Mathematical / Computational / Theoretical physics Books

Book SynopsisIn the 25 years since their introduction, Higgs bundles have seen a surprising number of interactions within different areas of mathematics and physics. There is a recent surge of interest following Ngô Bau Châu's proof of the Fundamental Lemma and the work of Kapustin and Witten on the Geometric Langlands program. The program on The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles, was held at the Institute for Mathematical Sciences at the National University of Singapore during 2014. It hosted a number of lectures on recent topics of importance related to Higgs bundles, and it is the purpose of this volume to collect these lectures in a form accessible to graduate students and young researchers interested in learning more about this field.

Read more less

Book SynopsisThis volume presents a collection of problems and solutions in differential geometry with applications. Both introductory and advanced topics are introduced in an easy-to-digest manner, with the materials of the volume being self-contained. In particular, curves, surfaces, Riemannian and pseudo-Riemannian manifolds, Hodge duality operator, vector fields and Lie series, differential forms, matrix-valued differential forms, Maurer-Cartan form, and the Lie derivative are covered.Readers will find useful applications to special and general relativity, Yang-Mills theory, hydrodynamics and field theory. Besides the solved problems, each chapter contains stimulating supplementary problems and software implementations are also included. The volume will not only benefit students in mathematics, applied mathematics and theoretical physics, but also researchers in the field of differential geometry.

Read more less

Book SynopsisThis book is devoted to the Einstein's field equations of general relativity for self-gravitating massive scalar fields. We formulate the initial value problem when the initial data set is a perturbation of an asymptotically flat, spacelike hypersurface in Minkowski spacetime. We then establish the existence of an Einstein development associated with this initial data set, which is proven to be an asymptotically flat and future geodesically complete spacetime.

Read more less

Book SynopsisThis volume contains contributions to the workshop, which was largely focused on the strong coupling gauge theories in search for theories beyond the standard model, particularly, the LHC experiments and lattice studies of conformal fixed point. The main topics include walking technicolor and the role of conformality in view of the 125 GeV Higgs as a light composite Higgs (technidilaton, and other composite Higgs, etc.). Nonperturbative studies like lattice simulations and stringy/holographic approaches are extensively discussed in close relation to the phenomenological studies.After the discovery of 125 GeV Higgs at LHC, the central issue of particle physics is now to reveal the dynamical origin of the Higgs itself. One of the possibilities would be the composite Higgs based on the strong coupling gauge theory in the TeV region, such as the technidilaton predicted in walking technicolor with infrared conformality. The volume contains, among others, many of the latest important reports on walking technicolor and related subjects in the general context of conformality, in a way of direct relevance to the LHC phenomenology as well as the lattice studies. It is very timely to study full theoretical implications in the exciting era when the LHC is vigorously working. This volume is of great importance for that purpose.Speakers of 40 talks (plus posters) include K-I Aoki, Y Aoki, K Bamba, E Bennett, R S Chivukula, H Georgi, A Hasenfratz, D-K Hong, K Itoh, D Elander, G Fleming, H Fukano, Y Iwasaki, M Jarvinen, D Kadoh, S Kim, R Kitano, K-I Kondo, J Kuti, D Lin, N Maru, H Matsufuru, S Matsuzaki, K-I Nagai, C Nonaka, H Ohki, E Pallante, M Rho, E Rinaldi, F Sannino, D Schaich, A Shibata, R E Shrock, E H Simmons, K Tuominen, C H Wong, N Yamada, M J S Yang, and K Yamawaki.

Read more less

Book SynopsisThis volume, recording the 10th international symposium honoring noted French mathematical physicist Jean-Pierre Vigier surveys and continues to develop Unified Field Mechanics (UFM) from the perspective of Multiverse cosmology and Topological Field Theory. UFM represents a developing paradigm shift with many new parameters extending the Standard Model to a 3rd regime of Natural Science beyond Quantum Mechanics. UFM is now experimentally testable, thus putatively able to demonstrate the existence of large-scale additional dimensionality (LSXD), test for QED violating phenomena and surmount the quantum uncertainty principle leading to a new 'Age of Discovery' paling all prior ages in the historical progression: Classical Mechanics (3D) to Quantum Mechanics (4D) and now to the birth of the 3rd regime of UFM in additional dimensionality correlating with M-Theory. Many still consider the Planck-scale as the 'basement of reality'. This could only be considered true under the limitations of the Standard Model. As we methodically enter the new regime a profound understanding of the multiverse and additional dimensionality beckons.

Read more less

Book SynopsisThis volume presents a collection of problems and solutions in differential geometry with applications. Both introductory and advanced topics are introduced in an easy-to-digest manner, with the materials of the volume being self-contained. In particular, curves, surfaces, Riemannian and pseudo-Riemannian manifolds, Hodge duality operator, vector fields and Lie series, differential forms, matrix-valued differential forms, Maurer-Cartan form, and the Lie derivative are covered.Readers will find useful applications to special and general relativity, Yang-Mills theory, hydrodynamics and field theory. Besides the solved problems, each chapter contains stimulating supplementary problems and software implementations are also included. The volume will not only benefit students in mathematics, applied mathematics and theoretical physics, but also researchers in the field of differential geometry.

Read more less

Book SynopsisThe book aims to analyze and explore deep and profound relations between string field theory, higher spin gauge theories and holography — the disciplines that have been on the cutting edge of theoretical high energy physics and other fields. These intriguing relations and connections involve some profound ideas in number theory, which appear to be part of a unifying language to describe these connections.

Read more less

Book SynopsisThis unique book complements traditional textbooks by providing a visual yet rigorous survey of the mathematics used in theoretical physics beyond that typically covered in undergraduate math and physics courses. The exposition is pedagogical but compact, and the emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints. Certain topics which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations, are avoided. The primary physical models targeted are general relativity, spinors, and gauge theories, with notable chapters on Riemannian geometry, Clifford algebras, and fiber bundles.Table of ContentsMathematical Structures; Abstract Algebra; Vector Algebras; Topological Spaces; Algebraic Topology; Manifolds; Lie Groups; Clifford Groups; Riemannian Manifolds; Fiber Bundles; Categories and Functors;

Read more less

Book SynopsisThis volume presents lecture notes of the 12th International School of Theoretical Physics held in 2016 in Rzeszów, Poland. The lectures serve as an introduction for young physicists starting their career in condensed matter theoretical physics. The book provides a comprehensive overview of modern ideas and advances in theories and experiments of new materials, quantum nanostructures as well as new mathematical methods.This lecture note is an essential source of reference for physicists and materials scientists. It is also a suitable reading for graduate students.

Read more less

Book SynopsisSymbolic dynamics is a coarse-grained description of dynamics. It has been a long-studied chapter of the mathematical theory of dynamical systems, but its abstract formulation has kept many practitioners of physical sciences and engineering from appreciating its simplicity, beauty, and power. At the same time, symbolic dynamics provides almost the only rigorous way to understand global systematics of periodic and, especially, chaotic motion in dynamical systems. In a sense, everyone who enters the field of chaotic dynamics should begin with the study of symbolic dynamics. However, this has not been an easy task for non-mathematicians. On one hand, the method of symbolic dynamics has been developed to such an extent that it may well become a practical tool in studying chaotic dynamics, both on computers and in laboratories. On the other hand, most of the existing literature on symbolic dynamics is mathematics-oriented. This book is an attempt at partially filling up this apparent gap by emphasizing the applied aspects of symbolic dynamics without mathematical rigor.

Read more less

Book SynopsisThe book presents the main approaches in study of algebraic structures of symmetries in models of theoretical and mathematical physics, namely groups and Lie algebras and their deformations. It covers the commonly encountered quantum groups (including Yangians). The second main goal of the book is to present a differential geometry of coset spaces that is actively used in investigations of models of quantum field theory, gravity and statistical physics. The third goal is to explain the main ideas about the theory of conformal symmetries, which is the basis of the AdS/CFT correspondence.The theory of groups and symmetries is an important part of theoretical physics. In elementary particle physics, cosmology and related fields, the key role is played by Lie groups and algebras corresponding to continuous symmetries. For example, relativistic physics is based on the Lorentz and Poincare groups, and the modern theory of elementary particles — the Standard Model — is based on gauge (local) symmetry with the gauge group SU(3) x SU(2) x U(1). This book presents constructions and results of a general nature, along with numerous concrete examples that have direct applications in modern theoretical and mathematical physics.

Read more less

Book SynopsisThis volume provides an in-depth treatment of several equations and systems of mathematical physics, describing the propagation and interaction of nonlinear waves as different modifications of these: the KdV equation, Fornberg-Whitham equation, Vakhnenko equation, Camassa-Holm equation, several versions of the NLS equation, Kaup-Kupershmidt equation, Boussinesq paradigm, and Manakov system, amongst others, as well as symmetrizable quasilinear hyperbolic systems arising in fluid dynamics.Readers not familiar with the complicated methods used in the theory of the equations of mathematical physics (functional analysis, harmonic analysis, spectral theory, topological methods, a priori estimates, conservation laws) can easily be acquainted here with different solutions of some nonlinear PDEs written in a sharp form (waves), with their geometrical visualization and their interpretation. In many cases, explicit solutions (waves) having specific physical interpretation (solitons, kinks, peakons, ovals, loops, rogue waves) are found and their interactions are studied and geometrically visualized. To do this, classical methods coming from the theory of ordinary differential equations, the dressing method, Hirota's direct method and the method of the simplest equation are introduced and applied. At the end, the paradifferential approach is used.This volume is self-contained and equipped with simple proofs. It contains many exercises and examples arising from the applications in mechanics, physics, optics and, quantum mechanics.

Read more less

Book Synopsis'The third edition of this book is designed to carefully and coherently introduce fractional calculus to physicists, by applying the ideas to two distinct applications: classical problems and multi-particle quantum problems. There remain many open questions and the field remains an active area of research. Dr Herrmann’s book is an excellent introduction to this field of study.'Contemporary PhysicsThe book presents a concise introduction to the basic methods and strategies in fractional calculus which enables the reader to catch up with the state-of-the-art in this field and to participate and contribute in the development of this exciting research area.This book is devoted to the application of fractional calculus on physical problems. The fractional concept is applied to subjects in classical mechanics, image processing, folded potentials in cluster physics, infrared spectroscopy, group theory, quantum mechanics, nuclear physics, hadron spectroscopy up to quantum field theory and will surprise the reader with new intriguing insights.This new, extended edition includes additional chapters about numerical solution of the fractional Schrödinger equation, self-similarity and the geometric interpretation of non-isotropic fractional differential operators. Motivated by the positive response, new exercises with elaborated solutions are added, which significantly support a deeper understanding of the general aspects of the theory.Besides students as well as researchers in this field, this book will also be useful as a supporting medium for teachers teaching courses devoted to this subject.

Read more less

Book Synopsis'This is an excellent, well-written and very comprehensive book covering many topics of mathematics and physics. An exhaustive collection of problems with detailed solutions that may be valuable to students and young researchers in several fields, ranging from Mathematics to Quantum Physics is presented … I found the book helpful in regards to several subjects that are not covered in other mathematical physics introductory textbooks.'Contemporary PhysicsThis updated and extended edition of the book combines the topics provided in the two parts of the previous editions as well as new topics. It is a comprehensive compilation covering most areas in mathematical and theoretical physics. The book provides a collection of problems together with their detailed solutions which will prove to be valuable to students as well as to researchers in the fields of mathematics, physics, engineering and other sciences.Each chapter provides a short introduction with the relevant definitions and notations. All relevant definitions are given. The topics range in difficulty from elementary to advanced. Almost all problems are solved in detail and most of the problems are self-contained. Stimulating supplementary problems are also provided in each chapter. Students can learn important principles and strategies required for problem solving. Teachers will also find this text useful as a supplement, since important concepts and techniques are developed in the problems. Introductory problems for both undergraduate and advanced undergraduate students are provided. More advanced problems together with their detailed solutions are collected, to meet the needs of graduate students and researchers. Problems included cover new fields in theoretical and mathematical physics such as tensor product, Lax representation, Bäcklund transformation, soliton equations, Hilbert space theory, uncertainty relation, entanglement, spin systems, Lie groups, Bose system, Fermi systems differential forms, Lie algebra valued differential forms, metric tensor fields, Hirota technique, Painlevé test, Bethe ansatz, Yang-Baxter relation, wavelets, gauge theory, differential geometry, string theory, chaos, fractals, complexity, ergodic theory, etc. A number of software implementations are also provided.

Read more less

Book Synopsis'This is an excellent, well-written and very comprehensive book covering many topics of mathematics and physics. An exhaustive collection of problems with detailed solutions that may be valuable to students and young researchers in several fields, ranging from Mathematics to Quantum Physics is presented … I found the book helpful in regards to several subjects that are not covered in other mathematical physics introductory textbooks.'Contemporary PhysicsThis updated and extended edition of the book combines the topics provided in the two parts of the previous editions as well as new topics. It is a comprehensive compilation covering most areas in mathematical and theoretical physics. The book provides a collection of problems together with their detailed solutions which will prove to be valuable to students as well as to researchers in the fields of mathematics, physics, engineering and other sciences.Each chapter provides a short introduction with the relevant definitions and notations. All relevant definitions are given. The topics range in difficulty from elementary to advanced. Almost all problems are solved in detail and most of the problems are self-contained. Stimulating supplementary problems are also provided in each chapter. Students can learn important principles and strategies required for problem solving. Teachers will also find this text useful as a supplement, since important concepts and techniques are developed in the problems. Introductory problems for both undergraduate and advanced undergraduate students are provided. More advanced problems together with their detailed solutions are collected, to meet the needs of graduate students and researchers. Problems included cover new fields in theoretical and mathematical physics such as tensor product, Lax representation, Bäcklund transformation, soliton equations, Hilbert space theory, uncertainty relation, entanglement, spin systems, Lie groups, Bose system, Fermi systems differential forms, Lie algebra valued differential forms, metric tensor fields, Hirota technique, Painlevé test, Bethe ansatz, Yang-Baxter relation, wavelets, gauge theory, differential geometry, string theory, chaos, fractals, complexity, ergodic theory, etc. A number of software implementations are also provided.

Read more less

Book SynopsisProfessor Robert Delbourgo, an illustrious physicist, shares his experience in this collection of papers with commentary. This book covers a wide range of topics, ranging from chaos theory through to unified field theories and gravitation. It illustrates that important contributions to physics can be achieved without chasing fashionable or popular topics. Also, science readers and researchers who live in far flung places and who feel isolated may perhaps gain inspiration and advance their careers from this volume. Being geographically distant from well-known centres of research, internet notwithstanding, need not be a hindrance as Professor Delbourgo can well attest.

Read more less

Book SynopsisIt is not uncommon for the Principle of Complementarity to be invoked in either Science or Philosophy, viz. the ancient oriental philosophy of Yin and Yang whose symbolic representation is portrayed on the cover of the book. Or Niels Bohr's use of it as the basis for the so-called Copenhagen interpretation of Quantum Mechanics. This book arose as an outgrowth of the author's previous book entitled 'Knots, Braids and Moebius Strips,' published by World Scientific in 2015, wherein the Principle itself was discovered to be expressible as a simple 2x2 matrix that summarizes the algebraic essence of both the well-known Microbiology of DNA and the author's version of the elementary particles of physics. At that point, the possibility of an even wider utilization of that expression of Complementarity arose.The current book, features Complementarity, in which the matrix algebra is extended to characterize not only DNA itself but the well-known process of its replication, a most gratifying outcome. The book then goes on to explore Complementarity, with and without its matrix expression, as it occurs, not only in much of physics but in its extension to cosmology as well.

Read more less

Book SynopsisThis book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer.Two different approaches to the subject are described. The first, a “bottom-up” approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of Bloch-Esnault-Kreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, “top-down” approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a Riemann-Hilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry.The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area.Table of ContentsThis book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. The main question is whether residues of Feynman integrals always evaluate to periods of mixed Tate motives, as appears to be the case from extensive computations of Feynman integrals carried out by Broadhurst and Kreimer. Two different approaches to the subject are described. The first, a "bottom-up" approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach grew out of work of Bloch-Esnault-Kreimer and suggests that, while the algebraic varieties associated to the Feynman graphs can be arbitrarily complicated as motives, the part that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, "top-down" approach to the problem, developed in the work of Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization with those formed by mixed Tate motives. The book draws connections between these two approaches and gives an overview of various ongoing directions of research in the field. The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it cal also be used by graduate students interested in working in this area.

Read more less

Book SynopsisThis book is suitable for use in any graduate course on analytical methods and their application to representation theory. Each concept is developed with special emphasis on lucidity and clarity. The book also shows the direct link of Cauchy-Pochhammer theory with the Hadamard-Reisz-Schwartz-Gel'fand et al. regularization. The flaw in earlier works on the Plancheral formula for the universal covering group of SL(2,R) is pointed out and rectified. This topic appears here for the first time in the correct form.Existing treatises are essentially magnum opus of the experts, intended for other experts in the field. This book, on the other hand, is unique insofar as every chapter deals with topics in a way that differs remarkably from traditional treatment. For example, Chapter 3 presents the Cauchy-Pochhammer theory of gamma, beta and zeta function in a form which has not been presented so far in any treatise of classical analysis.

Read more less

Book SynopsisThis book is suitable for use in any graduate course on analytical methods and their application to representation theory. Each concept is developed with special emphasis on lucidity and clarity. The book also shows the direct link of Cauchy-Pochhammer theory with the Hadamard-Reisz-Schwartz-Gel'fand et al. regularization. The flaw in earlier works on the Plancheral formula for the universal covering group of SL(2,R) is pointed out and rectified. This topic appears here for the first time in the correct form.Existing treatises are essentially magnum opus of the experts, intended for other experts in the field. This book, on the other hand, is unique insofar as every chapter deals with topics in a way that differs remarkably from traditional treatment. For example, Chapter 3 presents the Cauchy-Pochhammer theory of gamma, beta and zeta function in a form which has not been presented so far in any treatise of classical analysis.Table of ContentsConvergence, Analytic Functions, Complex Integration, Residue Theorem, Cauchy-Pochhammer Theory of Gamma, Beta and Zeta Function; Bargman-Segal Spaces, Elements of the Theory of Generalized Functions; Regularizations and Cauchy's Theory of Analytic Continuation; Gel'fand-Shilov Formulas for Gamma and Beta Function; Lie Group and Invariant Measure; Representations and Unitary Representation; Wigner-Eckart Theorem; SU(2) Group; Elements of SU(3); Gell-Mann Basis and λ-Matrices; Gell-Mann Neeman Octet Model and Mass Formula; Locally Compact Groups: SL(2,R) (SU(1,1)); Principal Exceptional, Positive and Negative Discreet Series and Their Canonical Carrier Spaces; The Clebsch-Gordan Problem: D+ X D+,c; Dc X Dc,e; Group Ring and Invariant Definition of Character; Plancherel Formula as a Completeness Condition of Character; The Group SL(2,C) and Its Unitary Representations; Group Ring and Character; Plancherel Formula; SU(1,1) Content of SL(2,C); Heisenberg-Weyl Group and Its Representations; Coherent-States and Bergman-Segal Spaces; Bargmann's Integral Transform; SU(1,1) Coherent States and Integral Transforms Connecting Well-Known Carrier Spaces of SU(1,1).

Read more less

Book SynopsisThis is the fifth, expanded edition of the comprehensive textbook published in 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations.In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions.The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena.Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders.Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern-Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.The relevance of path integrals to financial markets is discussed, and improvements of the famous Black-Scholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions.Table of ContentsFundamentals; Path Integrals - Elementary Properties and Simple Solutions; External Sources, Correlations, and Perturbation Theory; Semiclassical Time Evolution Amplitude; Variational Perturbation Theory; Path Integrals with Topological Constraints; Many Particle Orbits - Statistics and Second Quantization; Path Integrals in Polar and Spherical Coordinates; Wave Functions; Spaces with Curvature and Torsion; Schrodinger Equation in General Metric-Affine Spaces; New Path Integral Formula for Singular Potentials; Path Integral of Coulomb System; Solution of Further Path Integrals by Duru-Kleinert Method; Path Integrals in Polymer Physics; Polymers and Particle Orbits in Multiply Connected Spaces; Tunneling; Nonequilibrium Quantum Statistics; Relativistic Particle Orbits; Path Integrals and Financial Markets.

Read more less

Book SynopsisThis is the fifth, expanded edition of the comprehensive textbook published in 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations.In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions.The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena.Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders.Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern-Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.The relevance of path integrals to financial markets is discussed, and improvements of the famous Black-Scholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions.Table of ContentsFundamentals; Path Integrals - Elementary Properties and Simple Solutions; External Sources, Correlations, and Perturbation Theory; Semiclassical Time Evolution Amplitude; Variational Perturbation Theory; Path Integrals with Topological Constraints; Many Particle Orbits - Statistics and Second Quantization; Path Integrals in Polar and Spherical Coordinates; Wave Functions; Spaces with Curvature and Torsion; Schrodinger Equation in General Metric-Affine Spaces; New Path Integral Formula for Singular Potentials; Path Integral of Coulomb System; Solution of Further Path Integrals by Duru-Kleinert Method; Path Integrals in Polymer Physics; Polymers and Particle Orbits in Multiply Connected Spaces; Tunneling; Nonequilibrium Quantum Statistics; Relativistic Particle Orbits; Path Integrals and Financial Markets.

Read more less

Book SynopsisThis book provides a comprehensive collection of problems together with their detailed solutions in the field of Theoretical and Mathematical Physics. All modern fields in Theoretical and Mathematical Physics are covered. It is the only book which covers all the new techniques and methods in theoretical and mathematical physics.Third edition updated with: Exercises in: Hilbert space theory, Lie groups, Matrix-valued differential forms, Bose-Fermi operators and string theory. All other chapters have been updated with new problems and materials. Most chapters contain an introduction to the subject discussed in the text.Table of ContentsVolume I: Complex Numbers and Functions; Sums and Product; Discrete Fourier Transform; Algebraic and Transcendental Equations; Vector and Matrix Calculations; Matrices and Groups; Matrices and Eigenvalue Problems; Functions of Matrices; Transformations; L'Hospital's Rule; Lagrange Multiplier Method; Linear Difference Equations; Linear Differential Equations; Integration; Continuous Fourier Transform; Complex Analysis; Special Functions; Inequalities; Functional Analysis; Combinatorics; Convex Sets and Functions; Optimization; Volume II: Groups; Kronecker and Tensor Products; Nambu Mechanics; Gateaux and Frechet Derivatives; Stability and Bifurcations; Nonlinear Ordinary Difference Equations; Nonlinear Ordinary Differential Equations; Lax Representations in Classical Mechanics; Hilbert Spaces; Generalized Functions; Linear Partial Differential Equations; Nonlinear Partial Differential Equations; Symmetries and Group Theoretical Reductions; Soliton Equations; Backlund Transformations; Lax Pairs for Partial Differential Equations; Hirota Technique; Painleve Test; Lie Algebras; Lie Groups; Differential Forms; Matrix-Valued Differential Forms; Lie Derivative; Metric Tensor Fields; Killing Vector Fields; Inequalities; Ising Model and Heisenberg Model; Number Theory; Combinatorial Problems; Fermi Operators; Bose Operators; Bose-Fermi Systems; Lax Representations and Bethe Ansatz; Gauge Transformations; Chaos, Fractals and Complexity.

Read more less

Book SynopsisThis book provides a comprehensive collection of problems together with their detailed solutions in the field of Theoretical and Mathematical Physics. All modern fields in Theoretical and Mathematical Physics are covered. It is the only book which covers all the new techniques and methods in theoretical and mathematical physics.Third edition updated with: Exercises in: Hilbert space theory, Lie groups, Matrix-valued differential forms, Bose-Fermi operators and string theory. All other chapters have been updated with new problems and materials. Most chapters contain an introduction to the subject discussed in the text.Table of ContentsVolume I: Complex Numbers and Functions; Sums and Product; Discrete Fourier Transform; Algebraic and Transcendental Equations; Vector and Matrix Calculations; Matrices and Groups; Matrices and Eigenvalue Problems; Functions of Matrices; Transformations; L'Hospital's Rule; Lagrange Multiplier Method; Linear Difference Equations; Linear Differential Equations; Integration; Continuous Fourier Transform; Complex Analysis; Special Functions; Inequalities; Functional Analysis; Combinatorics; Convex Sets and Functions; Optimization; Volume II: Groups; Kronecker and Tensor Products; Nambu Mechanics; Gateaux and Frechet Derivatives; Stability and Bifurcations; Nonlinear Ordinary Difference Equations; Nonlinear Ordinary Differential Equations; Lax Representations in Classical Mechanics; Hilbert Spaces; Generalized Functions; Linear Partial Differential Equations; Nonlinear Partial Differential Equations; Symmetries and Group Theoretical Reductions; Soliton Equations; Backlund Transformations; Lax Pairs for Partial Differential Equations; Hirota Technique; Painleve Test; Lie Algebras; Lie Groups; Differential Forms; Matrix-Valued Differential Forms; Lie Derivative; Metric Tensor Fields; Killing Vector Fields; Inequalities; Ising Model and Heisenberg Model; Number Theory; Combinatorial Problems; Fermi Operators; Bose Operators; Bose-Fermi Systems; Lax Representations and Bethe Ansatz; Gauge Transformations; Chaos, Fractals and Complexity.

Read more less

Book SynopsisThis book provides a comprehensive collection of problems together with their detailed solutions in the field of Theoretical and Mathematical Physics. All modern fields in Theoretical and Mathematical Physics are covered. It is the only book which covers all the new techniques and methods in theoretical and mathematical physics.Third edition updated with: Exercises in: Hilbert space theory, Lie groups, Matrix-valued differential forms, Bose-Fermi operators and string theory. All other chapters have been updated with new problems and materials. Most chapters contain an introduction to the subject discussed in the text.

Read more less

Book SynopsisThis book provides a comprehensive collection of problems together with their detailed solutions in the field of Theoretical and Mathematical Physics. All modern fields in Theoretical and Mathematical Physics are covered. It is the only book which covers all the new techniques and methods in theoretical and mathematical physics.Third edition updated with: Exercises in: Hilbert space theory, Lie groups, Matrix-valued differential forms, Bose-Fermi operators and string theory. All other chapters have been updated with new problems and materials. Most chapters contain an introduction to the subject discussed in the text.Table of ContentsVolume I: Complex Numbers and Functions; Sums and Product; Discrete Fourier Transform; Algebraic and Transcendental Equations; Vector and Matrix Calculations; Matrices and Groups; Matrices and Eigenvalue Problems; Functions of Matrices; Transformations; L'Hospital's Rule; Lagrange Multiplier Method; Linear Difference Equations; Linear Differential Equations; Integration; Continuous Fourier Transform; Complex Analysis; Special Functions; Inequalities; Functional Analysis; Combinatorics; Convex Sets and Functions; Optimization; Volume II: Groups; Kronecker and Tensor Products; Nambu Mechanics; Gateaux and Frechet Derivatives; Stability and Bifurcations; Nonlinear Ordinary Difference Equations; Nonlinear Ordinary Differential Equations; Lax Representations in Classical Mechanics; Hilbert Spaces; Generalized Functions; Linear Partial Differential Equations; Nonlinear Partial Differential Equations; Symmetries and Group Theoretical Reductions; Soliton Equations; Backlund Transformations; Lax Pairs for Partial Differential Equations; Hirota Technique; Painleve Test; Lie Algebras; Lie Groups; Differential Forms; Matrix-Valued Differential Forms; Lie Derivative; Metric Tensor Fields; Killing Vector Fields; Inequalities; Ising Model and Heisenberg Model; Number Theory; Combinatorial Problems; Fermi Operators; Bose Operators; Bose-Fermi Systems; Lax Representations and Bethe Ansatz; Gauge Transformations; Chaos, Fractals and Complexity.

Read more less

Book SynopsisThis book attempts to bridge the gap between the principles of pure mathematics and the applications in physical science. After the Möbius inversion formula had been considered as purely academic, or beyond what was useful in the physics community for more than 150 years, the apparently obscure result in classical mathematics suddenly appears to be connected to a variety of important inverse problems in physical science.This book only requires readers to have some background in elementary calculus and general physics, and the prerequisite knowledge of number theory is not needed. It will be attractive to our multidisciplinary readers interested in the Möbius technique, which is a tiny but important part of the number-theoretic methods. It will inspire many students and researchers in both physics and mathematics. In a practical problem, continuity and discreteness are often correlated, and few textbooks have given attention to this wide and important field as this one.Clearly, this book will be an essential supplement for many existing courses such as mathematical physics, elementary number theory and discrete mathematics.Table of ContentsBasics of Mobius Inversion Formulas; Inverse Problems in Boson Systems; Inverse Problems in Fermion Systems; Arithmetic Fourier Transform; Inverse Lattice Problems in Low Dimensions; Inverse Lattice Problems.

Read more less

Book SynopsisThe book surveys mathematical relations between classical and quantum mechanics, gravity, time and thermodynamics from various points of view and many sources (with appropriate attribution). The emergence theme is developed with an emphasis on the meaning via mathematics. A background theme of Bohemian mechanics and connections to the quantum equivalence principle of Matone et al. is also developed in great detail. Some original work relating the quantum potential and Ricci flow is also included.Table of ContentsSome Quantum Background; Some Geometric Aspects; Aspects of Emergence; Kaluza-Klein and Cosmology; Remarks on Thermodynamics and Gravity; Geometry and Mechanics; On Time and the Universe; Gravity and the Quantum Potential.

Read more less

Book SynopsisThis volume contains a selection of contributions by prominent mathematicians from the many interesting presentations delivered at the Conference of Mathematics and Mathematical Physics that was held in Fez, Morocco duing the period of 28-30 October, 2008.Readers will find that this volume merges different approaches in nonlinear analysis, and covers, in a broad and balanced fashion, both the theoretical and numerical aspects of the subject. Graduate students, researchers and professionals with interest in the subject will find it useful while keeping abreast with the latest advancements in this field.Table of ContentsM/v (A Bahri); Nonlinear Elliptic Equations of Infinite Order (A Benkirane et al.); Parabolic Inclusions with Nonlocal Conditions (A Boucherif); Analysis of a New Mixed Formulation of the Obstacle Problem (B Seddoug et al.); Existence of Solutions for Variational Degenerated Unilateral Problems (L Aharouch et al.); A Survey on Potential Theory on Orlicz Spaces (N Aissaoui); Back on Stochastic Model for Sandpile (N Igbida); Combined Fuzzy Sliding Mode Controller for a Class of Nonlinear Systems (Y Alaoui Hafidi et al.); and other papers;

Read more less

Book SynopsisThis book provides an interdisciplinary introduction to the notion of fractal time, starting from scratch with a philosophical and perceptual puzzle. How subjective duration varies, depending on the way we embed current content into contexts, is explained.The complexity of our temporal perspective depends on the number of nestings performed, i.e. on the number of contexts taken into account. This temporal contextualization is described against the background of the notion of fractal time. Our temporal interface, the Now, is portrayed as a fractal structure which arises from the distribution of content and contexts in two dimensions: the length and the depth of time. The leitmotif of the book is the notion of simultaneity, which determines the temporal structure of our interfaces. Recent research results are described which present and discuss a number of distorted temporal perspectives. It is suggested that dynamical diseases arise from unsuccessful nesting attempts, i.e. from failed contextualization. Successful nesting, by contrast, manifests itself in a “win-win handshake” between the observer-participant and his chosen context. The answer as to why a watched kettle never boils has repercussions in many a discipline. It would be of immense interest to anyone who works in the fields of cognitive and complexity sciences, psychology and the neurosciences, social medicine, philosophy and the arts.Table of ContentsWhen Time Slows Down; Subjective Duration; The Fractal Structure of the Now: Time's Length, Depth and Density; Fractal Temporal Perspectives; Corrective Distortions; The View from Within: In-Forming Boundaries; Contextualization: Extended Observer-Participants; Temporal Binding: Synchronizing Perceptions; Nesting Speed: Global vs Local Perspectives; Duration: Distributing Content and Context; Modifying Duration I: Nesting and De-Nesting; Modifying Duration II: Time Condensation; Defining Boundaries: Why is It Always Now?; Outlook: Here There be Dragons.;

Read more less

Book SynopsisThis book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer.Two different approaches to the subject are described. The first, a “bottom-up” approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of Bloch-Esnault-Kreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, “top-down” approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a Riemann-Hilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry.The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area.Table of ContentsPerturbative Quantum Field Theory and Feynman Diagrams; Motives and Periods; Feynman Integrals and Algebraic Varieties; Feynman Integrals, Singularities, Hodge Structures; Connes-Kreimer Theory in a Nutshell; The Riemann-Hilbert Correspondence; Renormalization and Singularities; Beyond Scalar Theories.

Read more less

Book SynopsisThe geometric formulation of autonomous Hamiltonian mechanics in the terms of symplectic and Poisson manifolds is generally accepted. The literature on this subject is extensive. The present book provides the geometric formulation of non-autonomous mechanics in a general setting of time-dependent coordinate and reference frame transformations. This formulation of mechanics as like as that of classical field theory lies in the framework of general theory of dynamic systems, and Lagrangian and Hamiltonian formalisms on fiber bundles. The reader will find a strict mathematical exposition of non-autonomous dynamic systems, Lagrangian and Hamiltonian non-relativistic mechanics, relativistic mechanics, quantum non-autonomous mechanics, together with a number of advanced models — superintegrable systems, non-autonomous constrained systems, theory of Jacobi fields, mechanical systems with time-dependent parameters, non-adiabatic Berry phase theory, instantwise quantization, and quantization relative to different reference frames.Table of ContentsDynamic Equations; Lagrangian Mechanics; Hamiltonian Mechanics; Algebraic Quantization; Geometric Quantization; Constrained Systems; Integrable Hamiltonian Systems; Jacobi Fields; Systems with Parameters: Holonomy Operators; Relativistic Mechanics.

Read more less

Book SynopsisThis book deals with equations of mathematical physics as the different modifications of the KdV equation, the Camassa-Holm type equations, several modifications of Burger's equation, the Hunter-Saxton equation and others. The equations originate from physics but are proposed here for their investigation via purely mathematical methods in the frames of university courses. More precisely, the authors propose classification theorems for the traveling wave solutions for a sufficiently large class of third order nonlinear PDE when the corresponding profiles develop different kind of singularities (cusps, peaks). The orbital stability of the periodic solutions of traveling type for mKdV equations are also studied. Of great interest too is the interaction of peakon type solutions of the Camassa-Holm equation and the solvability of the classical and generalized Cauchy problem for the Hunter-Saxton equation. The Riemann problem for special systems of conservation laws and the corresponding d-shocks are also considered. At the end of the book the authors study the interaction of two piecewise smooth waves in the case of two space variables and they verify the appearance of logarithmic singularities. As it concerns numerical methods in the case of periodic waves the authors apply Cellular Neural Network (CNN) approach.Table of ContentsCompact Traveling Waves, Peakons, Cuspons, Solitons, Kinks and Periodic Solutions of Several Third Order Nonlinear PDE, including Camassa-Holm, Korteweg-De Vries, Burger's Equations and Their Modifications; Cellular Neural Network Realization; Fluxon and Breathon Solutions of the Sin-Gordon Equation and Their Interaction; Stability of Periodic Traveling Wave Solutions for Some Classes of KdV Type Equations; Interaction of Peakon Type Solutions of the Camassa-Holm Equation; Classical and Continuous Weak Solutions of the Cauchy Problem for the Hunter-Saxton Equation, Peakon Type Solutions; Weak Continuous Solutions for the Scalar Conservation Law and Existence Of δ-Shocks; Logarithmic Singularities and Microlocal Approach in Studying the Propagation of Nonlinear Waves.

Read more less

Book SynopsisThe Maxwell, Einstein, Schrödinger and Dirac equations are considered the most important equations in all of physics. This volume aims to provide new eight- and twelve-dimensional complex solutions to these equations for the first time in order to reveal their richness and continued importance for advancing fundamental Physics. If M-Theory is to keep its promise of defining the ultimate structure of matter and spacetime, it is only through the topological configurations of additional dimensionality (or degrees of freedom) that this will be possible. Stretching the exploration of complex space through all of the main equations of Physics should help tighten the noose on “the” fundamental theory. This kind of exploration of higher dimensional spacetime has for the most part been neglected by M-theorists and physicists in general and is taken to its penultimate form here.Table of ContentsOrbiting the Moons of Pluto; Formalism for Complexification of Maxwell's Equations; Relativistic Formulation of Maxwell's Equations in Complex Form; Solutions to the Schrodinger Equation in Complex 8-Space; Sub- and Superluminal Transformations of the Complex Vector Potential; The Dirac Equation: Spinors, Twistors and Quaternions in Complex Space; Complex Unified Geometrodynamics; The Arrow of Time and the Nature of Reality; Synopticon of the Plenum and the Structure of Matter; A New Medical Modality: A Revolution in Biological Science; Immanent FTL Warp Drive Technology from Extended Geometrodynamics; and other chapters.

Read more less

Book SynopsisThis distinctive volume presents a clear, rigorous grounding in modern nonlinear integrable dynamics theory and applications in mathematical physics, and an introduction to timely leading-edge developments in the field — including some innovations by the authors themselves — that have not appeared in any other book.The exposition begins with an introduction to modern integrable dynamical systems theory, treating such topics as Liouville-Arnold and Mischenko-Fomenko integrability. This sets the stage for such topics as new formulations of the gradient-holonomic algorithm for Lax integrability, novel treatments of classical integration by quadratures, Lie-algebraic characterizations of integrability, and recent results on tensor Poisson structures. Of particular note is the development via spectral reduction of a generalized de Rham-Hodge theory, related to Delsarte-Lions operators, leading to new Chern type classes useful for integrability analysis. Also included are elements of quantum mathematics along with applications to Whitham systems, gauge theories, hadronic string models, and a supplement on fundamental differential-geometric concepts making this volume essentially self-contained.This book is ideal as a reference and guide to new directions in research for advanced students and researchers interested in the modern theory and applications of integrable (especially infinite-dimensional) dynamical systems.Table of ContentsThis distinctive volume presents a clear, rigorous grounding in modern nonlinear integrable dynamics theory and applications in mathematical physics, and an introduction to timely leading-edge developments in the field - including some innovations by the authors themselves - that have not appeared in any other book. The exposition begins with an introduction modern integrable dynamical systems theory, treating such topics as Liouville-Arnold and Mischenko-Fomenko integrability. This sets the stage for such topics as new formulations of the gradient-holonomic algorithm for Lax integrability, novel treatments of classical integration by quadratures, Lie-algebraic characterizations of integrability, and recent results on tensor Poisson structures. Of particular note is the development via spectral reduction of a generalized de Rham-Hodge theory, related to Delsarte-Lions operators, leading to new Chern type classes useful for integrability analysis. Also included are elements of quantum mathematics along with applications to Whitham systems, gauge theories, hadronic string models models, and a supplement on fundamental differential-geometric concepts making this volume essentially self-contained. This book is ideal as a reference and guide to new directions in research for advanced students and researchers interested in the modern theory and applications of integrable (especially infinite-dimensional) dynamical systems.

Read more less

Book SynopsisNew Edition: Fractional Calculus: An Introduction for Physicists (3rd Edition)Fractional calculus is undergoing rapid and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics.This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory.Table of ContentsFunctions; The Fractional Derivatives; Friction Forces; Fractional Calculus; The Fraction Harmonic Oscillator; Wave Equations and Parity; Nonlocality and Memory Effects; Quantum Mechanics; Fractional Spin; Factorization; Symmetries; The Fractional Symmetric Rigid Rotor; Fractional Spectroscopy of Hadrons; Higher Dimensional Fractional Rotation Groups; Fractional Fields.

Read more less

Book SynopsisThe aim of the book is to provide a new and fruitful approach to the challenging problems of modern physics, astrophysics, and cosmology. The well-known observations of the flat rotation curves of spiral galaxies and of the gravitational lensing effect greatly exceeding the expectations based on the classical GRT can be explained without bringing in the notion of dark matter. The Tully-Fisher law and the unusual features of globular clusters' motion then become clearer. It also turns out that new features appear in the cosmological picture that involves the Universe expansion and acceleration.The theory and the first observational results of the specific galactic scale experiment based on the optical-metrical parametric resonance are also discussed in the book. Instead of the direct measurements of the extremely small gravitational waves, it appears sufficient just to register their action on the radiation of the space masers for special cases when the source of the gravitational wave is strictly periodic and presents a close binary system. When the amount of data obtained in such observations is large enough, it would be possible to judge upon the geometrical properties of the space-time region enveloping our galaxy, the Milky Way.The foundations of the new approach stem from the equivalence principle which is the basics of the classical GRT. In order to make the presentation self-contained, the roots of century-old ideas are discussed again. This makes the book interesting not only to the specialists in the field but also to graduates and ambitious undergraduate students.Table of ContentsPhysics and Geometry - Brief History of Relations; Causality; Relativity and Covariance; Paradoxical Observations and Notions; Anisotropic Metric and Space; Gravitation Potential and Classical GRT Tests on the Galaxy Scale; Cosmological Consequences; Theory and Observations of the Optical-Metrical Parametric Resonance.

Read more less

Book SynopsisThis unique volume summarizes with a historical perspective several of the major scientific achievements of Ludwig Faddeev, with a foreword by Nobel Laureate C N Yang. The volume that spans over fifty years of Faddeev's career begins where he started his own scientific research, in the subject of scattering theory and the three-body problem. It then continues to describe Faddeev's contributions to automorphic functions, followed by an extensive account of his many fundamental contributions to quantum field theory including his original article on ghosts with Popov. Faddeev's contributions to soliton theory and integrable models are then described, followed by a survey of his work on quantum groups. The final scientific section is devoted to Faddeev's contemporary research including articles on his long-term interest in constructing knotted solitons and understanding confinement. The volume concludes with his personal view on science and mathematical physics in particular.Table of ContentsScattering Theory; Automorphic Functions; Field THeory; Theory of Solitons; Quantum Groups; Knots; General Questions.

Read more less

Book SynopsisMemory is a universal function of organized matter. What is the mathematics of memory? How does memory affect the space-time behaviour of spatially extended systems? Does memory increase complexity? This book provides answers to these questions.It focuses on the study of spatially extended systems, i.e., cellular automata and other related discrete complex systems. Thus, arrays of locally connected finite state machines, or cells, update their states simultaneously, in discrete time, by the same transition rule. The classical dynamics in these systems is Markovian: only the actual configuration is taken into account to generate the next one.Generalizing the conventional view on spatially extended discrete dynamical systems evolution by allowing cells (or nodes) to be featured by some trait state computed as a function of its own previous state-values, the transition maps of the classical systems are kept unaltered, so that the effect of memory can be easily traced.The book demonstrates that discrete dynamical systems with memory are not only priceless tools for modeling natural phenomena but unique mathematical and aesthetic objects.Table of ContentsCellular Automata and Memory; Average Type Memory; Other Memories; Asynchrony and Probabilistic Rules; Cycles and Random Sequences; Three State Automata; Reversible Dynamics; Block Cellular Automata; Structurally Dynamic Systems; Boolean Networks; Coupled Layers; Continuous State Variable; Spatial Games.

Read more less

Book SynopsisQuantum computing and quantum information are two of the fastest growing and most exciting research fields in physics. Entanglement, teleportation and the possibility of using the non-local behavior of quantum mechanics to factor integers in random polynomial time have also added to this new interest. This book supplies a huge collection of problems in quantum computing and quantum information together with their detailed solutions, which will prove to be invaluable to students as well as researchers in these fields. All the important concepts and topics such as quantum gates and quantum circuits, product Hilbert spaces, entanglement and entanglement measures, deportation, Bell states, Bell inequality, Schmidt decomposition, quantum Fourier transform, magic gate, von Neumann entropy, quantum cryptography, quantum error corrections, number states and Bose operators, coherent states, squeezed states, Gaussian states, POVM measurement, quantum optics networks, beam splitter, phase shifter and Kerr Hamilton operator are included. The topics range in difficulty from elementary to advanced. Almost all problems are solved in detail and most of the problems are self-contained.Table of ContentsFinite-Dimensional Hilbert Spaces: Qubits; Kronecker Product and Tensor Product; Matrix Properties; Density Operators; Partial Trace; Unitary Transforms and Quantum Gates; Entropy; Measurement; Entanglement; Bell Inequality; Teleportation; Cloning; Quantum Algorithms; Quantum Error Correction; Quantum Cryptography; Infinite-Dimensional Hilbert Spaces: Qubits; Harmonic Oscillator and Bose Operators; Coherent States; Squeezed States; Entanglement; Teleportation; Swapping and Cloning; Hamilton Operators.

Read more less

Book SynopsisThis book tries to point out the mathematical importance of the Partial Differential Equations of First Order (PDEFO) in Physics and Applied Sciences. The intention is to provide mathematicians with a wide view of the applications of this branch in physics, and to give physicists and applied scientists a powerful tool for solving some problems appearing in Classical Mechanics, Quantum Mechanics, Optics, and General Relativity. This book is intended for senior or first year graduate students in mathematics, physics, or engineering curricula.This book is unique in the sense that it covers the applications of PDEFO in several branches of applied mathematics, and fills the theoretical gap between the formal mathematical presentation of the theory and the pure applied tool to physical problems that are contained in other books.Improvements made in this second edition include corrected typographical errors; rewritten text to improve the flow and enrich the material; added exercises in all chapters; new applications in Chapters 1, 2, and 5 and expanded examples.Table of ContentsGeometric Concepts and Generalities; Partial Differential Equations of First Order; Physical Applications I; Nonlinear Partial Differential Equations of First Order; Physical Applications II; Characteristic Surfaces of Linear Partial Differential Equation of Second Order.

Read more less

Book SynopsisAlthough contact geometry and topology is briefly discussed in V I Arnol'd's book “Mathematical Methods of Classical Mechanics ”(Springer-Verlag, 1989, 2nd edition), it still remains a domain of research in pure mathematics, e.g. see the recent monograph by H Geiges “An Introduction to Contact Topology” (Cambridge U Press, 2008). Some attempts to use contact geometry in physics were made in the monograph “Contact Geometry and Nonlinear Differential Equations” (Cambridge U Press, 2007). Unfortunately, even the excellent style of this monograph is not sufficient to attract the attention of the physics community to this type of problems. This book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be rewritten in the language of contact geometry and topology: from mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity; from physics of liquid crystals to quantum mechanics and quantum computers, etc. The book is written in the style of famous Landau-Lifshitz (L-L) multivolume course in theoretical physics. This means that its readers are expected to have solid background in theoretical physics (at least at the level of the L-L course). No prior knowledge of specialized mathematics is required. All needed new mathematics is given in the context of discussed physical problems. As in the L-L course some problems/exercises are formulated along the way and, again as in the L-L course, these are always supplemented by either solutions or by hints (with exact references). Unlike the L-L course, though, some definitions, theorems, and remarks are also presented. This is done with the purpose of stimulating the interest of our readers in deeper study of subject matters discussed in the text.Table of ContentsMotivation and Background; From Ideal Magnetohydrodynamics to String and Knot Theory; All About and Around Woltjer's Theorem; Topologically Massive Gauge Theories and Force-Free Fields; Contact Geometry and Physics (Fundamentals); Sub-Riemannian Geometry, Heisenberg Manifolds and Quantum Mechanics of Landau Levels; Abrikosov Lattices, TGB Phases in Liquid Crystals and Heisenberg Group; Sub-Riemannian Geometry, Spin Dynamics and Quantum Classical Optimal Control; Legendrian and Transverse Knots and Links and Their Physical Uses; Appendices: A) Heisenberg Group and Its Relatives in the Context of Sub-Riemannian Geometry and Optimal Control; B) Sub-Riemannian Dynamics of Josephson Junctions; C) Quantum Computers and Quantum Random Walks (Almost a Glossary).

Read more less

Book SynopsisThis volume showcases selected recent work presented at the 13th Regional Conference on Mathematical Physics held in Antalya, Turkey in 2010. The conference was dedicated to the memory of Faheem Hussain, one of the initiators of the Regional Conference series, and one of the organizers of the 12th Regional Conference. The “region”, originally comprised of Turkey, Iran and Pakistan, extends now to Bangladesh and Central Asia. However, the contributing researchers are not only limited to this region.Prominent contributors include B Ahmedov (Tashkent), F Ardalan (Tehran), N Dadhich (Pune), D A Demir (İzmir), R L Hall (Montreal), M Hortaçsu (İstanbul), M Koca (Oman), C S Lim (Kobe), F Mahomed (Johannesburg), A Qadir (Rawalpindi), M A Rashid (Rawalpindi), M Sakamoto (Kobe), M Sharif (Lahore), F Toppan (Rio), N Ünal (Antalya), amongst others. They sample a number of topics in the formal aspects of mathematical physics, general relativity and cosmology, quantum gravity, quantum field theory, and even applied physics.Table of ContentsFormal Aspects; General Relativity and Cosmology; Quantum Gravity; Quantum Field Theory; Applied Physics.

Read more less

Book SynopsisBased on lectures held at the 7th Villa de Leyva summer school, this book presents an introduction to topics of current interest in the interface of geometry, topology and physics. It is aimed at graduate students in physics or mathematics with interests in geometric, algebraic as well as topological methods and their applications to quantum field theory.This volume contains the written notes corresponding to lectures given by experts in the field. They cover current topics of research in a way that is suitable for graduate students of mathematics or physics interested in the recent developments and interactions between geometry, topology and physics. The book also contains contributions by younger participants, displaying the ample range of topics treated in the school. A key feature of the present volume is the provision of a pedagogical presentation of rather advanced topics, in a way which is suitable for both mathematicians and physicists.Table of ContentsGeneralized Euler Characteristics, Graph Hypersurfaces and Feynman Periods; Noncommutative Spacetimes and Quantum Physics; Index Theory of Non-Compact G-Manifolds; Compactifications of String Theory and Generalized Geometry; Spectral Geometry; Noncommutative Geometry Models for Particle Physics; Integrability and the AdS/CFT Correspondence; Introductory Lectures on Chern - Simons Theories in Physics; Grothendieck Ring Class of Banana and Flower Graphs; Groupoids and Poisson Sigma Models with Boundary; On the Geometry Underlying a Real Lie Algebra Representation; A Survey on Orbifold String Topology.

Read more less

Book SynopsisIn this second edition, a comprehensive review is given for path integration in two- and three-dimensional (homogeneous) spaces of constant and non-constant curvature, including an enumeration of all the corresponding coordinate systems which allow separation of variables in the Hamiltonian and in the path integral. The corresponding path integral solutions are presented as a tabulation. Proposals concerning interbasis expansions for spheroidal coordinate systems are also given. In particular, the cases of non-constant curvature Darboux spaces are new in this edition.The volume also contains results on the numerical study of the properties of several integrable billiard systems in compact domains (i.e. rectangles, parallelepipeds, circles and spheres) in two- and three-dimensional flat and hyperbolic spaces. In particular, the discussions of integrable billiards in circles and spheres (flat and hyperbolic spaces) and in three dimensions are new in comparison to the first edition.In addition, an overview is presented on some recent achievements in the theory of the Selberg trace formula on Riemann surfaces, its super generalization, their use in mathematical physics and string theory, and some further results derived from the Selberg (super-) trace formula.Table of ContentsIntroduction; Path Integrals in Quantum Mechanics; Separable Coordinate Systems; Path Integrals in Pseudo-Euclidean Geometry; Path Integrals in Euclidean Spaces; Path Integrals on Spheres; Path Integrals on Hyperboloids; Path Integrals on the Complex Sphere; Path Integrals on Hermitian Hyperbolic Space; Path Integrals on Darboux Spaces; Path Integrals on Single-Sheeted Hyperboloids; Path Integration in Homogeneous Spaces; Billiard Systems and Periodic Orbit Theory; The Selberg Trace Formula; The Selberg Super-Trace Formula; Summary and Discussion.

Read more less

Book SynopsisThis volume is based on lectures given during the program “Complex Quantum Systems” held at the National University of Singapore's Institute for Mathematical Sciences from 17 February to 27 March 2010. It guides the reader through two introductory expositions on large Coulomb systems to five of the most important developments in the field: derivation of mean field equations, derivation of effective Hamiltonians, alternative high precision methods in quantum chemistry, modern many-body methods originating from quantum information, and — the most complex — semirelativistic quantum electrodynamics.These introductions are written by leaders in their fields; amongst them are Volker Bach, Rafael Benguria, Thomas Chen, and Jan Philip Solovej. Together, they fill a gap between current textbooks and the vast modern literature on complex quantum systems.Table of ContentsStability of Matter (Rafael Benguria and Benjamin Loewe); Mathematical Density and Density Matrix Functional Theory (Volker Bach); On the Dynamics of a Fermi Gas in a Random Medium with Dynamical Hartree - Fock Interactions (Thomas Chen); On the Minimization of Hamiltonians over Pure Gaussian States (Jan Derezinski, Marcin Napiorkowski and Jan Philip Solovej); Variational Approach to Electronic Structure Calculations on Second-Order Reduced Density Matrices and the N-Representability Problem (Maho Nakata, Mituhiro Fukuda, and Katsuki Fujisawa); Fermionic Many-Body Systems: A Quantum Information Approach (Christina Kraus); Hydrogen-Like Atoms in Relativistic QED (Martin Konenberg, Oliver Matte and Edgardo Stockmeyer).

Read more less

Book SynopsisThis volume deals with the fundamental concepts of space, time and matter. It presents a novel reformulation of both the special and general theory of relativity, in which time does not constitute the fourth dimension in a conventional 4-dimensional space-time. Instead, the role of time is played by the flow of a vector field on a 3-dimensional space. The standard models of de Sitter, Schwarzschild and Kerr space-times are reformulated in a purely 3-dimensional manifold.The volume also presents a theory of matter in which the fundamental particles, such as baryons and leptons, appear as a result of an interaction between left-handed and right-handed 2-component Weyl neutrinos. The Appendices contain a comprehensive treatment of classical mechanics in terms of Hamiltonian vector fields on symplectic manifolds. Graduate students of mathematical physics or theoretical physics, as well as academics, will find this volume of interest.Table of ContentsRelativistic Kinematics and Dynamics on 3-Manifolds; Gauss - Einstein Equations on 3-Manifolds; The de Sitter, Schwarzschild and Kerr Space-Times; A New Solution of the Vacuum Einstein Field Equations; Weyl Neutrinos and the Photon; A Neutrino Theory of Matter; Dynamical Vector Fields of Classical Mechanics.

Read more less