Quantized Partial Differential Equations

Author : Agostino Prastaro
Publisher : World Scientific
Page : 500 pages
File Size : 52,8 Mb
Release : 2004
Category : Mathematics
ISBN : 9789812562517

Get Book

Quantized Partial Differential Equations by Agostino Prastaro Pdf

This book presents, for the first time, a systematic formulation ofthe geometric theory of noncommutative PDE''s which is suitable enoughto be used for a mathematical description of quantum dynamics andquantum field theory. A geometric theory of supersymmetric quantumPDE''s is also considered, in order to describe quantumsupergravity. Covariant and canonical quantizations of (super) PDE''sare shown to be founded on the geometric theory of PDE''s and toproduce quantum (super) PDE''s by means of functors from the categoryof commutative (super) PDE''s to the category of quantum (super)PDE''s. Global properties of solutions to (super) (commutative) PDE''sare obtained by means of their integral bordism groups.

Author : Vladimir E. Nazaikinskii,B.-W. Schulze,Boris Yu. Sternin
Publisher : CRC Press
Page : 368 pages
File Size : 41,7 Mb
Release : 2002-05-16
Category : Mathematics
ISBN : 9781482265033

Get Book

Quantization Methods in the Theory of Differential Equations by Vladimir E. Nazaikinskii,B.-W. Schulze,Boris Yu. Sternin Pdf

This volume presents a systematic and mathematically rigorous exposition of methods for studying linear partial differential equations. It focuses on quantization of the corresponding objects (states, observables and canonical transformations) in the phase space. The quantization of all three types of classical objects is carried out in a unified w

Author : Vladimir E. Nazaikinskii,B.-W. Schulze,Boris Yu. Sternin
Publisher : CRC Press
Page : 372 pages
File Size : 50,7 Mb
Release : 2002-05-16
Category : Mathematics
ISBN : 0415273641

Get Book

Quantization Methods in the Theory of Differential Equations by Vladimir E. Nazaikinskii,B.-W. Schulze,Boris Yu. Sternin Pdf

This volume presents a systematic and mathematically rigorous exposition of methods for studying linear partial differential equations. It focuses on quantization of the corresponding objects (states, observables and canonical transformations) in the phase space. The quantization of all three types of classical objects is carried out in a unified way with the use of a special integral transform. This book covers recent as well as established results, treated within the framework of a universal approach. It also includes applications and provides a useful reference text for graduate and research-level readers.

Author : Dorothea Bahns,Wolfram Bauer,Ingo Witt
Publisher : Birkhäuser
Page : 314 pages
File Size : 44,9 Mb
Release : 2016-02-11
Category : Mathematics
ISBN : 9783319224077

Get Book

Quantization, PDEs, and Geometry by Dorothea Bahns,Wolfram Bauer,Ingo Witt Pdf

This book presents four survey articles on different topics in mathematical analysis that are closely linked to concepts and applications in physics. Specifically, it discusses global aspects of elliptic PDEs, Berezin-Toeplitz quantization, the stability of solitary waves, and sub-Riemannian geometry. The contributions are based on lectures given by distinguished experts at a summer school in Göttingen. The authors explain fundamental concepts and ideas and present them clearly. Starting from basic notions, these course notes take the reader to the point of current research, highlighting new challenges and addressing unsolved problems at the interface between mathematics and physics. All contributions are of interest to researchers in the respective fields, but they are also accessible to graduate students.

Author : A Prástaro
Publisher : World Scientific
Page : 500 pages
File Size : 53,6 Mb
Release : 2004-04-06
Category : Science
ISBN : 9789814483186

Get Book

Quantized Partial Differential Equations by A Prástaro Pdf

' This book presents, for the first time, a systematic formulation of the geometric theory of noncommutative PDE's which is suitable enough to be used for a mathematical description of quantum dynamics and quantum field theory. A geometric theory of supersymmetric quantum PDE's is also considered, in order to describe quantum supergravity. Covariant and canonical quantizations of (super) PDE's are shown to be founded on the geometric theory of PDE's and to produce quantum (super) PDE's by means of functors from the category of commutative (super) PDE's to the category of quantum (super) PDE's. Global properties of solutions to (super) (commutative) PDE's are obtained by means of their integral bordism groups. Contents: Quantized PDE's I: Noncommutative ManifoldsQuantized PDE's II: Noncommutative PDE'sQuantized PDE's III: Quantizations of Commutative PDE'sAddendum I: Bordism Groups and the (NS)-ProblemAddendum II: Bordism Groups and Variational PDE's Readership: Researchers and graduate students in the fields of partial differential equations, mathematical physics and theoretical physics. Keywords:Noncommutative Manifolds;Noncommutative PDE''s;(Co)Bordism Groups in (Noncommutative) PDE''s;(Quantum) Navier–Stokes Equations;(Quantum) Super Yang–Mills Equations;Quantum Supergravity;Global Existence Solutions of (Quantum) PDE''s'

Author : John Von Neumann,William Arveson,Thomas Branson,Irving Ezra Segal
Publisher : American Mathematical Soc.
Page : 224 pages
File Size : 50,8 Mb
Release : 1996
Category : Science
ISBN : 9780821803813

Get Book

Quantization, Nonlinear Partial Differential Equations, and Operator Algebra by John Von Neumann,William Arveson,Thomas Branson,Irving Ezra Segal Pdf

Recent inroads in higher-dimensional nonlinear quantum field theory and in the global theory of relevant nonlinear wave equations have been accompanied by very interesting cognate developments. These developments include symplectic quantization theory on manifolds and in group representations, the operator algebraic implementation of quantum dynamics, and differential geometric, general relativistic, and purely algebraic aspects. Quantization and Nonlinear Wave Equations thus was highly appropriate as the theme for the first John von Neumann Symposium (June 1994) held at MIT. The symposium was intended to treat topics of emerging signifigance underlying future mathematical developments. This book describes the outstanding recent progress in this important and challenging field and presents general background for the scientific context and specifics regarding key difficulties. Quantization is developed in the context of rigorous nonlinear quantum field theory in four dimensions and in connection with symplectic manifold theory and random Schrodinger operators. Nonlinear wave equations are exposed in relation to recent important progress in general relativity, in purely mathematical terms of microlocal analysis, and as represented by progress on the relativistic Boltzmann equation. Most of the developments in this volume appear in book form for the first time. The resulting work is a concise and informative way to explore the field and the spectrum of methods available for its investigation.

Author : S.I. Andersson,H.-D. Doebner
Publisher : Springer
Page : 344 pages
File Size : 46,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540386957

Get Book

Non-linear Partial Differential Operators and Quantization Procedures by S.I. Andersson,H.-D. Doebner Pdf

Author : Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft
Publisher : Springer
Page : 214 pages
File Size : 45,8 Mb
Release : 2008-08-15
Category : Mathematics
ISBN : 9783540682684

Get Book

Pseudo-Differential Operators by Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft Pdf

Pseudo-differential operators were initiated by Kohn, Nirenberg and Hörmander in the sixties of the last century. Beside applications in the general theory of partial differential equations, they have their roots also in the study of quantization first envisaged by Hermann Weyl thirty years earlier. Thanks to the understanding of the connections of wavelets with other branches of mathematical analysis, quantum physics and engineering, such operators have been used under different names as mathematical models in signal analysis since the last decade of the last century. The volume investigates the mathematics of quantization and signals in the context of pseudo-differential operators, Weyl transforms, Daubechies operators, Wick quantization and time-frequency localization operators. Applications to quantization, signal analysis and the modern theory of PDE are highlighted.

Author : Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft
Publisher : Springer
Page : 214 pages
File Size : 55,6 Mb
Release : 2009-08-29
Category : Mathematics
ISBN : 3540863974

Get Book

Pseudo-Differential Operators by Hans G. Feichtinger,Bernard Helffer,Michael Lamoureux,Nicolas Lerner,Joachim Toft Pdf

Pseudo-differential operators were initiated by Kohn, Nirenberg and Hörmander in the sixties of the last century. Beside applications in the general theory of partial differential equations, they have their roots also in the study of quantization first envisaged by Hermann Weyl thirty years earlier. Thanks to the understanding of the connections of wavelets with other branches of mathematical analysis, quantum physics and engineering, such operators have been used under different names as mathematical models in signal analysis since the last decade of the last century. The volume investigates the mathematics of quantization and signals in the context of pseudo-differential operators, Weyl transforms, Daubechies operators, Wick quantization and time-frequency localization operators. Applications to quantization, signal analysis and the modern theory of PDE are highlighted.

Author : A. M. Vinogradov
Publisher : American Mathematical Soc.
Page : 268 pages
File Size : 45,5 Mb
Release : 2001-10-16
Category : Mathematics
ISBN : 0821897993

Get Book

Cohomological Analysis of Partial Differential Equations and Secondary Calculus by A. M. Vinogradov Pdf

This book is dedicated to fundamentals of a new theory, which is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This theory grew up from the classical geometry of PDE's originated by S. Lie and his followers by incorporating some nonclassical ideas from the theory of integrable systems, the formal theory of PDE's in its modern cohomological form given by D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). The main result of this synthesis is Secondary Calculus on diffieties, new geometrical objects which are analogs of algebraic varieties in the context of (nonlinear) PDE's. Secondary Calculus surprisingly reveals a deep cohomological nature of the general theory of PDE's and indicates new directions of its further progress. Recent developments in quantum field theory showed Secondary Calculus to be its natural language, promising a nonperturbative formulation of the theory. In addition to PDE's themselves, the author describes existing and potential applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants, theory of geometric structures, variational calculus, control theory, etc. This book, focused mainly on theoretical aspects, forms a natural dipole with Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Volume 182 in this same series, Translations of Mathematical Monographs, and shows the theory "in action".

Author : S. I. Andersson,Heinz-Dietrich Doebner
Publisher : Unknown
Page : 334 pages
File Size : 42,5 Mb
Release : 1983
Category : Electronic
ISBN : OCLC:802798285

Get Book

Non-linear Partial Differential Operators and Quantization Procedures by S. I. Andersson,Heinz-Dietrich Doebner Pdf

Author : Claus Gerhardt
Publisher : Springer
Page : 200 pages
File Size : 52,6 Mb
Release : 2018-04-14
Category : Science
ISBN : 9783319773711

Get Book

The Quantization of Gravity by Claus Gerhardt Pdf

​A unified quantum theory incorporating the four fundamental forces of nature is one of the major open problems in physics. The Standard Model combines electro-magnetism, the strong force and the weak force, but ignores gravity. The quantization of gravity is therefore a necessary first step to achieve a unified quantum theory. In this monograph a canonical quantization of gravity has been achieved by quantizing a geometric evolution equation resulting in a gravitational wave equation in a globally hyperbolic spacetime. Applying the technique of separation of variables we obtain eigenvalue problems for temporal and spatial self-adjoint operators where the temporal operator has a pure point spectrum with eigenvalues $\lambda_i$ and related eigenfunctions, while, for the spatial operator, it is possible to find corresponding eigendistributions for each of the eigenvalues $\lambda_i$, if the Cauchy hypersurface is asymptotically Euclidean or if the quantized spacetime is a black hole with a negative cosmological constant. The hyperbolic equation then has a sequence of smooth solutions which are products of temporal eigenfunctions and spatial eigendistributions. Due to this "spectral resolution" of the wave equation quantum statistics can also be applied to the quantized systems. These quantum statistical results could help to explain the nature of dark matter and dark energy.

Author : John Von Neumann William Arveson Thomas Branson Irving Ezra Segal
Publisher : American Mathematical Soc.
Page : 240 pages
File Size : 43,9 Mb
Release : 1996-05-07
Category : Differential equations, Nonlinear
ISBN : 0821868322

Get Book

Quantization, nonlinear partial differential equations, and operator algebra by John Von Neumann William Arveson Thomas Branson Irving Ezra Segal Pdf

Recent inroads in higher-dimensional nonlinear quantum field theory and in the global theory of relevant nonlinear wave equations have been accompanied by very interesting cognate developments. These developments include symplectic quantization theory on manifolds and in group representations, the operator algebraic implementation of quantum dynamics, and differential geometric, general relativistic, and purely algebraic aspects. Quantization and Nonlinear Wave Equations thus was highly appropriate as the theme for the first John von Neumann Symposium (June 1994) held at MIT. The symposium was intended to treat topics of emerging signifigance underlying future mathematical developments. This book describes the outstanding recent progress in this important and challenging field and presents general background for the scientific context and specifics regarding key difficulties. Quantization is developed in the context of rigorous nonlinear quantum field theory in four dimensions and in connection with symplectic manifold theory and random Schrodinger operators. Nonlinear wave equations are exposed in relation to recent important progress in general relativity, in purely mathematical terms of microlocal analysis, and as represented by progress on the relativistic Boltzmann equation. Most of the developments in this volume appear in book form for the first time. The resulting work is a concise and informative way to explore the field and the spectrum of methods available for its investigation.

Author : Min-You Qi,L Rodino
Publisher : CRC Press
Page : 248 pages
File Size : 51,7 Mb
Release : 1996-05-20
Category : Mathematics
ISBN : 0582292123

Get Book

General Theory of Partial Differential Equations and Microlocal Analysis by Min-You Qi,L Rodino Pdf

Author : S. I. Andersson,H. D. Doebner
Publisher : Unknown
Page : 344 pages
File Size : 47,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662214253

Get Book

Non-Linear Partial Differential Operators and Quantization Procedures by S. I. Andersson,H. D. Doebner Pdf