Mathematical Feynman Path Integrals And Their Applications (Second Edition)

Mathematical Feynman Path Integrals And Their Applications (Second Edition) PDF Author: Sonia Mazzucchi
Publisher: World Scientific
ISBN: 9811214808
Category : Science
Languages : en
Pages : 360

Get Book

Book Description
Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics.This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added.This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists.

Mathematical Theory of Feynman Path Integrals

Mathematical Theory of Feynman Path Integrals PDF Author: Sergio Albeverio
Publisher: Springer Science & Business Media
ISBN: 3540769544
Category : Mathematics
Languages : en
Pages : 182

Get Book

Book Description
The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. An entire new chapter on the current forefront of research has been added. Except for this new chapter and the correction of a few misprints, the basic material and presentation of the first edition has been maintained. At the end of each chapter the reader will also find notes with further bibliographical information.

Rigorous Time Slicing Approach to Feynman Path Integrals

Rigorous Time Slicing Approach to Feynman Path Integrals PDF Author: Daisuke Fujiwara
Publisher: Springer
ISBN: 4431565531
Category : Science
Languages : en
Pages : 333

Get Book

Book Description
This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schrödinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives of order equal to or higher than two are bounded. The semi-classical asymptotic formula up to the second term of the fundamental solution is also proved by a method different from that of Birkhoff. A bound of the remainder term is also proved.The Feynman path integral is a method of quantization using the Lagrangian function, whereas Schrödinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger's method, there is still much to be done concerning rigorous mathematical treatment of Feynman's method. Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces which is obtained by dividing the time interval into small pieces. This method is called the time slicing approximation method or the time slicing method.This book consists of two parts. Part I is the main part. The time slicing method is performed step by step in detail in Part I. The time interval is divided into small pieces. Corresponding to each division a finite-dimensional integral is constructed following Feynman's famous paper. This finite-dimensional integral is not absolutely convergent. Owing to the assumption of the potential, it is an oscillatory integral. The oscillatory integral techniques developed in the theory of partial differential equations are applied to it. It turns out that the finite-dimensional integral gives a finite definite value. The stationary phase method is applied to it. Basic properties of oscillatory integrals and the stationary phase method are explained in the book in detail.Those finite-dimensional integrals form a sequence of approximation of the Feynman path integral when the division goes finer and finer. A careful discussion is required to prove the convergence of the approximate sequence as the length of each of the small subintervals tends to 0. For that purpose the book uses the stationary phase method of oscillatory integrals over a space of large dimension, of which the detailed proof is given in Part II of the book. By virtue of this method, the approximate sequence converges to the limit. This proves that the Feynman path integral converges. It turns out that the convergence occurs in a very strong topology. The fact that the limit is the fundamental solution of the Schrödinger equation is proved also by the stationary phase method. The semi-classical asymptotic formula naturally follows from the above discussion.A prerequisite for readers of this book is standard knowledge of functional analysis. Mathematical techniques required here are explained and proved from scratch in Part II, which occupies a large part of the book, because they are considerably different from techniques usually used in treating the Schrödinger equation.

Wave Packet Analysis of Feynman Path Integrals

Wave Packet Analysis of Feynman Path Integrals PDF Author: Fabio Nicola
Publisher: Springer Nature
ISBN: 3031061861
Category : Science
Languages : en
Pages : 220

Get Book

Book Description
The purpose of this monograph is to offer an accessible and essentially self-contained presentation of some mathematical aspects of the Feynman path integral in non-relativistic quantum mechanics. In spite of the primary role in the advancement of modern theoretical physics and the wide range of applications, path integrals are still a source of challenging problem for mathematicians. From this viewpoint, path integrals can be roughly described in terms of approximation formulas for an operator (usually the propagator of a Schrödinger-type evolution equation) involving a suitably designed sequence of operators. In keeping with the spirit of harmonic analysis, the guiding theme of the book is to illustrate how the powerful techniques of time-frequency analysis - based on the decomposition of functions and operators in terms of the so-called Gabor wave packets – can be successfully applied to mathematical path integrals, leading to remarkable results and paving the way to a fruitful interaction. This monograph intends to build a bridge between the communities of people working in time-frequency analysis and mathematical/theoretical physics, and to provide an exposition of the present novel approach along with its basic toolkit. Having in mind a researcher or a Ph.D. student as reader, we collected in Part I the necessary background, in the most suitable form for our purposes, following a smooth pedagogical pattern. Then Part II covers the analysis of path integrals, reflecting the topics addressed in the research activity of the authors in the last years.

Proceedings of the Norbert Wiener Centenary Congress, 1994

Proceedings of the Norbert Wiener Centenary Congress, 1994 PDF Author: Norbert Wiener Centenary Congress (1994 : Michigan State University)
Publisher: American Mathematical Soc.
ISBN: 0821804529
Category : Mathematics
Languages : en
Pages : 566

Get Book

Book Description
One of the great mathematicians of this century, Norbert Wiener was a universal thinker of colossal proportions. This book contains the proceedings of the Norbert Wiener Centenary Congress held at Michigan State University on November 27 - December 3, 1994. The aim of the Congress was to reveal the depth and strong coherence of thought that runs through Wiener's legacy, and to exhibit its continuation in ongoing research.This volume brings together the great minds who have furthered Wiener's ideas in physics, stochastics, harmonic analysis, philosophy, prosthesis and cybernetics. The presentations coherently lay out the developments of the subjects from their inception. This volume provides an excellent pathway for new investigators who may wish to pursue these developments by following the footsteps of world experts. There is no other book available in which experts in the various fields in which Weiner worked have presented his thoughts and contributions in such a coherent and lucid manner.

Landscapes of Time-Frequency Analysis

Landscapes of Time-Frequency Analysis PDF Author: Paolo Boggiatto
Publisher: Springer Nature
ISBN: 3030560058
Category : Mathematics
Languages : en
Pages : 208

Get Book

Book Description
This contributed volume features chapters based on talks given at the second international conference titled Aspects of Time-Frequency Analysis (ATFA 19), held at Politecnico di Torino from June 25th to June 27th, 2019. Written by experts in harmonic analysis and its applications, these chapters provide a valuable overview of the state-of-the-art of this active area of research. New results are collected as well, making this a valuable resource for readers seeking to be brought up-to-date. Topics covered include: Signal analysis Quantum theory Modulation space theory Applications to the medical industry Wavelet transform theory Anti-Wick operators Landscapes of Time-Frequency Analysis: ATFA 2019 will be of particular interest to researchers and advanced students working in time-frequency analysis and other related areas of harmonic analysis.

Quantum Field Theory

Quantum Field Theory PDF Author: Bertfried Fauser
Publisher: Springer Science & Business Media
ISBN: 376438736X
Category : Science
Languages : en
Pages : 436

Get Book

Book Description
The present volume emerged from the 3rd `Blaubeuren Workshop: Recent Developments in Quantum Field Theory', held in July 2007 at the Max Planck Institute of Mathematics in the Sciences in Leipzig/Germany. All of the contributions are committed to the idea of this workshop series: To bring together outstanding experts working in the field of mathematics and physics to discuss in an open atmosphere the fundamental questions at the frontier of theoretical physics.

Mathematical Theory of Feynman Path Integrals

Mathematical Theory of Feynman Path Integrals PDF Author: Sergio A. Albeverio
Publisher: Springer
ISBN: 354038250X
Category : Mathematics
Languages : en
Pages : 186

Get Book

Book Description
Feynman path integrals integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ideas based on Feynman path integrals have also played an important role in areas of mathematics like low dimensional topology and differential geometry, algebraic geometry, infinite dimensional analysis and geometry, and number theory. The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. To take care of the many developments which have occurred since then, an entire new chapter about the current forefront of research has been added. Except for this new chapter, the basic material and presentation of the first edition was mantained, a few misprints have been corrected. At the end of each chapter the reader will also find notes with further bibliographical information.

Mathematical Feynman Path Integrals And Their Applications (Second Edition)

Mathematical Feynman Path Integrals And Their Applications (Second Edition) PDF Author: Sonia Mazzucchi
Publisher: World Scientific
ISBN: 9811214808
Category : Science
Languages : en
Pages : 360

View

Book Description
Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics.This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added.This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists.

Mathematical Theory of Feynman Path Integrals

Mathematical Theory of Feynman Path Integrals PDF Author: Sergio Albeverio
Publisher: Springer Science & Business Media
ISBN: 3540769544
Category : Mathematics
Languages : en
Pages : 182

View

Book Description
The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. An entire new chapter on the current forefront of research has been added. Except for this new chapter and the correction of a few misprints, the basic material and presentation of the first edition has been maintained. At the end of each chapter the reader will also find notes with further bibliographical information.

Mathematical Feynman Path Integrals and Their Applications

Mathematical Feynman Path Integrals and Their Applications PDF Author:
Publisher:
ISBN: 9814469270
Category :
Languages : en
Pages :

View

Book Description


Rigorous Time Slicing Approach to Feynman Path Integrals

Rigorous Time Slicing Approach to Feynman Path Integrals PDF Author: Daisuke Fujiwara
Publisher: Springer
ISBN: 4431565531
Category : Science
Languages : en
Pages : 333

View

Book Description
This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schrödinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives of order equal to or higher than two are bounded. The semi-classical asymptotic formula up to the second term of the fundamental solution is also proved by a method different from that of Birkhoff. A bound of the remainder term is also proved.The Feynman path integral is a method of quantization using the Lagrangian function, whereas Schrödinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger's method, there is still much to be done concerning rigorous mathematical treatment of Feynman's method. Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces which is obtained by dividing the time interval into small pieces. This method is called the time slicing approximation method or the time slicing method.This book consists of two parts. Part I is the main part. The time slicing method is performed step by step in detail in Part I. The time interval is divided into small pieces. Corresponding to each division a finite-dimensional integral is constructed following Feynman's famous paper. This finite-dimensional integral is not absolutely convergent. Owing to the assumption of the potential, it is an oscillatory integral. The oscillatory integral techniques developed in the theory of partial differential equations are applied to it. It turns out that the finite-dimensional integral gives a finite definite value. The stationary phase method is applied to it. Basic properties of oscillatory integrals and the stationary phase method are explained in the book in detail.Those finite-dimensional integrals form a sequence of approximation of the Feynman path integral when the division goes finer and finer. A careful discussion is required to prove the convergence of the approximate sequence as the length of each of the small subintervals tends to 0. For that purpose the book uses the stationary phase method of oscillatory integrals over a space of large dimension, of which the detailed proof is given in Part II of the book. By virtue of this method, the approximate sequence converges to the limit. This proves that the Feynman path integral converges. It turns out that the convergence occurs in a very strong topology. The fact that the limit is the fundamental solution of the Schrödinger equation is proved also by the stationary phase method. The semi-classical asymptotic formula naturally follows from the above discussion.A prerequisite for readers of this book is standard knowledge of functional analysis. Mathematical techniques required here are explained and proved from scratch in Part II, which occupies a large part of the book, because they are considerably different from techniques usually used in treating the Schrödinger equation.

Wave Packet Analysis of Feynman Path Integrals

Wave Packet Analysis of Feynman Path Integrals PDF Author: Fabio Nicola
Publisher: Springer Nature
ISBN: 3031061861
Category : Science
Languages : en
Pages : 220

View

Book Description
The purpose of this monograph is to offer an accessible and essentially self-contained presentation of some mathematical aspects of the Feynman path integral in non-relativistic quantum mechanics. In spite of the primary role in the advancement of modern theoretical physics and the wide range of applications, path integrals are still a source of challenging problem for mathematicians. From this viewpoint, path integrals can be roughly described in terms of approximation formulas for an operator (usually the propagator of a Schrödinger-type evolution equation) involving a suitably designed sequence of operators. In keeping with the spirit of harmonic analysis, the guiding theme of the book is to illustrate how the powerful techniques of time-frequency analysis - based on the decomposition of functions and operators in terms of the so-called Gabor wave packets – can be successfully applied to mathematical path integrals, leading to remarkable results and paving the way to a fruitful interaction. This monograph intends to build a bridge between the communities of people working in time-frequency analysis and mathematical/theoretical physics, and to provide an exposition of the present novel approach along with its basic toolkit. Having in mind a researcher or a Ph.D. student as reader, we collected in Part I the necessary background, in the most suitable form for our purposes, following a smooth pedagogical pattern. Then Part II covers the analysis of path integrals, reflecting the topics addressed in the research activity of the authors in the last years.

Path-integral methods and their applications

Path-integral methods and their applications PDF Author:
Publisher: Allied Publishers
ISBN: 9788177642315
Category :
Languages : en
Pages : 364

View

Book Description


Proceedings of the Norbert Wiener Centenary Congress, 1994

Proceedings of the Norbert Wiener Centenary Congress, 1994 PDF Author: Norbert Wiener Centenary Congress (1994 : Michigan State University)
Publisher: American Mathematical Soc.
ISBN: 0821804529
Category : Mathematics
Languages : en
Pages : 566

View

Book Description
One of the great mathematicians of this century, Norbert Wiener was a universal thinker of colossal proportions. This book contains the proceedings of the Norbert Wiener Centenary Congress held at Michigan State University on November 27 - December 3, 1994. The aim of the Congress was to reveal the depth and strong coherence of thought that runs through Wiener's legacy, and to exhibit its continuation in ongoing research.This volume brings together the great minds who have furthered Wiener's ideas in physics, stochastics, harmonic analysis, philosophy, prosthesis and cybernetics. The presentations coherently lay out the developments of the subjects from their inception. This volume provides an excellent pathway for new investigators who may wish to pursue these developments by following the footsteps of world experts. There is no other book available in which experts in the various fields in which Weiner worked have presented his thoughts and contributions in such a coherent and lucid manner.

Landscapes of Time-Frequency Analysis

Landscapes of Time-Frequency Analysis PDF Author: Paolo Boggiatto
Publisher: Springer Nature
ISBN: 3030560058
Category : Mathematics
Languages : en
Pages : 208

View

Book Description
This contributed volume features chapters based on talks given at the second international conference titled Aspects of Time-Frequency Analysis (ATFA 19), held at Politecnico di Torino from June 25th to June 27th, 2019. Written by experts in harmonic analysis and its applications, these chapters provide a valuable overview of the state-of-the-art of this active area of research. New results are collected as well, making this a valuable resource for readers seeking to be brought up-to-date. Topics covered include: Signal analysis Quantum theory Modulation space theory Applications to the medical industry Wavelet transform theory Anti-Wick operators Landscapes of Time-Frequency Analysis: ATFA 2019 will be of particular interest to researchers and advanced students working in time-frequency analysis and other related areas of harmonic analysis.

Quantum Field Theory

Quantum Field Theory PDF Author: Bertfried Fauser
Publisher: Springer Science & Business Media
ISBN: 376438736X
Category : Science
Languages : en
Pages : 436

View

Book Description
The present volume emerged from the 3rd `Blaubeuren Workshop: Recent Developments in Quantum Field Theory', held in July 2007 at the Max Planck Institute of Mathematics in the Sciences in Leipzig/Germany. All of the contributions are committed to the idea of this workshop series: To bring together outstanding experts working in the field of mathematics and physics to discuss in an open atmosphere the fundamental questions at the frontier of theoretical physics.

Mathematical Theory of Feynman Path Integrals

Mathematical Theory of Feynman Path Integrals PDF Author: Sergio A. Albeverio
Publisher: Springer
ISBN: 354038250X
Category : Mathematics
Languages : en
Pages : 186

View

Book Description
Feynman path integrals integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ideas based on Feynman path integrals have also played an important role in areas of mathematics like low dimensional topology and differential geometry, algebraic geometry, infinite dimensional analysis and geometry, and number theory. The 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. To take care of the many developments which have occurred since then, an entire new chapter about the current forefront of research has been added. Except for this new chapter, the basic material and presentation of the first edition was mantained, a few misprints have been corrected. At the end of each chapter the reader will also find notes with further bibliographical information.