Spectral Methods In Quantum Field Theory

Author : Noah Graham,Markus Quandt,Herbert Weigel
Publisher : Springer
Page : 182 pages
File Size : 42,7 Mb
Release : 2009-08-29
Category : Science
ISBN : 3642002463

Get Book

Spectral Methods in Quantum Field Theory by Noah Graham,Markus Quandt,Herbert Weigel Pdf

In this monograph we apply scattering theory methods to calculations in quantum ?eld theory, with a particular focus on properties of the quantum vacuum. These methods will provide e?cient and reliable solutions to a - riety of problems in quantum ?eld theory. Our approach will also elucidate in a concrete context many of the subtleties of quantum ?eld theory, such as divergences, regularization, and renormalization, by connecting them to more familiar results in quantum mechanics. We will use tools of scattering theory to characterize the spectrum of energyeigenstatesinapotentialbackground,hencethetermspectralmethods. This mode spectrum comprises both discrete bound states and a continuum of scattering states. We develop a powerful formalism that parameterizes the e?ects of the continuum by the density of states, which we compute from scattering data. Summing the zero-point energies of these modes gives the energy of the quantum vacuum, which is one of the central quantities we study.Althoughthemostcommonlystudiedbackgroundpotentialsarisefrom static soliton solutions to the classical equations of motion, these methods are not limited to such cases.

Author : Noah Graham,Markus Quandt,Herbert Weigel
Publisher : Springer Science & Business Media
Page : 187 pages
File Size : 41,8 Mb
Release : 2009-05-08
Category : Science
ISBN : 9783642001383

Get Book

Spectral Methods in Quantum Field Theory by Noah Graham,Markus Quandt,Herbert Weigel Pdf

In this monograph we apply scattering theory methods to calculations in quantum ?eld theory, with a particular focus on properties of the quantum vacuum. These methods will provide e?cient and reliable solutions to a - riety of problems in quantum ?eld theory. Our approach will also elucidate in a concrete context many of the subtleties of quantum ?eld theory, such as divergences, regularization, and renormalization, by connecting them to more familiar results in quantum mechanics. We will use tools of scattering theory to characterize the spectrum of energyeigenstatesinapotentialbackground,hencethetermspectralmethods. This mode spectrum comprises both discrete bound states and a continuum of scattering states. We develop a powerful formalism that parameterizes the e?ects of the continuum by the density of states, which we compute from scattering data. Summing the zero-point energies of these modes gives the energy of the quantum vacuum, which is one of the central quantities we study.Althoughthemostcommonlystudiedbackgroundpotentialsarisefrom static soliton solutions to the classical equations of motion, these methods are not limited to such cases.

Author : Dmitri Fursaev,Dmitri Vassilevich
Publisher : Springer Science & Business Media
Page : 294 pages
File Size : 45,7 Mb
Release : 2011-06-25
Category : Science
ISBN : 9789400702059

Get Book

Operators, Geometry and Quanta by Dmitri Fursaev,Dmitri Vassilevich Pdf

This book gives a detailed and self-contained introduction into the theory of spectral functions, with an emphasis on their applications to quantum field theory. All methods are illustrated with applications to specific physical problems from the forefront of current research, such as finite-temperature field theory, D-branes, quantum solitons and noncommutativity. In the first part of the book, necessary background information on differential geometry and quantization, including less standard material, is collected. The second part of the book contains a detailed description of main spectral functions and methods of their calculation. In the third part, the theory is applied to several examples (D-branes, quantum solitons, anomalies, noncommutativity). This book addresses advanced graduate students and researchers in mathematical physics with basic knowledge of quantum field theory and differential geometry. The aim is to prepare readers to use spectral functions in their own research, in particular in relation to heat kernels and zeta functions.

Author : I︠U︡riĭ Makarovich Berezanskiĭ,I︠U︡riĭ Grigorʹevich Kondratʹev
Publisher : Springer Science & Business Media
Page : 448 pages
File Size : 46,5 Mb
Release : 1995
Category : Degree of freedom
ISBN : 0792328485

Get Book

Spectral methods in infinite-dimensional analysis. 2 (1995) by I︠U︡riĭ Makarovich Berezanskiĭ,I︠U︡riĭ Grigorʹevich Kondratʹev Pdf

Author : Hernan Ocampo,Sylvie Paycha,Andres F Reyes-lega
Publisher : World Scientific
Page : 530 pages
File Size : 52,7 Mb
Release : 2001-04-30
Category : Science
ISBN : 9789814492829

Get Book

Geometric Methods For Quantum Field Theory by Hernan Ocampo,Sylvie Paycha,Andres F Reyes-lega Pdf

Both mathematics and mathematical physics have many active areas of research where the interplay between geometry and quantum field theory has proved extremely fruitful. Duality, gauge field theory, geometric quantization, Seiberg-Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups offer some striking recent examples of modern topics which stand on the borderline between geometry and analysis on the one hand and quantum field theory on the other, where the physicist's and the mathematician's perspective complement each other, leading to new mathematical and physical concepts and results.This volume introduces the reader to some basic mathematical and physical tools and methods required to follow the recent developments in some active areas of mathematical physics, including duality, gauge field theory, geometric quantization, Seiberg-Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups. It comprises seven self-contained lectures, which should progressively give the reader a precise idea of some of the techniques used in these areas, as well as a few short communications presented by young participants at the school.

Author : Yu.M. Berezansky,Y.G. Kondratiev
Publisher : Springer Science & Business Media
Page : 983 pages
File Size : 46,6 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9789401105095

Get Book

Spectral Methods in Infinite-Dimensional Analysis by Yu.M. Berezansky,Y.G. Kondratiev Pdf

The Russian edition of this book appeared 5 years ago. Since that time, many results have been improved upon and new approaches to the problems investigated in the book have appeared. But the greatest surprise for us was to discover that there exists a large group of mathematicians working in the area of the so-called White Noise Analysis which is closely connected with the essential part of our book, namely, with the theory of generalized functions of infinitely many variables. The first papers dealing with White Noise Analysis were written by T. Hida in Japan in 1975. Later, this analysis was devel oped intensively in Japan, Germany, U.S.A., Taipei, and in other places. The related problems of infinite-dimensional analysis have been studied in Kiev since 1967, and the theory of generalized functions of infinitely many variables has been in vestigated since 1973. However, due to the political system in the U.S.S.R., contact be tween Ukrainian and foreign mathematicians was impossible for a long period of time. This is why, to our great regret, only at the end of 1988 did one of the authors meet L. Streit who told him about the existence of White Noise Analysis. And it become clear that many results in these two theories coincide and that, in fact, there exists a single theory and not two distinct ones.

Author : Valter Moretti
Publisher : Springer
Page : 950 pages
File Size : 51,9 Mb
Release : 2018-01-30
Category : Mathematics
ISBN : 9783319707068

Get Book

Spectral Theory and Quantum Mechanics by Valter Moretti Pdf

This book discusses the mathematical foundations of quantum theories. It offers an introductory text on linear functional analysis with a focus on Hilbert spaces, highlighting the spectral theory features that are relevant in physics. After exploring physical phenomenology, it then turns its attention to the formal and logical aspects of the theory. Further, this Second Edition collects in one volume a number of useful rigorous results on the mathematical structure of quantum mechanics focusing in particular on von Neumann algebras, Superselection rules, the various notions of Quantum Symmetry and Symmetry Groups, and including a number of fundamental results on the algebraic formulation of quantum theories. Intended for Master's and PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book also benefits established researchers by organizing and presenting the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly."

Author : Gerald Teschl
Publisher : American Mathematical Soc.
Page : 322 pages
File Size : 40,7 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821846605

Get Book

Mathematical Methods in Quantum Mechanics by Gerald Teschl Pdf

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. It is well suited for self-study and includes numerous exercises (many with hints).

Author : Leonardo Cano,Alexander Cardona,Hern Ocampo,Andr F Reyes Lega
Publisher : World Scientific
Page : 384 pages
File Size : 53,6 Mb
Release : 2016-09-06
Category : Mathematics
ISBN : 9789814730891

Get Book

Geometric, Algebraic and Topological Methods for Quantum Field Theory by Leonardo Cano,Alexander Cardona,Hern Ocampo,Andr F Reyes Lega Pdf

Based on lectures held at the 8th edition of the series of summer schools in Villa de Leyva since 1999, this book presents an introduction to topics of current interest at the interface of geometry, algebra, analysis, topology and theoretical physics. It is aimed at graduate students and researchers in physics or mathematics, and offers an introduction to the topics discussed in the two weeks of the summer school: operator algebras, conformal field theory, black holes, relativistic fluids, Lie groupoids and Lie algebroids, renormalization methods, spectral geometry and index theory for pseudo-differential operators.

Author : Valter Moretti
Publisher : Springer Science & Business Media
Page : 742 pages
File Size : 41,6 Mb
Release : 2013-04-02
Category : Mathematics
ISBN : 9788847028357

Get Book

Spectral Theory and Quantum Mechanics by Valter Moretti Pdf

This book pursues the accurate study of the mathematical foundations of Quantum Theories. It may be considered an introductory text on linear functional analysis with a focus on Hilbert spaces. Specific attention is given to spectral theory features that are relevant in physics. Having left the physical phenomenology in the background, it is the formal and logical aspects of the theory that are privileged. Another not lesser purpose is to collect in one place a number of useful rigorous statements on the mathematical structure of Quantum Mechanics, including some elementary, yet fundamental, results on the Algebraic Formulation of Quantum Theories. In the attempt to reach out to Master's or PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book should benefit established researchers to organise and present the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly.

Author : Alekse? Alekseevich Abrikosov,Lev Petrovich Gor?kov,I. E. Dzyaloshinski,Igor? Ekhiel?evich Dzi?aloshinski?
Publisher : Courier Corporation
Page : 383 pages
File Size : 52,7 Mb
Release : 1975-10-01
Category : Science
ISBN : 9780486632285

Get Book

Methods of Quantum Field Theory in Statistical Physics by Alekse? Alekseevich Abrikosov,Lev Petrovich Gor?kov,I. E. Dzyaloshinski,Igor? Ekhiel?evich Dzi?aloshinski? Pdf

This comprehensive introduction to the many-body theory was written by three renowned physicists and acclaimed by American Scientist as "a classic text on field theoretic methods in statistical physics."

Author : Ben-yu Guo
Publisher : World Scientific
Page : 360 pages
File Size : 45,8 Mb
Release : 1998-05-05
Category : Mathematics
ISBN : 9789814496643

Get Book

Spectral Methods And Their Applications by Ben-yu Guo Pdf

This book presents the basic algorithms, the main theoretical results, and some applications of spectral methods. Particular attention is paid to the applications of spectral methods to nonlinear problems arising in fluid dynamics, quantum mechanics, weather prediction, heat conduction and other fields.The book consists of three parts. The first part deals with orthogonal approximations in Sobolev spaces and the stability and convergence of approximations for nonlinear problems, as the mathematical foundation of spectral methods. In the second part, various spectral methods are described, with some applications. It includes Fourier spectral method, Legendre spectral method, Chebyshev spectral method, spectral penalty method, spectral vanishing viscosity method, spectral approximation of isolated solutions, multi-dimensional spectral method, spectral method for high-order equations, spectral-domain decomposition method and spectral multigrid method. The third part is devoted to some recent developments of spectral methods, such as mixed spectral methods, combined spectral methods and spectral methods on the surface.

Author : Sylvie Paycha
Publisher : American Mathematical Soc.
Page : 272 pages
File Size : 43,9 Mb
Release : 2007
Category : Algebraic topology
ISBN : 9780821840627

Get Book

Geometric and Topological Methods for Quantum Field Theory by Sylvie Paycha Pdf

This volume, based on lectures and short communications at a summer school in Villa de Leyva, Colombia (July 2005), offers an introduction to some recent developments in several active topics at the interface between geometry, topology and quantum field theory. It is aimed at graduate students in physics or mathematics who might want insight in the following topics (covered in five survey lectures): Anomalies and noncommutative geometry, Deformation quantisation and Poisson algebras, Topological quantum field theory and orbifolds. These lectures are followed by nine articles on various topics at the borderline of mathematics and physics ranging from quasicrystals to invariant instantons through black holes, and involving a number of mathematical tools borrowed from geometry, algebra and analysis.

Author : Eberhard Zeidler
Publisher : Springer Science & Business Media
Page : 1060 pages
File Size : 47,8 Mb
Release : 2007-04-18
Category : Science
ISBN : 9783540347644

Get Book

Quantum Field Theory I: Basics in Mathematics and Physics by Eberhard Zeidler Pdf

This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists, at levels ranging from advanced undergraduate students to professional scientists. The book bridges the acknowledged gap between the different languages used by mathematicians and physicists. For students of mathematics the author shows that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which goes beyond the usual curriculum in physics.

Author : Bernard Shizgal
Publisher : Springer
Page : 431 pages
File Size : 45,7 Mb
Release : 2015-01-07
Category : Science
ISBN : 9789401794541

Get Book

Spectral Methods in Chemistry and Physics by Bernard Shizgal Pdf

This book is a pedagogical presentation of the application of spectral and pseudospectral methods to kinetic theory and quantum mechanics. There are additional applications to astrophysics, engineering, biology and many other fields. The main objective of this book is to provide the basic concepts to enable the use of spectral and pseudospectral methods to solve problems in diverse fields of interest and to a wide audience. While spectral methods are generally based on Fourier Series or Chebychev polynomials, non-classical polynomials and associated quadratures are used for many of the applications presented in the book. Fourier series methods are summarized with a discussion of the resolution of the Gibbs phenomenon. Classical and non-classical quadratures are used for the evaluation of integrals in reaction dynamics including nuclear fusion, radial integrals in density functional theory, in elastic scattering theory and other applications. The subject matter includes the calculation of transport coefficients in gases and other gas dynamical problems based on spectral and pseudospectral solutions of the Boltzmann equation. Radiative transfer in astrophysics and atmospheric science, and applications to space physics are discussed. The relaxation of initial non-equilibrium distributions to equilibrium for several different systems is studied with the Boltzmann and Fokker-Planck equations. The eigenvalue spectra of the linear operators in the Boltzmann, Fokker-Planck and Schrödinger equations are studied with spectral and pseudospectral methods based on non-classical orthogonal polynomials. The numerical methods referred to as the Discrete Ordinate Method, Differential Quadrature, the Quadrature Discretization Method, the Discrete Variable Representation, the Lagrange Mesh Method, and others are discussed and compared. MATLAB codes are provided for most of the numerical results reported in the book - see Link under 'Additional Information' on the the right-hand column.