, and would be best used for additional support and information after a topic has been introduced in class rather than for introductory reading. The inclusion of superstrings and supersymmetry illustrates another strength: the book is up to date. The discovery of the top quark is in, as are the scrapping of the American Superconducting Super Collider project and the development of the Large Hadron Collider at CERN, and the as yet undiscovered Higgs particle (but no WIMPs). Further strengths include a good discussion of forces (`one end of an interaction'), a nice summary of the ways matter and radiation can interact and a helpful explanation of why heavy nuclei are relatively neutron rich - to name but a few. This is, in summary, a worthwhile book, which should appeal particularly to students (and teachers) with a theoretical turn of mind. The professor should approve.

We continue our study of generalized probability from the viewpoint of category theory. Assuming that each generalized probability measure is a morphism, we model basic probabilistic notions within the category cogenerated by its range. It is known that the closed unit interval I = [0, 1], carrying a suitable difference structure, cogenerates the category ID in which the classical and fuzzy probability theories can be modeled. We study generalized probability theories modeled within two different categories cogenerated by a simplex S n = {(x 1, x 2, …, x n) ∈ I n: $$ \mathop \sum \limits_{i = 1}^n $$ x i ≤ 1}, carrying a suitable difference structure; since I and S 1 coincide, for n = 1 we get the fuzzy probability theory as a special case. In the first case, when the morphisms preserve the so-called pure elements, the resulting category S n D, n > 1, and ID are isomorphic and the generalized probability theories modeled in ID and S n D are “the same”. In the second case, when the morphisms map each maximal element to a maximal element, the resulting categories WS n D, n > 1, lead to different models of generalized probability theories. We define basic notions of the corresponding simplex-valued probability theories and mention some applications.

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications. Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one. Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications. Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.

These lectures are intended to be a pedagogical introduction to some of the ideas and concepts concerning scaling phenomena which arise in nuclear and particle physics. The level will be fairly modest and is aimed at the graduate student who has some familiarity with Feynman diagrams and the rudiments of quantum field theory. In actual fact, the first two-thirds of these lectures really only requires a standard background in non-relativistic quantum mechanics.KeywordsStructure FunctionDimensional AnalysisOperator Product ExpansionRenormalization Group EquationAsymptotic FreedomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

<p>Wilson admitted in private correspondence in 1923 with Francis Ysidro Edgeworth that he should not have attempted his review of the A Treatise on Probability (1921) in 1923.The reason Wilson gave to Edgeworth in his correspondence was exactly the same as the reason Edgeworth gave Wilson as the reason Edgeworth was writing to Wilson-a lack of understanding about what Keynes was doing in Part II of the A Treatise on Probability. What Keynes had done, of course, was revolutionary. Keynes had built on the earlier, logical theory of Boole .Keynes erected a general theory of probability where numerical probabilities appeared as a special case when the weight of the evidence, w, defined on the unit interval[0,1],where a w=1 defined a complete evidence set upon which to base a decision maker’s estimates of probability, equaled ,approximated or approached 1 .w measured the completeness of the evidence. If the weight of the evidence equaled 1 and the decision maker had linear probability preferences, then precise, exact, numerical probabilities could be calculated.</p><p><br />Keynes ‘s axiom system in Part II of the A Treatise on Probability was set up to deal with both interval valued probabilities ,which are non additive, and numerical(cardinal) probabilities, which are additive .Non additive probabilities are not subject to the mathematical laws of the probability calculus, the addition rule and the multiplication rule. Keynes used interval valued estimates to deal with the problems of uncertainty and ignorance, where missing relevant information, data, knowledge, or evidence meant that a decision maker could not use numerical probabilities because Keynes’s weight of the evidence variable, w, was less than 1.Only in the case of a complete information set could numerical probabilities or a probability distribution be specified.</p><p><br />In 1934, Wilson published a note titled, Boole’s Challenge Problem, in the Journal of the American Statistical Association. It is actually a disguised, camouflaged second review of J M Keynes’s A Treatise on Probability that deals with Keynes’s Part II again. This time, Wilson gets the technical analysis correct right.<br /><br />Wilson’s note, although he himself had no idea about why dealing with conditions of uncertainty and ignorance was relevant when making a decision, totally and completely destroys the logical and intellectual foundations erected by G. L. S.Shackle and Paul Davidson ,as well as Jochen Runde, Shiela Dow,Rod O’Donnell,Tony Lawson,Anna Carabelli,Robert Skidelsky, Gay Meeks,Bradley Bateman,Donald Gillies ,and many other Post Keynesians,Institutionalists,and Keynesian Fundamentalists,who have claimed, on the basis of papers published by Frank Ramsey and Richard Braithwaite, for nearly 40 years that J M Keynes’s theory in the A Treatise on Probability is an ordinal theory that could only be applied some of the time or has to be interpreted as a relative frequency theory to make an sense out of it (Davis, Winslow).</p><p><br />The two books that supposedly claim to deal with Keynes’s approach to uncertainty, The Philosophy of Keynes’ Economics: Probability, Uncertainty, and Convention,2003,edited by Runde and Mizuhara,and Fundamental Uncertainty,2011,edited by Marzetti Dall’Aste Brandolini and Scazzieri contain no mention of either George Boole or interval valued probability,Thus,they have nothing to say about Keynes’s interval valued approach to probability.</p>

An Analysis of Edwin B. Wilson's Secret, Second Review of the 'A Treatise on Probability' in 1934: How It Demonstrated That Keynes's Theory of Probability Was an Interval Valued Approach to Probability and Not an Ordinal Theory

On Keynes's Realization That His Concept of Uncertainty in the General Theory, Defined as an Inverse Function of Weight from the 'A Treatise on Probability', Would Not Have Any Impact at All on Economists Thinking

Quantum field thory is a fundamental branch of theoretical physics, which in the last 20 years, has led to spectacular progress in our understanding of phase transitions and elementary particle physics. This textbook emphasizes the underlying unity of the concepts and methods used in both domains, and presents in clear language topics such as the pertubative expansion, Feynman diagrams, renormalization, and the renormalization group. It contains detailed applications of critical phenomena to condensed matter physics, such as the calculation of critical exponents and a discussion of the XY model. Applications to particle physics include quantum electrodynamics and chromodynamics, electroweak interactions, and lattice gauge theories. The book is based on courses given over several years on statistical mechanics and field theory, and is written at graduate level. It attempts to guide the reader through a somewhat difficult and sometimes intricate subject. Further it is intended to lead the student up to the level of more advanced textbooks and research articles. A large number of problems are given to measure the reader's understanding.

В статье рассматривается проблема расширения стандартной модели физики элементарных частиц. Основное внимание уделено неминимальной суперсимметричной модели. На сегодняшний день еe рассмотрение является наиболее приемлемым по сравнению с другими разрабатываемыми суперсимметричными теориями. Мы также рассматриваем нарушение СР-инвариантности в секторе Хиггса. При нахождении физических состояний бозонов Хиггса мы находим локальный минимум потенциала Хиггса. Эту процедуру необходимо выполнить для того, чтобы получить стабильное состояние вакуума. Далее произведены расчеты ширины распада бозона Хиггса на два фотона. Мы используем метод квантово-полевой теории возмущений совместно с фейнмановским диаграммным подходом. Результаты расчeтов представлены графически. Показано влияние СР-нарушающей фазы на величину ширины распада. Также приведены зависимости масс бозонов Хиггса от фазы СР-нарушения в древесном приближении. Некоторые параметры модели не определены на эксперименте, а заключены в некотором диапазоне значений, что дает возможность рассматривать различные сценарии нескольких бозонов Хиггса. В частности, в случае СР-нарушения возможно существование бозона Хиггса с массой меньше 125 ГэВ. Это объясняется отсутствием определенного СР-состояния этой частицы.

A Comparison of J. M. Keynes’s Logical Approach to Probability and Any ‘Objective Bayesian’ Approach to Probability Needs to Incorporate All Five Parts of Keynes’s a Treatise on Probability, Not Just Part I

A general algebraic system M is considered with two binary operations. The family of all measurable functions with values in the unit interval is a motivating example. A state is a morphism from M to the unit interval, an observable is a morphism from the family of Borel sets to M. A joint distribution of two observables is constructed. It is applied for the construction of the sum of observables and for a representation of conditional probability.

There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various physical systems in view, this book is a collection of topical survey articles by leading researchers in both fields, working on the mathematical description of growth phenomena in the broadest sense. The main aim of the book is to foster interaction between researchers in probability and analysis, and to inspire joint efforts to attack important physical problems. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic analysis, multi-scale techniques, and homogenization of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose–Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe.

Analytical and numerical methods for massive two-loop self-energy diagrams

This report describes the importance of oscillations in brain function and introduces metaphors to quantum dynamics. In order to analyze scattering processes at the level of elementary particles Werner Heisenberg proposed the use of the so-called S-Matrix to understand nuclear interactions by studying ingoing at outgoing particles. Later, Richard Feynman developed useful schemes in order to visualize processes of elementary particle interactions. In the present report a metaphor to Feynman Diagrams is developed in order to model the “Ongoing Brain Activity and Event Related Oscillations”. The response susceptibility of the brain shows a probabilistic causality similar to uncertain processes in elementary particle physics. A new grammar called Brain Feynman Diagrams is proposed in order to show brain oscillatory responses as a more visible construct in comparison to conventional compound brain potentials.

Information exchange between inanimate objects (like individual physical particles, or systems) involves special approaches, due to the peculiarity that conscious information emitters/recipients are excluded from the process. This paper aims at answering a part of some questions arising by such approaches. One can ask the question, whether is it possible to speak about physical information when there is no live recipient to accept, evaluate, and use it? Can one speak about “physical information” (e.g., signal exchange) between inanimate physical objects at all? (cf., Feynman diagrams.) If yes, what is the nature of that information? Is (physical) information a passive phenomenon, or its existence presumes activity? What does a signal represent if it is not observed and used at the other end, and where is that other end when one can say that the signal in question was lost without observation or use? I try to illustrate my personal answers with a few examples quoted from the history of 20th c. physics. My answers to the questions are not intended to be revelations and to provide final solutions, rather they serve as arguments and indicate that nothing is closed, the discussion is open.