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• Introduction--
• 1. Two lattices--
• 2. Geometric transformations of Grassmannians--
• 3. Lattices of closed subspaces--
• 4. Wigner's theorem and its generalizations--
• 5. Compatibility relation--
• 6. Applications-- References-- Index.
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Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.
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• Introduction Basic definitions and simple examples Topological structures Topological spinor bundles Further examples Tangent cone measures and the tangential Clifford section The topology of discrete and singular fermion systems Basic examples Spinors on singular spaces Bibliography.
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Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential topology is worked out. The constructions are illustrated by many simple examples such as the Euclidean plane, the two-dimensional Minkowski space, a conical singularity, a lattice system as well as the curvature singularity of the Schwarzschild space-time. As further examples, it is shown how complex and Kahler structures can be encoded in Riemannian fermion systems.
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• Part I. Theory:
• 1. The wave function--
• 2. Time-independent Schrodinger equation--
• 3. Formalism--
• 4. Quantum mechanics in three dimensions--
• 5. Identical particles--
• 6. Symmetry-- Part II. Application:
• 7. Time-independent perturbation theory--
• 8. The variational principle--
• 9. The WKB approximation--
• 10. Scattering--
• 11. Quantum dynamics--
• 12. Afterword-- Appendix A. Linear algebra-- Index.
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Changes and additions to the new edition of this classic textbook include: new chapter on Symmetries, new problems and examples, improved explanations, more numerical problems to be worked on a computer, new applications to solid state physics, consolidated treatment of time-dependent potentials.
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• Preface / Amir-Kian Kashani-Poor, Ruben Minasian, Nikita Nekrasov, Boris Pioline
• Three-dimensional N = 4 gauge theories in omega background / Mathew Bullimore
• 3d supersymmetric gauge theories and Hilbert series / Stefano Cremonesi
• Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras / Ryosuke Kodera and Hiraku Nakajima
• Supersymmetric field theories and geometric Langlands : the other side of the coin / Aswin Balasubramanian and Jörg Teschner
• A journey from the Hitchin section to the oper moduli / Olivia Dumitrescu
• S-duality of boundary conditions and the Geometric Langlands program / Davide Gaiotto
• Pure SU(2) gauge theory partition function and generalized Bessel kernel / P. Gavrylenko and O. Lisovyy
• Reduction for SL(3) pre-buildings / Ludmil Katzarkov, Pranav Pandit, and Carlos Simpson
• Conformal nets are factorization algebras / André Henriques
• Contracting the Weierstrass locus to a point / Alexander Polishchuk
• Spectral theory and mirror symmery / Marcos Mariño.

This volume contains the proceedings of the conference String-Math 2016, which was held from June 27-July 2, 2016, at College de France, Paris, France. String-Math is an annual conference covering the most significant progress at the interface of string theory and mathematics. The two fields have had a very fruitful dialogue over the last thirty years, with string theory contributing key ideas which have opened entirely new areas of mathematics and modern mathematics providing powerful concepts and tools to deal with the intricacies of string and quantum field theory. The papers in this volume cover topics ranging from supersymmetric quantum field theories, topological strings, and conformal nets to moduli spaces of curves, representations, instantons, and harmonic maps, with applications to spectral theory and to the geometric Langlands program.
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• Alice and Bob: Mathematical Aspects of Quantum Information: Notation and basic conceptsElementary convex analysisThe mathematics of quantum information theoryQuantum mechanics for mathematiciansBanach and His spaces: Asymptotic Geometric Analysis Miscellany: More convexityMetric entropy and concentration of measure in classical spacesGaussian processes and random matricesSome tools from asymptotic geometric analysisThe Meeting: AGA and QIT: Entanglement of pure states in high dimensionsGeometry of the set of mixed statesRandom quantum statesBell inequalities and the Grothendieck-Tsirelson inequalityPOVMs and the distillability problemGaussian measures and Gaussian variablesClassical groups and manifoldsExtreme maps between Lorentz cones and the $S$-lemmaPolarity and the Santalo point via duality of conesHints to exercisesBibliographyNotationIndex.
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The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, especially the part that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.
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• * I. Losev, Rational Cherednik algebras and categorification* O. Dudas, M. Varagnolo, and E. Vasserot, Categorical actions on unipotent representations of finite classical groups* J. Brundan and N. Davidson, Categorical actions and crystals* A. M. Licata, On the 2-linearity of the free group* M. Ehrig, C. Stroppel, and D. Tubbenhauer, The Blanchet-Khovanov algebras* G. Lusztig, Generic character sheaves on groups over $k[\epsilon]/(\epsilon^r)$* D. Berdeja Suarez, Integral presentations of quantum lattice Heisenberg algebras* Y. Qi and J. Sussan, Categorification at prime roots of unity and hopfological finiteness* B. Elias, Folding with Soergel bimodules* L. T. Jensen and G. Williamson, The $p$-canonical basis for Hecke algebras.
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The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory. This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory. The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.
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• * B. Webster, Geometry and categorification* Y. Li, A geometric realization of modified quantum algebras* T. Lawson, R. Lipshitz, and S. Sarkar, The cube and the Burnside category* S. Chun, S. Gukov, and D. Roggenkamp, Junctions of surface operators and categorification of quantum groups* R. Rouquier, Khovanov-Rozansky homology and 2-braid groups* I. Cherednik and I. Danilenko, DAHA approach to iterated torus links.
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The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorification is a powerful tool for relating various branches of mathematics and exploiting the commonalities between fields. It provides a language emphasizing essential features and allowing precise relationships between vastly different fields. This volume focuses on the role categorification plays in geometry, topology, and physics. These articles illustrate many important trends for the field including geometric representation theory, homotopical methods in link homology, interactions between higher representation theory and gauge theory, and double affine Hecke algebra approaches to link homology. The companion volume (Contemporary Mathematics, Volume 683) is devoted to categorification and higher representation theory.
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• * Introduction* Physical motivations* Basic mathematical properties of superoscillating sequences* Function spaces of holomorphic functions with growth* Schrodinger equation and superoscillations* Superoscillating functions and convolution equations* Superoscillating functions and operators* Superoscillations in $SO(3)$* Bibliography* Index.
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In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. The purpose of this work is twofold: on one hand the authors provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, they obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of analytically uniform spaces. In particular, the authors will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrodinger equation and other equations.
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• * Three motivating problems* Back to the beginning* Some elementary number theory* A second view of modular arithmetic: $\mathbb{Z}_n$ and $U_n$* Public-key cryptography and RSA* A little more algebra* Curves in affine and projective space* Applications of elliptic curves* Deeper results and concluding thoughts* Answers to selected exercises* Bibliography* Index.
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This book offers the beginning undergraduate student some of the vista of modern mathematics by developing and presenting the tools needed to gain an understanding of the arithmetic of elliptic curves over finite fields and their applications to modern cryptography. This gradual introduction also makes a significant effort to teach students how to produce or discover a proof by presenting mathematics as an exploration, and at the same time, it provides the necessary mathematical underpinnings to investigate the practical and implementation side of elliptic curve cryptography (ECC). Elements of abstract algebra, number theory, and affine and projective geometry are introduced and developed, and their interplay is exploited. Algebra and geometry combine to characterize congruent numbers via rational points on the unit circle, and group law for the set of points on an elliptic curve arises from geometric intuition provided by Bezout's theorem as well as the construction of projective space. The structure of the unit group of the integers modulo a prime explains RSA encryption, Pollard's method of factorization, Diffie-Hellman key exchange, and ElGamal encryption, while the group of points of an elliptic curve over a finite field motivates Lenstra's elliptic curve factorization method and ECC. The only real prerequisite for this book is a course on one-variable calculus; other necessary mathematical topics are introduced on-the-fly. Numerous exercises further guide the exploration.
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• About the Authors ix Preface xi
• 1 General Properties of the Schrodinger Equation 1
• 2 Operators 15
• 3 Bound States 47
• 4 Heisenberg Principle 80
• 5 Current and Energy Flux Densities 101
• 6 Density of States 128
• 7 Transfer Matrix 166
• 8 Scattering Matrix 205
• 9 Perturbation Theory 228
• 10 Variational Approach 245
• 11 Electron in a Magnetic Field 261
• 12 Electron in an Electromagnetic Field and Optical Properties of Nanostructures 281
• 13 Time-Dependent Schrodinger Equation 292 A Postulates of Quantum Mechanics 314 B Useful Relations for the One-Dimensional Harmonic Oscillator 317 C Properties of Operators 319 D The Pauli Matrices and their Properties 322 E Threshold Voltage in a High Electron Mobility Transistor Device 325 F Peierls s Transformation 329 G Matlab Code 332 Index 343.
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This topical and timely textbook is a collection of problems for students, researchers, and practitioners interested in state-of-the-art material and device applications in quantum mechanics. Most problem are relevant either to a new device or a device concept or to current research topics which could spawn new technology. It deals with the practical aspects of the field, presenting a broad range of essential topics currently at the leading edge of technological innovation. Includes discussion on: Properties of Schroedinger Equation Operators Bound States in Nanostructures Current and Energy Flux Densities in Nanostructures Density of States Transfer and Scattering Matrix Formalisms for Modelling Diffusive Quantum Transport Perturbation Theory, Variational Approach and their Applications to Device Problems Electrons in a Magnetic or Electromagnetic Field and Associated Phenomena Time-dependent Perturbation Theory and its Applications Optical Properties of Nanostructures Problems in Quantum Mechanics: For Material Scientists, Applied Physicists and Device Engineers is an ideal companion to engineering, condensed matter physics or materials science curricula. It appeals to future and present engineers, physicists, and materials scientists, as well as professionals in these fields needing more in-depth understanding of nanotechnology and nanoscience.
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• * R. Fleming, From Helgermites to Lipschitz: Remembering Jim Jamison* T. Abe, A Mazur-Ulam theorem for normed gyrolinear spaces* S. Aryal, H. Choi, and F. Jafari, Sparse Hamburger moment multisequences* S. Bandyopadhyay and A. I. Singh, Polynomial representation of quantum entanglement* S. Basu, On span of small combination of slices points in Banach spaces* F. Botelho and J. Jamison, Surjective isometries on absolutely continuous vector valued function spaces* I. Chalendar and J. R. Partington, Compactness, differentiability and similarity to isometry of composition semigroups* F. Colonna and M. Tjani, Weighted composition operators from Banach spaces of analytic functions into Bloch-type spaces* C. C. Cowen and E. A. Gallardo-Gutierrez, A new proof of a Nordgren, Rosenthal and Wintrobe theorem on universal operators* C. Farsi, E. Gillaspy, A. Julien, S. Kang, and J. Packer, Wavelets and spectral triples for fractal representations of Cuntz algebras* N. J. Gal, The isometric equivalence problem* O. Hatori, Extension of isometries in generalized gyrovector spaces of the positive cones* D. Ilisevic, Generalized $n$-circular projections on JB*-triples* R. King, Hermitian operators on $H^1_{\mathcal{H}}$* B. Miller, Kernels of adjoints of composition operators with rational symbols of degree two* T. Miura and H. Takagi, Surjective isometries on the Banach space of continuously differentiable functions* L. Molnar, The arithmetic, geometric and harmonic means in operator algebras and transformations among them* B. Randrianantoanina, On sign embeddings and narrow operators on $L_2$* T. S. S. R. K. Rao, Into isometries that preserve finite dimensional structure of the range* J. E. Stovall and W. A. Feldman, Associating linear and nonlinear operators* D. Thompson, Normality properties of weighted composition operators on $H^2$.
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This volume contains the proceedings of the Workshop on Problems and Recent Methods in Operator Theory, held at the University of Memphis, Memphis, TN, from October 15-16, 2015 and the AMS Special Session on Advances in Operator Theory and Applications, in Memory of James Jamison, held at the University of Memphis, Memphis, TN, from October 17-18, 2015. Operator theory is at the root of several branches of mathematics and offers a broad range of challenging and interesting research problems. It also provides powerful tools for the development of other areas of science including quantum theory, physics and mechanics. Isometries have applications in solid-state physics. Hermitian operators play an integral role in quantum mechanics very much due to their "nice" spectral properties. These powerful connections demonstrate the impact of operator theory in various branches of science. The articles in this volume address recent problems and research advances in operator theory. Highlighted topics include spectral, structural and geometric properties of special types of operators on Banach spaces, with emphasis on isometries, weighted composition operators, multi-circular projections on function spaces, as well as vector valued function spaces and spaces of analytic functions. This volume gives a succinct overview of state-of-the-art techniques from operator theory as well as applications to classical problems and long-standing open questions.
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• Discovery and understanding (1900-1927). Planck, Einstein, Bohr : experiments and early ideas ; Heisenberg, Dirac, Schrödinger : quantum mechanics and the quantum atom ; Application : six hundred million watts!
• Interpretation and mind-boggling implications (1916-2016). The essential features of quantum mechanics ; Clash of titans : what is real? Uncertainty, entanglement, John Bell, and many worlds ; What does it all mean? : quantum mechanics, mathematics, and the nature of science ; Applications : quantum computing, code cracking, teleportation, and encryption
• Our world of relativity and the quantum, from the Big Bang to the galaxies. Galaxies, black holes, gravity waves, matter, the forces of nature, the Higgs boson, dark matter, dark energy, and string theory
• The many-electron atom and the foundations of chemistry and materials science. Energy, momentum, and the spatial states of the electrons in the hydrogen atom ; Spin and magnetism ; Exclusion and the Periodic table ; The physics underlying the chemistry of the elements ; A few types of chemical bonds, for example ; The makeup of solid materials ; Insulators and electrical conduction in normal metals and semiconductors
• Quantum wonders in materials and devices, large and small. Superconductors I : definition, and applications in transportation, medicine, and computing ; Fusion for electrical power, and lasers also for defense ; Magnetism, magnets, magnetic materials, and their applications ; Graphene, nanotubes, and one "dream" application ; Semiconductors and electronic applications ; Superconductors II : large scale applications in science, power generation, and transmission
• Appendix A. The nature and spectrum of electromagnetic waves
• Appendix B. Empirical development of the Periodic Table of Elements
• Appendix C. Quantum computer development
• Appendix D. The atomic sizes and chemistries of the elements
• Appendix E. The production of X-rays.

"A lucid introduction to the basics of quantum physics for the lay reader, showing how much the scientific understanding of reality has changed in the last century and how important this science is to technology, the economy, and modern life"-- Provided by publisher.

• Introduction: A revolution in time
• Wheeler's watch
• The only particle in the universe
• All the roads not to paradise
• The hidden paths of ghosts
• The island and the mountains: mapping the particle landscape
• Life as an amoeba in the foamy sea of possibilities
• Time's arrow and the mysterious Mr. X
• Minds, machines, and the cosmos
• Conclusion: The way of the labyrinth
• Epilogue: Encounters with Wheeler.

In 1939, Richard Feynman, a brilliant graduate of MIT, arrived in John Wheeler's Princeton office to report for duty as his teaching assistant. A lifelong friendship and enormously productive collaboration was born, despite sharp differences in personality. The soft-spoken Wheeler, though conservative in appearance, was a raging nonconformist full of wild ideas about the universe. The boisterous Feynman was a cautious physicist who believed only what could be tested. Yet they were complementary spirits. Their collaboration led to a complete rethinking of the nature of time and reality. It enabled Feynman to show how quantum reality is a combination of alternative, contradictory possibilities, and inspired Wheeler to develop his landmark concept of wormholes, portals to the future and past. Together, Feynman and Wheeler made sure that quantum physics would never be the same again.
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• What are the Issues raised by Quantum Mechanics?.- The First Mystery: Interference.- "Philosophical" Intermezzo I: What is Determinism?.- How do Physicists deal with Interference?.- Schroedinger's Cat and Hidden Variables.- "Philosophical" Intermezzo II: What is Wrong with "Observations"?.- The Second Mystery: Nonlocality.- How to do "The Impossible", a Quantum Mechanics without Observers: The de Broglie-Bohm Theory.- Many Worlds.- A Revised History of Quantum Mechanics.- The Cultural Impact of Quantum Mechanics.- Summary of the Main Theses of this Book.
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Permeated by the author's delightful humor, this little book explains, with nearly no mathematics, the main conceptual issues associated with quantum mechanics: The issue of determinism. Does quantum mechanics signify the end of a deterministic word-view? The role of the human subject or of the "observer" in science. Since Copernicus, science has increasingly tended to dethrone Man from his formerly held special position in the Universe. But quantum mechanics, with its emphasis on the notion of observation, may once more have given a central role to the human subject. The issue of locality. Does quantum mechanics imply that instantaneous actions at a distance exist in Nature? In these pages the author offers a variety of views and answers - bad as well as good - to these questions. The reader will be both entertained and enlightened by Jean Bricmont's clear and incisive arguments.
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• K. Becker and M. Becker, Superstring compactifications to all orders in $\alpha'$F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfacesH.-L. Chang, J. Li, W.-P. Li, and C.-C. M. Liu, On the mathematics and physics of Mixed Spin P-fieldsC.-H. Cho, Homological mirror functors via Maurer-Cartan formalismC. F. Doran, A. Harder, and A. Thompson, Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifoldsM. Han, SL(2, $\mathbb{C}$) Chern-Simons theory and four-dimensional quantum geometryY.-P. Lee, H.-W. Lin, and C.-L. Wang, Quantum cohomology under birational maps and transitionsG. W. Moore, A. B. Royston, and D. Van den Bleeken, $L^2$-kernels of Dirac-type operators on monopole moduli spacesN. Nekrasov, $\mathcal{BPS/CFT}$ correspondence: Instantons at crossroads and gauge origamiX. Wang and Y. Zhang, Balanced embedding of degenerating Abelian varietiesN. Yui, The modularity/automorphy of Calabi-Yau varieties of CM type.
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This volume contains the proceedings of the conference String-Math 2015, which was held from December 31, 2015-January 4, 2016, at Tsinghua Sanya International Mathematics Forum in Sanya, China. Two of the main themes of this volume are frontier research on Calabi-Yau manifolds and mirror symmetry and the development of non-perturbative methods in supersymmetric gauge theories. The articles present state-of-the-art developments in these topics. String theory is a broad subject, which has profound connections with broad branches of modern mathematics. In the last decades, the prosperous interaction built upon the joint efforts from both mathematicians and physicists has given rise to marvelous deep results in supersymmetric gauge theory, topological string, M-theory and duality on the physics side, as well as in algebraic geometry, differential geometry, algebraic topology, representation theory and number theory on the mathematics side.
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• * Introduction and outline* Floquet-Bloch and Fourier analysis* Dirac points of 1D periodic structures* Domain wall modulated periodic Hamiltonian and formal derivation of topologically protected bound states* Main Theorem--Bifurcation of topologically protected states* Proof of the Main Theorem* Appendix A. A variant of Poisson summation* Appendix B. 1D Dirac points and Floquet-Bloch eigenfunctions* Appendix C. Dirac points for small amplitude potentials* Appendix D. Genericity of Dirac points - 1D and 2D cases* Appendix E. Degeneracy lifting at Quasi-momentum zero* Appendix F. Gap opening due to breaking of inversion symmetry* Appendix G. Bounds on leading order terms in multiple scale expansion* Appendix H. Derivation of key bounds and limiting relations in the Lyapunov-Schmidt reduction* References.
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The authors study a class of periodic Schrodinger operators, which in distinguished cases can be proved to have linear band-crossings or "Dirac points". They then show that the introduction of an "edge", via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized "edge states". These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The authors' model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states the authors construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.
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• Summary
• Preface
• Prologue
• Prelude
• Part I. The butterfly fractal
• 0. Kiss precise. Apollonian gaskets and integer wonderlands
• Appendix. An Apollonian sand painting--the world's largest artwork
• 1. The fractal family. The Mandelbrot set
• The Feigenbaum set
• Classic fractals
• The Hofstadter set
• Appendix. Harper's equation as an iterative mapping
• 2. Geometry, number theory, and the butterfly : friendly numbers and kissing circles. Ford circles, the Farey tree, and the butterfly
• A butterfly at every scale--butterfly recursions
• Scaling and universality
• The butterfly and a hidden trefoil symmetry
• Closing words : physics and number theory
• Appendix A. Hofstadter recursions and butterfly generations
• Appendix B. Some theorems of number theory
• Appendix C. Continued-fraction expansions
• Appendix D. Nearest-integer continued fraction expansion
• Appendix E. Farey paths and some comments on universality
• 3. The Apollonian-butterfly connection (ABC). Integral Apollonian gaskets (IAG) and the butterfly
• The kaleidoscopic effect and trefoil symmetry
• Beyond Ford Apollonian gaskets and fountain butterflies
• Appendix. Quadratic Diophantine equations and IAGs --

• 4. Quasiperiodic patterns and the butterfly. A tale of three irrationals
• Self-similar butterfly hierarchies
• The diamond, golden, and silver hierarchies, and Hofstadter recursions
• Symmetries and quasiperiodicities
• Appendix. Quasicrystals
• Part II. Butterfly in the quantum world
• 5. The quantum world. Wave or particle--what is it?
• Quantization
• What is waving?--The Schrödinger picture
• Quintessentially quantum
• Quantum effects in the macroscopic world
• 6. A quantum-mechanical marriage and its unruly child. Two physical situations joined in a quantum-mechanical marriage
• The marvelous pure number [phi]
• Harper's equation, describing Bloch electrons in a magnetic field
• Harper's equation as a recursion relation
• On the key role of inexplicable artistic intuitions in physics
• Discovering the strange eigenvalue spectrum of Harper's equation
• Continued fractions and the looming nightmare of discontinuity
• Polynomials that dance on several levels at once
• A short digression on INT and on perception of visual patterns
• The spectrum belonging to irrational values of [phi] and the "ten-martini problem"
• In which continuity (of a sort) is finally established
• Infinitely recursively scalloped wave functions : cherries on the doctoral sundae
• Closing words
• Appendix. Supplementary material on Harper's equation --

• Part III. Topology and the butterfly
• 7. A different kind of quantization : the quantum Hall effect. What is the Hall effect? Classical and quantum answers
• A charged particle in a magnetic field : cyclotron orbits and their quantization
• Landau levels in the Hofstadter butterfly
• Topological insulators
• Appendix A. Excerpts from the 1985 Nobel Prize press release
• Appendix B. Quantum mechanics of electrons in a magnetic field
• Appendix C. Quantization of the Hall conductivity
• 8. Topology and topological invariants : preamble to the topological aspects of the quantum Hall effect
• A puzzle : the precision and the quantization of Hall conductivity
• Topological invariants
• Anholonomy : parallel transport and the Foucault pendulum
• Geometrization of the Foucault pendulum
• Berry magnetism--effective vector potential and monopoles
• The ESAB effect as an example of anholonomy
• Appendix. Classical parallel transport and magnetic monopoles
• 9. The Berry phase and the quantum Hall effect. The Berry phase
• Examples of Berry phase
• Chern numbers in two-dimensional electron gases
• Conclusion : the quantization of Hall conductivity
• Closing words : topology and physical phenomena
• Appendix A. Berry magnetism and the Berry phase
• Appendix B. The Berry phase and 2 x 2 matrices
• Appendix C. What causes Berry curvature? Dirac strings, vortices, and magnetic monopoles
• Appendix D. The two-band lattice model for the quantum Hall effect --

• 10. The kiss precise and precise quantization. Diophantus gives us two numbers for each swath in the butterfly
• Chern labels not just for swaths but also for bands
• A topological map of the butterfly
• Apollonian-butterfly connection : where are the Chern numbers?
• A topological landscape that has trefoil symmetry
• Chern-dressed wave functions
• Summary and outlook
• Part IV. Catching the butterfly
• 11. The art of tinkering. The most beautiful physics experiments
• 12. The butterfly in the laboratory
• Two-dimensional electron gases, superlattices, and the butterfly revealed
• Magical carbon : a new net for the Hofstadter butterfly
• A potentially sizzling hot topic in ultracold atom laboratories
• Appendix. Excerpts from the 2010 Physics Nobel Prize press release
• 13. The butterfly gallery : variations on a theme of Philip G Harper
• 14. Divertimento
• 15. Gratitude
• 16. Poetic math & science
• 17. Coda.

"Butterfly in the Quantum World by Indu Satija, with contributions by Douglas Hofstadter, is the first book to tell the story of the "Hofstadter butterfly", a beautiful and fascinating graph lying at the heart of the quantum theory of matter. The butterfly came out of a simple-sounding question: What happens if you immerse a crystal in a magnetic field? What energies can the electrons take on? From 1930 onwards, physicists struggled to answer this question, until 1974, when graduate student Douglas Hofstadter discovered that the answer was a graph consisting of nothing but copies of itself nested down infinitely many times. This wild mathematical object caught the physics world totally by surprise, and it continues to mesmerize physicists and mathematicians today. The butterfly plot is intimately related to many other important phenomena in number theory and physics, including Apollonian gaskets, the Foucault pendulum, quasicrystals, the quantum Hall effect and many more. Its story reflects the magic, the mystery and the simplicity of the laws of nature, and Indu Satija, in a wonderfully personal style, relates this story, enriching it with a vast number of lively historical anecdotes, many photographs, beautiful visual images and even poems, making her book a great feast for the eyes, for the mind and for the soul."--Page 4 of cover.

• I The Cellular Automaton Interpretation as a general doctrine: Motivation for this work.- Deterministic models in quantum notation.- Interpreting quantum mechanics.- Deterministic quantum mechanics.- Concise description of the CA Interpretation.- Quantum gravity.- Information loss.- More problems.- Alleys to be further investigated and open questions.- Conclusions.- II Calculation Techniques: Introduction to part II.- More on cogwheels.- The continuum limit of cogwheels, harmonic rotators and oscillators.- Locality.- Fermions.- PQ theory.- Models in two space-time dimensions without interactions.- Symmetries.- The discretised Hamiltonian formalism in PQ theory.- Quantum Field Theory.- The cellular automaton.- The problem of quantum locality.- Conclusions of part II.- Some remarks on gravity in 2+1 dimensions.- A summary of our views on Conformal Gravity.- Abbreviations.
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This book presents the deterministic view of quantum mechanics developed by Nobel Laureate Gerard 't Hooft. Dissatisfied with the uncomfortable gaps in the way conventional quantum mechanics meshes with the classical world, 't Hooft has revived the old hidden variable ideas, but now in a much more systematic way than usual. In this, quantum mechanics is viewed as a tool rather than a theory. The author gives examples of models that are classical in essence, but can be analysed by the use of quantum techniques, and argues that even the Standard Model, together with gravitational interactions, might be viewed as a quantum mechanical approach to analysing a system that could be classical at its core. He shows how this approach, even though it is based on hidden variables, can be plausibly reconciled with Bell's theorem, and how the usual objections voiced against the idea of `superdeterminism' can be overcome, at least in principle. This framework elegantly explains - and automatically cures - the problems of the wave function collapse and the measurement problem. Even the existence of an "arrow of time" can perhaps be explained in a more elegant way than usual. As well as reviewing the author's earlier work in the field, the book also contains many new observations and calculations. It provides stimulating reading for all physicists working on the foundations of quantum theory.
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• Acknowledgements ix Preface xi Are fashion, faith, or fantasy relevant to fundamental science? xi
• 1 Fashion 1 1.1 Mathematical elegance as a driving force 1 1.2 Some fashionable physics of the past 10 1.3 Particle-physics background to string theory 17 1.4 The superposition principle in QFT 20 1.5 The power of Feynman diagrams 25 1.6 The original key ideas of string theory 32 1.7 Time in Einstein's general relativity 42 1.8 Weyl's gauge theory of electromagnetism 52 1.9 Functional freedom in Kaluza-Klein and string models 59 1.10 Quantum obstructions to functional freedom? 69 1.11 Classical instability of higher-dimensional string theory 77 1.12 The fashionable status of string theory 82 1.13 M-theory 90 1.14 Supersymmetry 95 1.15 AdS/CFT 104 1.16 Brane-worlds and the landscape 117
• 2 Faith 121 2.1 The quantum revelation 121 2.2 Max Planck's E = hnu 126 2.3 The wave-particle paradox 133 2.4 Quantum and classical levels: C, U, and R 138 2.5 Wave function of a point-like particle 145 2.6 Wave function of a photon 153 2.7 Quantum linearity 158 2.8 Quantum measurement 164 2.9 The geometry of quantum spin 174 2.10 Quantum entanglement and EPR effects 182 2.11 Quantum functional freedom 188 2.12 Quantum reality 198 2.13 Objective quantum state reduction: a limit to the quantum faith? 204
• 3 Fantasy 216 3.1 The Big Bang and FLRW cosmologies 216 3.2 Black holes and local irregularities 230 3.3 The second law of thermodynamics 241 3.4 The Big Bang paradox 250 3.5 Horizons, comoving volumes, and conformal diagrams 258 3.6 The phenomenal precision in the Big Bang 270 3.7 Cosmological entropy? 275 3.8 Vacuum energy 285 3.9 Inflationary cosmology 294 3.10 The anthropic principle 310 3.11 Some more fantastical cosmologies 323
• 4 A New Physics for the Universe? 334 4.1 Twistor theory: an alternative to strings? 334 4.2 Whither quantum foundations? 353 4.3 Conformal crazy cosmology? 371 4.4 A personal coda 391 Appendix A Mathematical
• Appendix 397 A.1 Iterated exponents 397 A.2 Functional freedom of fields 401 A.3 Vector spaces 407 A.4 Vector bases, coordinates, and duals 413 A.5 Mathematics of manifolds 417 A.6 Manifolds in physics 425 A.7 Bundles 431 A.8 Functional freedom via bundles 439 A.9 Complex numbers 445 A.10 Complex geometry 448 A.11 Harmonic analysis 458 References 469 Index 491.
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What can fashionable ideas, blind faith, or pure fantasy possibly have to do with the scientific quest to understand the universe? Surely, theoretical physicists are immune to mere trends, dogmatic beliefs, or flights of fancy? In fact, acclaimed physicist and bestselling author Roger Penrose argues that researchers working at the extreme frontiers of physics are just as susceptible to these forces as anyone else. In this provocative book, he argues that fashion, faith, and fantasy, while sometimes productive and even essential in physics, may be leading today's researchers astray in three of the field's most important areas--string theory, quantum mechanics, and cosmology. Arguing that string theory has veered away from physical reality by positing six extra hidden dimensions, Penrose cautions that the fashionable nature of a theory can cloud our judgment of its plausibility. In the case of quantum mechanics, its stunning success in explaining the atomic universe has led to an uncritical faith that it must also apply to reasonably massive objects, and Penrose responds by suggesting possible changes in quantum theory. Turning to cosmology, he argues that most of the current fantastical ideas about the origins of the universe cannot be true, but that an even wilder reality may lie behind them. Finally, Penrose describes how fashion, faith, and fantasy have ironically also shaped his own work, from twistor theory, a possible alternative to string theory that is beginning to acquire a fashionable status, to "conformal cyclic cosmology, " an idea so fantastic that it could be called "conformal crazy cosmology." The result is an important critique of some of the most significant developments in physics today from one of its most eminent figures.
(source: Nielsen Book Data)

• QM and QM/MM Methods Polarizable continuum models for (bio)molecular electrostatics: Basic theory and recent developments for macromolecules and simulations, J. M. Herbert, A. W. Lange A modified divide-and-conquer linear-scaling quantum force field with multipolar charge densities, T. J. Giese, D. M. York Explicit polarization theory, Y. Wang, M. J. M. Mazack, D. G. Truhlar, J. Gao Effective fragment potential method, L. Slipchenko Quantum mechanical methods for quantifying and analyzing non-covalent interactions and for force-field development, D. Sherrill, K. Merz Force field development with density-based energy decomposition analysis, N. Zhou, Q. Wu, Y. Zhang Atomistic Models Differential geometry-based solvation and electrolyte transport models for biomolecular modeling: a review, W. Guowei, N. Baker Explicit inclusion of induced polarization in atomistic force fields based on the classical Drude oscillator model, A. Savelyev, B. Roux, A. D. Mackerell, Jr. Multipolar force fields for atomistic simulations, M. Meuwly, T. Bereau Quantum mechanics based polarizable force field for proteins, C. Ji, Y. Mei, J. Zhang Status of the Gaussian electrostatic model, a density-based polarizable force field, G. A. Cisneros, J.-P. Piquemal Water models: Looking forward by looking backward, T. Ichiye Coarse-Grained Models A physics-based coarse-grained model with electric multipoles, G. Li Coarse-grained membrane force field based on Gay-Berne potential and electric multipoles, D. Lin, A. Grossfield Perspectives on the coarse-grained models of DNA, I. Echeverria, G. Papoian RNA coarse-grained model theory, D. Bell, P. Ren.
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As computational hardware continues to develop at a rapid pace, quantitative computations are playing an increasingly essential role in the study of biomolecular systems. One of the most important challenges that the field faces is to develop the next generation of computational models that strike the proper balance of computational efficiency and accuracy, so that the problems of increasing complexity can be tackled in a systematic and physically robust manner. In particular, properly treating intermolecular interactions is fundamentally important for the reliability of all computational models. In this book, contributions by leading experts in the area of biomolecular simulations discuss cutting-edge ideas regarding effective strategies to describe many-body effects and electrostatics at quantum, classical, and coarse-grained levels. The goal of the book is to not only provide an up-to-date snapshot of the current simulation field but also stimulate exchange of ideas across different sub-fields of modern computational (bio)chemistry. The text will be a useful reference for the biomolecular simulation community and help attract talented young students into this exciting frontier of research.
(source: Nielsen Book Data)