The book explains concepts and ideas of mathematics and physics that are relevant for advanced students and researchers of condensed matter physics.

Author: Siddhartha Sen

Publisher: World Scientific

ISBN: 9789814678186

Category: Science

Page: 220

View: 842

The book explains concepts and ideas of mathematics and physics that are relevant for advanced students and researchers of condensed matter physics. With this aim, a brief intuitive introduction to many-body theory is given as a powerful qualitative tool for understanding complex systems. The important emergent concept of a quasiparticle is then introduced as a way to reduce a many-body problem to a single particle quantum problem. Examples of quasiparticles in graphene, superconductors, superfluids and in a topological insulator on a superconductor are discussed. The mathematical idea of self-adjoint extension, which allows short distance information to be included in an effective long distance theory through boundary conditions, is introduced through simple examples and then applied extensively to analyse and predict new physical consequences for graphene. The mathematical discipline of topology is introduced in an intuitive way and is then combined with the methods of differential geometry to show how the emergence of gapless states can be understood. Practical ways of carrying out topological calculations are described. Contents:OverviewMany-Body TheoryTopology and GeometryBoundary Conditions and Self-Adjoint ExtensionsElectronic Properties of Graphene Readership: Graduate students and researchers in condensed matter physics and mathematical physics. Key Features:Topics are of current interest, e.g. graphene, topological insulators, Majorana fermionsIs self-contained and provides all the background material necessary to understand the physical or mathematical concepts discussedPractical ways of using topology, self-adjoint extensions as well as ways of making qualitative estimates in physics are explained and then illustrated by examplesKeywords:Condensed Matter Physics;Topology;Differential Geometry;Many-Body Problem;Graphene;Self-Adjoint Extensions;K-Theory;Quasiparticles;Superconductivity;Superfluidity;Topological Insulator;Mathematical Physics

... mechanical problems (controlled semiclassical limits, analogies to classical
mechanics, statistical mechanics, concepts of topology and geometry, etc.).
Similarly, the introduction of functional field integration into many-body physics
spawned ...

Author: Alexander Altland

Publisher: Cambridge University Press

ISBN: 9780521845083

Category: Science

Page: 624

View: 196

Primer, including problems and solutions, for graduate level courses on theoretical quantum condensed matter physics.

He gave an assignment to his assistants, Pauli, Jordan and Heisenberg, to make
improvements in Bohr's theory in accordance ... “Algebraic mean field theory (
AMFT) is a many-body physics modeling tool which firstly, is a generalization of ...

Author: Arkady L Kholodenko

Publisher: World Scientific

ISBN: 9789814412100

Category: Mathematics

Page: 492

View: 362

Although 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. Contents:Motivation and BackgroundFrom Ideal Magnetohydrodynamics to String and Knot TheoryAll About and Around Woltjer's TheoremTopologically Massive Gauge Theories and Force-Free FieldsContact Geometry and PhysicsSub-Riemannian Geometry, Heisenberg Manifolds and Quantum Mechanics of Landau LevelsAbrikosov Lattices, TGB Phases in Liquid Crystals and Heisenberg GroupSub-Riemannian Geometry, Spin Dynamics and Quantum-Classical Optimal ControlFrom Contact Geometry to Contact TopologyClosing Remarks:The Unreasonable Effectivenessof Contact Geometry and Topology in Physical SciencesAppendices:Heisenberg Group in the Context of Sub-Riemannian Geometry and Optimal ControlSub-Riemannian Dynamics of Josephson JunctionsQuantum Computers and Quantum Random WalksThe Measurement Protocol. Geometry and Topology of Entanglements Readership: Students in applied mathematics and theoretical physics. Keywords:Force-Free Fields;Contact and Sub-Riemannian Geometry;Optimal Control;Theoretical PhysicsKey Features:This book is the world's first book on contact/sub-Riemannian geometry and topology for physicistsUnlike books discussing mathematical methods for physicists, this book discusses physical problems first and only then uses new mathematics to solve these problems. Problems are selected from practically all branches of theoretical physicsThis is done with the purpose of demonstrating that contact geometry should be looked upon as a universal language/technical tool of theoretical physicsReviews: “This book is written in the style of the well-known Landau-Lifshitz multivolume course in theoretical physics and its prime goal, as the author puts it, is to show the diversity of applications of contact geometry and topology. I enjoyed reading this book, in which the author allows readers to see for themselves “the same forest behind different kinds of trees”. I strongly recommend this book to interested readers.” MathSciNet

A robust characterization of topological phases is provided by the ground-state
degeneracy on a periodic geometry (which has the topology of a torus). For the
Moore–Read state, the expected degeneracy appears clearly in the spectrum
even ...

Author: Christophe Salomon

Publisher: Oxford University Press

ISBN: 9780199661886

Category: Science

Page: 345

View: 256

This book provides authoritative tutorials on the most recent achievements in the field of quantum gases at the interface between atomic physics and quantum optics, condensed matter physics, nuclear and high-energy physics, non-linear physics, and quantum information.

... representations of vectors and efficient contraction schemes are needed. Here
approaches from many-body quantum physics for onedimensional and two-
dimensional systems (Matrix Product States and Projected Entangled Pair ...

Author:

Publisher: ScholarlyEditions

ISBN: 9781490111063

Category: Mathematics

Page: 698

View: 428

Issues in Algebra, Geometry, and Topology / 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about Topology. The editors have built Issues in Algebra, Geometry, and Topology: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Topology in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Algebra, Geometry, and Topology: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.

Meanwhile there is the Course in Mathematical Physics by W. Thirring, a large body of monographs and textbooks for ... minimalistic deductive style of a
sequence of theorems and proofs without much of commentary or even
motivating text.

Author: Helmut Eschrig

Publisher: Springer Science & Business Media

ISBN: 9783642146992

Category: Science

Page: 390

View: 815

A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.

On the other hand, the maximum in the spectrum of the black-body radiation
increases with temperature and reaches according to Planck's law at T = 220
MeV a value of 620 MeV. A priori one would not expect a dissociation of the
glueballs at ...

Author: Eike Bick

Publisher: Springer Science & Business Media

ISBN: 3540231250

Category: Mathematics

Page: 358

View: 892

Application of the concepts and methods of topology and geometry have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity and particle physics. This book can be considered an advanced textbook on modern applications and recent developments in these fields of physical research. Written as a set of largely self-contained extensive lectures, the book gives an introduction to topological concepts in gauge theories, BRST quantization, chiral anomalies, supersymmetric solitons and noncommutative geometry. It will be of benefit to postgraduate students, educating newcomers to the field and lecturers looking for advanced material.

Field Theory and Many-body Theory Joel S. Feldman, Richard Gerd Froese, Lon
M. Rosen. CRM Proceedings and Lecture Notes Volume 7, 1994 Conformal Field
Theory and Geometry of Strings Jiirg Frohlich and Krzysztof Gaw^dzki
ABSTRACT. ... The perturbative expansion around the classical solutions is built
by considering the CFT models on two-dimensional space-times of non-trivial topology.

Author: Joel S. Feldman

Publisher: American Mathematical Soc.

ISBN: 0821870483

Category: Science

Page: 234

View: 674

This book is the first volume of the proceedings of the Canadian Mathematical Society Annual Seminar on Mathematical Quantum Theory, held in Vancouver in August 1993. The seminar was run as a research-level summer school concentrating on two related areas of contemporary mathematical physics. The subject of the first session, quantum field theory and many-body theory, is covered in the present volume; papers from the second session, on Schrodinger operators, are in volume 2. Each session featured a series of minicourses, consisting of approximately four one-hour lectures, designed to introduce students to current research in a particular area. In addition, about thirty speakers gave one-hour expository lectures. With contributions by some of the top experts in the field, this book provides an overview of the state of the art in mathematical quantum field and many-body theory.

Author: Stuart G. WhittingtonPublish On: 1998-08-13

The behavior of physical knots depends on the topological types of the knots; that
is an experimental observation. Mathematical differences between knot types
and, more generally, the whole body of knot theory should help explain the ...

Author: Stuart G. Whittington

Publisher: Springer Science & Business Media

ISBN: 0387985808

Category: Mathematics

Page: 201

View: 308

This book contains contributions from a workshop on topology and geometry of polymers, held at the IMA in June 1996, which brought together topologists, combinatorialists, theoretical physicists and polymer scientists, with a common interest in polymer topology. Polymers can be highly self-entangled even in dilute solution. In the melt the inter- and intra-chain entanglements can dominate the rheological properties of these phenomena. Although the possibility of knotting in ring polymers has been recognized for more than thirty years it is only recently that the powerful methods of algebraic topology have been used in treating models of polymers. This book contains a series of chapters which review the current state of the field and give an up to date account of what is known and perhaps more importantly, what is still unknown. The field abounds with open problems. The book is of interest to workers in polymer statistical mechanics but will also be useful as an introduction to topological methods for polymer scientists, and will introduce mathematicians to an area of science where topological approaches are making a substantial contribution.

Leaving aside developments triggered by problems in classical physics (
classical dynamical systems , “ chaos ” , fluid ... many degrees of freedom ; “
quantum symmetries ; " non - relativistic quantum theory and many - body theory .
In these and other areas one has been led to deep questions and new results in
algebra , algebraic topology , algebraic geometry and analysis , and this is likely
to continue .

Author: Anthony Joseph

Publisher: Nelson Thornes

ISBN: 3764327995

Category: Mathematics

Page: 513

View: 672

Table of Contents: D. Duffie: Martingales, Arbitrage, and Portfolio Choice • J. Fröhlich: Mathematical Aspects of the Quantum Hall Effect • M. Giaquinta: Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings • U. Hamenstädt: Harmonic Measures for Leafwise Elliptic Operators Along Foliations • M. Kontsevich: Feynman Diagrams and Low-Dimensional Topology • S.B. Kuksin: KAM-Theory for Partial Differential Equations • M. Laczkovich: Paradoxical Decompositions: A Survey of Recent Results • J.-F. Le Gall: A Path-Valued Markov Process and its Connections with Partial Differential Equations • I. Madsen: The Cyclotomic Trace in Algebraic K-Theory • A.S. Merkurjev: Algebraic K-Theory and Galois Cohomology • J. Nekovár: Values of L-Functions and p-Adic Cohomology • Y.A. Neretin: Mantles, Trains and Representations of Infinite Dimensional Groups • M.A. Nowak: The Evolutionary Dynamics of HIV Infections • R. Piene: On the Enumeration of Algebraic Curves - from Circles to Instantons • A. Quarteroni: Mathematical Aspects of Domain Decomposition Methods • A. Schrijver: Paths in Graphs and Curves on Surfaces • B. Silverman: Function Estimation and Functional Data Analysis • V. Strassen: Algebra and Complexity • P. Tukia: Generalizations of Fuchsian and Kleinian Groups • C. Viterbo: Properties of Embedded Lagrange Manifolds • D. Voiculescu: Alternative Entropies in Operator Algebras • M. Wodzicki : Algebraic K-Theory and Functional Analysis • D. Zagier: Values of Zeta Functions and Their Applications

... FIELD THEORY MODELS C. Conformal Field Theory, Statistical Mechanics
Models and Quantum Field Theory GEOMETRY OF ... NON-RELATIVISTIC MANY-BODY THEORY J. Frohlich and U. M. Studer 2201 GEOMETRY OF
QUANTUM ...

PHYSICS. A series reporting advancesin theoretical molecular and material
sciences, including theoretical, ... involves the use of algebra and topology in the
analysis of molecular structures and reactions); molecular mechanics, molecular
... in rationalizing the geometric and electronic structures of molecular assemblies
and polymers, v clusters and crystals; surface, interface, solvent and solid-state
effects; excited-state Brillouin–Wigner Methods for Many-Body Systems: Aim and
Scope.

Author: Stephen Wilson

Publisher: Springer Science & Business Media

ISBN: 9789048133734

Category: Science

Page: 235

View: 124

Brillouin-Wigner Methods for Many-Body Systems gives an introduction to many-body methods in electronic structure theory for the graduate student and post-doctoral researcher. It provides researchers in many-body physics and theoretical chemistry with an account of Brillouin-Wigner methodology as it has been developed in recent years to handle the multireference correlation problem. Moreover, the frontiers of this research field are defined. This volume is of interest to atomic and molecular physicists, physical chemists and chemical physicists, quantum chemists and condensed matter theorists, computational chemists and applied mathematicians.

845 : A . Tannenbaum , Invariance and System Theory : Algebraic and Geometric
Aspects . X , 161 pages . 1981 . ... 849 : P . Major , Multiple Wiener - Ito Integrals .
VII , 127 pages . ... 870 : Shape Theory and Geometric Topology . Proceedings ...

Their relation to algebraic geometry , DAN SSSR , 219 : 3 , 1974 , 19 . 36. S. Yu .
... Airault , H. McKean , J. Moser , Rational and elliptic solutions of the KDV
equation and related many - body problems , Comm . Pure and Appl . Math . , 30
...

... D.Phil . , Algebraic topology F.R.S. ( Savilian Professor of Geometry ) Professor
J. F. C. Kingman , Probability theory ... Ph.D. Field theory and Green's function
methods in particle physics and nonrelativistic many - body theory M. J. Collins ...

This book developed out of a review we wrote for the journal Reports of Progress
in Physics (Taylor et al 2001). ... This is a common feature in the methods (some
would say pseudomethods) described later and is true also of many of the
descriptions of protein topology. Rather ... Even with this restriction, there is a
large body of work on the mechanical properties of proteins that we have not
ventured into.

Author: William R. Taylor

Publisher: CRC Press

ISBN: 9781420033632

Category: Science

Page: 348

View: 673

Using a geometric perspective, Protein Geometry, Classification, Topology, and Symmetry reviews and analyzes the structural principals of proteins with the goal of revealing the underlying regularities in their construction. It also reviews computer methods for structure analysis and the automatic comparison and classification of these structures with an analysis of the statistical significance of comparing different shapes. Following an analysis of the current state of protein classification, the authors explore more abstract geometric and topological representations, including the occurrence of knotted topologies. The book concludes with a consideration of the origin of higher-level symmetries in protein structure. The authors focus on simple geometric methods that are deterministic rather than probabilistic and on the more abstract simplifications of protein structure that allow a better understanding of the overall fold of the structure. Most of the methods described in this book have corresponding computer programs. These can be found (as C source code) at the ftp site of the Division of Mathematical Biology at the National Institute for Medical Research. This collection of ideas contains pedagogical material that make it ideal for post-graduate courses as well as new ideas and results essential for researchers investigating protein structures.

In these notes , only the topological aspects and few of its consequences in physics are investigated . The systematic study of the transverse Geometry as
well as the N - body problem ( 105 ) are not treated here and are left for future ...

Author: Alexander Cardona

Publisher: World Scientific

ISBN: 9812705066

Category: Algebraic topology

Page: 482

View: 248

This volume offers an introduction to recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications to index theory on one hand and the study of aperiodic solids on the other, geometry and topology of low dimensional manifolds with applications to topological field theory, Chern-Simons supergravity and the anti de Sitter/conformal field theory correspondence. It comprises seven lectures organized around three main topics, noncommutative geometry, topological field theory, followed by supergravity and string theory, complemented by some short communications by young participants of the school.