Matematisk fysik

"This graduate textbook covers the basic formalism of supergravity, as well as a wide range of its modern applications, suitable for a focused first course. Assuming a working knowledge of quantum field theory, it gives Ph.D. students the tools they need to do research that uses supergravity"--

Tiden er en gåde. For at løse den tager fysiker Troels Harmark os med på en rejse tilbage til antikkens græske filosoffer, videre op gennem naturvidenskabens historie til bl.a. Newton og Einstein og helt ud til forskningsfronten i moderne fysik og de største spørgsmål som, hvordan universet blev til. Tiden oprandt ved universets fødsel, og vi lever stadig i tiden her 13,8 milliarder år senere. Vi har en krop, der ældes, og kan ikke forestille os vores liv uden tid – vi indordner os under den, og tænker, at den bevæger sig regelmæssigt, som visernes præcise vandring over døgnets timer på uret. Men fysikken beretter en anden historie – om tidens relativitet: at den går langsommere tæt ved Jordens overflade end på toppen af et bjerg, at tiden står stille på kanten af sorte huller, og at universet udvider sig uendeligt. Tidens gang skal findes imellem de to modpoler: at alt er tilfældigt og at alt er uafvendeligt. På sporet af tidens gåde handler om, hvordan det hele hænger sammen, og hvordan det kan være, at man regner med, at tiden en dag vil stoppe med at gå. Hvordan skal vi forstå tiden?

This book provides a solution to the problem with the energy concept. This problem manifests itself in the fact that physicists clearly diverge regarding the question of what energy is. Some define it but others state that we do not know what it is. Although this is a problem for physicists who need to explain the concept, it is not a problem for physics that can be solved by laboratory means. Penetrating into the origin of the notion of energy, this book offers a clear idea of what was discovered and what was invented to interpret the findings.Following the development of the concept, it provides an explanation of the trends in contemporary textbooks. The author's repetition, in his "History and Philosophy of Physics Laboratory", of Joule¿s famous experiment ¿ the paddle wheel experiment ¿ with a calorimeter as originally used by Joule and with a calorimeter as proposed in textbooks, is presented, yielding new insight into the phenomenon. Thus, science teachers andstudents will benefit from reading the book as well as historians, philosophers, students of the history and philosophy of science, and all who are interested in knowing about what it is that we call energy.

This thesis describes two groundbreaking measurements in the precision frontier at the LHC: the first ever differential measurement of the Z-associated single top quark (tZq) production, and the luminosity measurement using Z boson production rate for the first time in CMS. Observed only in 2018, the tZq process is of great importance in probing top quark electroweak couplings. These couplings are natural places for new phenomena to happen in the top quark sector of the standard model. Yet, they are the least explored directly. One has to obtain a firm understanding of the modeling of sensitive distributions to new top-Z interactions. The present analysis marks a major milestone in this long-term effort. All distributions relevant for new phenomena, and/or modeling of tZq, are studied in full depth using advanced Machine Learning techniques.The luminosity and its uncertainty contributes to every physics result of the experiment. The method minutely developed in this thesis provides a complementary measurement that results in a significant overall reduction of uncertainties.

Time, space and entanglement are the main characters in this book. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we provide a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.

In recent decades, twistor theory has grown into an irreplaceable tool for the study of scattering amplitudes in gauge theory and gravity. This book introduces the reader to cutting-edge advances in twistor theory and its applications to general relativity. The problem of graviton scattering in four dimensions is shown to be dual to dramatically simpler computations in a two-dimensional CFT known as a twistor sigma model. Twistor sigma models are the first step toward a holographic description of gravity in asymptotically flat space-times. They underpin the infinitely many asymptotic symmetries of flat space physics discovered in celestial holography, and extend them to exciting new arenas like curved space-times. They also yield intrinsically mathematical results in the field of hyperkähler manifolds. This volume will be of broad interest to students and researchers looking for an accessible entry point into twistor geometry, scattering amplitudes, and celestial holography. It will also provide an invaluable reference for specialists by bringing together results from a host of different disciplines.

This book provides an advanced introduction to extended theories of quantum field theory and algebraic topology, including Hamiltonian quantization associated with some geometrical constraints, symplectic embedding and Hamilton-Jacobi quantization and Becchi-Rouet-Stora-Tyutin (BRST) symmetry, as well as de Rham cohomology. This extended new edition offers a multifaced insight into phenomenology of particles such as baryons and photons, in terms of extended objects. In particular, in the second edition, the baryons are described in hypersphere soliton model, and the photon properties are additionally included in stringy photon model and in Dirac type relativistic quantum mechanics for a photon.It offers a critical overview of the research in this area and unifies the existing literatures, employing a consistent notation. Although the results presented apply in principle to all alternative quantization schemes, special emphasis is placed on the BRST quantization and its de Rham cohomology group which contribute to a deep understanding of constrained physical theories. The book describes how solitons and other models subject to constraints include rigorous treatments of the geometrical constraints which affect the predictions themselves. The book is intended for use by any graduate-level student with quantum field and relativity theories, and it also serves as a useful reference for those working in the field. An extensive bibliography guides the reader toward the source literature on particular topics.

This book comprises the proceedings of the Conference and Exhibition on Non-Destructive Evaluation (NDE 2021) organised by the Indian Society for Non-destructive Testing (ISNT). This book covers topics from wide domains from conventional to advanced NDE, including conventional and advanced NDE methods, drone-based inspections, thermal wave imaging, NDT data fusion, material characterization, waveguide sensors, inspections of civil structures, medical applications such as bone density and cancer diagnosis, periodic maintenance, life estimation, as well as structural integrity and related areas. This book serves as a useful reference for students, researchers, and practitioners alike.

The Standard Model is the most comprehensive physical theory ever developed. This textbook conveys the basic elements of the Standard Model using elementary concepts, without the theoretical rigor found in most other texts on this subject. It contains examples of basic experiments, allowing readers to see how measurements and theory interplay in the development of physics. The author examines leptons, hadrons and quarks, before presenting the dynamics and the surprising properties of the charges of the different forces. The textbook concludes with a brief discussion on the discoveries of physics beyond the Standard Model, and its connections with cosmology. Quantitative examples are given, and the reader is guided through the necessary calculations. Each chapter ends in the exercises, and solutions to some problems are included in the book. Complete solutions are available to instructors at www.cambridge.org/9781107406094.

The book concerns with solving about 650 ordinary and partial differential equations. Each equation has at least one solution and each solution has at least one coloured graph. The coloured graphs reveal different features of the solutions. Some graphs are dynamical as for Clairaut differential equations. Thus, one can study the general and the singular solutions. All the equations are solved by Mathematica. The first chapter contains mathematical notions and results that are used later through the book. Thus, the book is self-contained that is an advantage for the reader. The ordinary differential equations are treated in Chapters 2 to 4, while the partial differential equations are discussed in Chapters 5 to 10. The book is useful for undergraduate and graduate students, for researchers in engineering, physics, chemistry, and others. Chapter 9 treats parabolic partial differential equations while Chapter 10 treats third and higher order nonlinear partial differential equations, both with modern methods. Chapter 10 discusses the Korteweg-de Vries, Dodd-Bullough-Mikhailov, Tzitzeica-Dodd-Bullough, Benjamin, Kadomtsev-Petviashvili, Sawada-Kotera, and Kaup-Kupershmidt equations.

This book focuses on nonextensive statistical mechanics, a current generalization of Boltzmann-Gibbs (BG) statistical mechanics.Conceived nearly 150 years ago by Maxwell, Boltzmann and Gibbs, the BG theory, one of the greatest monuments of contemporary physics, exhibits many impressive successes in physics, chemistry, mathematics, and computational sciences. Presently, several thousands of publications by scientists around the world have been dedicated to its nonextensive generalization. A variety of applications have emerged in complex systems and its mathematical grounding is by now well advanced.Since the first edition release thirteen years ago, there has been a vast amount of new results in the field, all of which have been incorporated in this comprehensive second edition. Heavily revised and updated with new sections and figures, the second edition remains the go-to text on the subject.A pedagogical introduction to the BG theory concepts and their generalizations ¿ nonlinear dynamics, extensivity of the nonadditive entropy, global correlations, generalization of the standard CLT¿s, complex networks, among others ¿ is presented in this book, as well as a selection of paradigmatic applications in various sciences together with diversified experimental verifications of some of its predictions. Introduction to Nonextensive Statistical Mechanics is suitable for students and researchers with an interest in complex systems and statistical physics.

This book proposes and develops the equilibrium compound nucleus post-fission theory, a powerful tool for studying the fission process and making numerical calculations of post-fission nuclear data. It begins with a detailed historical background on fission theory and covers fundamental concepts, such as the Bohr-Wheeler formula and time dependent nuclear density functional theory.The authors explain the kinematics of heavy-ion collisions and develop a heavy-ion spherical optical model. They also present the theoretical methods for calculating the yield, kinetic energy distribution, and angular distribution of fission fragments in the initial state of fission. In addition, readers are provided with the method for calculating the prompt neutron and prompt gamma-ray data as well as the proportion of the isomeric state nucleus and independent yield from the initial yield of the fission fragments. Using the nuclear decay data of the fission products, a method for calculating thecumulative yield and decay heat of the fission fragments is also given. A fission delayed neutron simplification model is proposed and the theoretical method for calculating the total contribution of three fission channels to post-fission nuclear data is provided.This book concludes with a step-by-step guide on numerical calculations for post-fission nuclear data and a Fortran program for optimizing the best theoretical model parameters. It is ideal for both junior and senior researchers in nuclear physics, as well as graduate students who are interested learning about the subject. Given the current interest in post-fission and the tremendous experimental and theoretical efforts in studying it, this book serves as a timely and comprehensive resource for the nuclear physics community.

This book provides an alternative approach to time-independent perturbation theory in non-relativistic quantum mechanics. It allows easy application to any initial condition because it is based on an approximation to the evolution operator and may also be used on unitary evolution operators for the unperturbed Hamiltonian in the case where the eigenvalues cannot be found. This flexibility sets it apart from conventional perturbation theory. The matrix perturbation method also gives new theoretical insights; for example, it provides corrections to the energy and wave function in one operation. Another notable highlight is the facility to readily derive a general expression for the normalization constant at m-th order, a significant difference between the approach within and those already in the literature. Another unique aspect of the matrix perturbation method is that it can be extended directly to the Lindblad master equation. The first and second-order corrections are obtained for this equation and the method is generalized for higher orders. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained. Throughout the book, several benchmark examples and practical applications underscore the potential, accuracy and good performance of this novel approach. Moreover, the method's applicability extends to some specific time-dependent Hamiltonians. This book represents a valuable addition to the literature on perturbation theory in quantum mechanics and is accessible to students and researchers alike.

One of the main unanswered question of modern Physics is "How does gravity behave at small scales?". The aim of this thesis is to illustrate in a comprehensive but accessible way how to look for deviations from Einstein's theory of General Relativity in this regime, looking at the simplest celestial bodies: static and spherically symmetric ones.With a conservative and bottom-up approach, at smaller scales the first corrections to the action of General Relativity are generally considered to be terms quadratic in the curvature tensors; while these modifications do not cure the inconsistency between gravity and quantum mechanics, the solutions of this theory are plausible candidates to be the first-order corrections of the classical ones.Even with such simple modifications, a striking picture emerges from the study of isolated objects: the unique Schwarzschild solution of General Relativity is only a rare bird in the set of solutions, with non-Schwarzschild black holes, wormholes and naked singularities appearing as possible substitutes.Tailored to graduate students and researchers entering this field, this thesis shows how to construct these new solutions from action principles, how to characterize their metric, how to study their physical properties, such as their stability or Thermodynamics, and how to look for phenomenological signatures.

"This concise textbook introduces an innovative computational approach to quantum mechanics. Emphasizing learning through implementation and simulation of quantum phenomena, it's packed with exercises and examples. Source code and data files are provided, along with solutions. Extra problems with locked solutions are provided online for instructors"--

This book presents the iterative beauty of fractals and fractal functions graphically with the aid of MATLAB programming. The fractal images generated using the MATLAB codes provide visual delight and highly encourage the fractal lovers for creative thinking. The book compiles five cutting-edge research chapters, each with state-of-the art fractal illustrations. It starts with the fundamental theory for the construction of fractal sets via the deterministic iteration algorithm. Incorporating the theoretical base, fractal illustrations of elementary fractal sets are provided with the explicit MATLAB code. The book gives examples of MATLAB codes to present the fractal surfaces.This book is contributed to all the research beginners as well as the professionals on the field of fractal analysis. As it covers basic fractals like Sierpinski triangle to advanced fractal functions with explicit MATLAB code, the presented fractal illustrations hopefully benefit even the non-field readers. The book is a useful course to all the research beginners on the fractal and fractal-related fields.

This open access book bridges a gap between introductory Quantum Field Theory (QFT) courses and state-of-the-art research in scattering amplitudes. It covers the path from basic definitions of QFT to amplitudes, which are relevant for processes in the Standard Model of particle physics. The book begins with a concise yet self-contained introduction to QFT, including perturbative quantum gravity. It then presents modern methods for calculating scattering amplitudes, focusing on tree-level amplitudes, loop-level integrands and loop integration techniques. These methods help to reveal intriguing relations between gauge and gravity amplitudes and are of increasing importance for obtaining high-precision predictions for collider experiments, such as those at the Large Hadron Collider, as well as for foundational mathematical physics studies in QFT, including recent applications to gravitational wave physics.These course-tested lecture notes include numerous exercises with solutions. Requiring only minimal knowledge of QFT, they are well-suited for MSc and PhD students as a preparation for research projects in theoretical particle physics. They can be used as a one-semester graduate level course, or as a self-study guide for researchers interested in fundamental aspects of quantum field theory.

This book presents a new way of thinking about quantum mechanics and machine learning by merging the two. Quantum mechanics and machine learning may seem theoretically disparate, but their link becomes clear through the density matrix operator which can be readily approximated by neural network models, permitting a formulation of quantum physics in which physical observables can be computed via neural networks. As well as demonstrating the natural affinity of quantum physics and machine learning, this viewpoint opens rich possibilities in terms of computation, efficient hardware, and scalability. One can also obtain trainable models to optimize applications and fine-tune theories, such as approximation of the ground state in many body systems, and boosting quantum circuits¿ performance. The book begins with the introduction of programming tools and basic concepts of machine learning, with necessary background material from quantum mechanics and quantum information also provided. This enables the basic building blocks, neural network models for vacuum states, to be introduced. The highlights that follow include: non-classical state representations, with squeezers and beam splitters used to implement the primary layers for quantum computing; boson sampling with neural network models; an overview of available quantum computing platforms, their models, and their programming; and neural network models as a variational ansatz for many-body Hamiltonian ground states with applications to Ising machines and solitons. The book emphasizes coding, with many open source examples in Python and TensorFlow, while MATLAB and Mathematica routines clarify and validate proofs. This book is essential reading for graduate students and researchers who want to develop both the requisite physics and coding knowledge to understand the rich interplay of quantum mechanics and machine learning.

This book contains a self-consistent treatment of a geometric averaging technique, induced by the Ricci flow, that allows comparing a given (generalized) Einstein initial data set with another distinct Einstein initial data set, both supported on a given closed n-dimensional manifold. This is a case study where two vibrant areas of research in geometric analysis, Ricci flow and Einstein constraints theory, interact in a quite remarkable way. The interaction is of great relevance for applications in relativistic cosmology, allowing a mathematically rigorous approach to the initial data set averaging problem, at least when data sets are given on a closed space-like hypersurface. The book does not assume an a priori knowledge of Ricci flow theory, and considerable space is left for introducing the necessary techniques. These introductory parts gently evolve to a detailed discussion of the more advanced results concerning a Fourier-mode expansion and a sophisticated heat kernel representation of the Ricci flow, both of which are of independent interest in Ricci flow theory. This work is intended for advanced students in mathematical physics and researchers alike.

Sind Sie fasziniert von den Fragen der Quantenphysik und wollen Sie Beiträge in den Medien zum Thema besser verstehen können? Wolfgang Osterhage gibt Ihnen in diesem kompakten Buch den Einstieg und Überblick, den Sie dafür brauchen - auch ohne quantenphysikalisches Vorwissen! Dieses Buch lädt Sie ein, mitzudenken, und die großen Probleme der Physik des Kleinsten kennenzulernen.Eine Reise durch die Quantenphysik ist und bleibt spannend: denn obwohl die Theorie fast 85 Jahre alt und eine der erfolgreichsten Theorien der Physik ist - hat sie doch viele ihrer Grundsatzfragen bis heute nicht endgültig beantworten können. Es scheint, als habe Richard Feynmanns Ausspruch ¿Ich glaube mit Sicherheit behaupten zu können, daß niemand die Quantenmechanik versteht¿ noch immer Bestand.Um Ihnen den Einstieg zu erleichtern, beginnt dieses Buch mit der Gedankenwelt der Quantenphysik, die manchmal quer zu den Erfahrungen der Alltagswirklichkeit zu stehen scheint. Über das Phänomen der Strahlung wird anschließend atom- und kernphysikalisches Grundwissen vermittelt, bevor es hineingeht in die fremde Welt der Operatoren, Wahrscheinlichkeiten und des Messproblems. Zuletzt münden alle Puzzelstücke in das Standardmodell der Elementarteilchen und in die Quantenfeldtheorie.In der zweiten Auflage wurde das Buch gründlich aktualisiert und u.a. durch die spannenden Themen Quantencomputer und Quanteninformation ergänzt.Eine kurz und bündige Einführung, die Ihnen den Überblick verschafft, den Sie für Ihr Weiterlesen brauchen - viel Freude bei Ihrem Studium Generale.