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WORLD SCIENTIFIC eBooks

Mathematical Feynman Path Integrals and Their Applications

Sonia Mazzucchi

Published May 1, 2009• 43 citations
Publisher

Abstract

Infinite Dimensional Oscillatory Integrals Feynman Path Integrals and The Schroedinger Equation The Stationary Phase Method and the Semiclassical Limit of Quantum Mechanics Quantum Open Systems Alternative Approaches and Further Results.
DOI
10.1142/7104

Related papers

Mathematical Feynman Path Integrals and Their Applications

Sonia Mazzucchi

"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"--

WORLD SCIENTIFIC eBooks•Nov 7, 2019•3 citations

Methods of bosonic path integrals representations : random systems on classical physics

Luiz C. L. Botelho

CONTENTS: Preface Bosonic Path Integrals Nelson's Stochastic Mechanics by Means of Feynman Path Integrals Feymman path-integrals for Brownian Classical Motions (1+1)-dimensional Burger turbulence in the framework of Hopf-Cole Transformation and Feymman Trajectories path-integrals Feynman-Wiener Path Integral Representations for the Quantum Harmonic Oscillator with Time-Dependent Stochastic Frequency: Quantum Brownian Motions and Navier-Stokes Weakly Turbulence Some studies on functional integrals representations for fluid motion with random conditions Statistics turbulence suppression by a strong magnetic field: A Feynman path integral argument A non-linear diffusion equation, Some existence and uniqueness results on weak solutions for random initial conditions and its path integral representation -- a mathematical methods approach The Exact Evalution of the Response Function for Wave Propagation in Random Media: A Functional Integral Approach On the Abstract Method of Integration for Wave Equation and some Applications for Wave Physics Feynman Path-Integral Studies for Quantum Particles Propagators Interacting with Environments The Quantum Fractional Dimensional Brownian Motion: A Path-Integral Study Ohm's Law: A Langevin Path-Integral Study at Classical and Quantum W.K.B. Regime On the Statistical of An Ideal Gas with Mass Fluctuations and the Boltzman Ergodic Theorem: The Combined Boltzman-Tsallis Statistics The Suppression of the Quantum Phenomena by an External Strong White-Noise Magnetic Field: A Path Integral Argument Index.

•Jan 1, 2006•5 citations

From Path Integration to the Schrödinger Equation

Lukong Cornelius Fai

We show how from the path integral we rederive the Schrödinger equation. This will confirm the Feynman path integration to be the formulation of quantum mechanics presenting an alternative way of examining the subject matter than the usual approaches with applications as vast as those of quantum mechanics itself.

CRC Press eBooks•Feb 19, 2021•0 citations

Quantum Mechanics for Mathematicians

Leon A. Takhtajan

This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.

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The Feynman Path Integral: An Historical Slice

John R. Klauder

Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.

arXiv:quant-ph/0303034•WORLD SCIENTIFIC eBooks•Jul 1, 2003•20 citations

Advanced Topics in Quantum Mechanics

Marcos Mariño

Quantum mechanics is one of the most successful theories in science, and is relevant to nearly all modern topics of scientific research. This textbook moves beyond the introductory and intermediate principles of quantum mechanics frequently covered in undergraduate and graduate courses, presenting in-depth coverage of many more exciting and advanced topics. The author provides a clearly structured text for advanced students, graduates and researchers looking to deepen their knowledge of theoretical quantum mechanics. The book opens with a brief introduction covering key concepts and mathematical tools, followed by a detailed description of the Wentzel–Kramers–Brillouin (WKB) method. Two alternative formulations of quantum mechanics are then presented: Wigner's phase space formulation and Feynman's path integral formulation. The text concludes with a chapter examining metastable states and resonances. Step-by-step derivations, worked examples and physical applications are included throughout.

Cambridge University Press eBooks•Dec 2, 2021•25 citations
"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"--
CONTENTS: Preface Bosonic Path Integrals Nelson's Stochastic Mechanics by Means of Feynman Path Integrals Feymman path-integrals for Brownian Classical Motions (1+1)-dimensional Burger turbulence in the framework of Hopf-Cole Transformation and Feymman Trajectories path-integrals Feynman-Wiener Path Integral Representations for the Quantum Harmonic Oscillator with Time-Dependent Stochastic Frequency: Quantum Brownian Motions and Navier-Stokes Weakly Turbulence Some studies on functional integrals representations for fluid motion with random conditions Statistics turbulence suppression by a strong magnetic field: A Feynman path integral argument A non-linear diffusion equation, Some existence and uniqueness results on weak solutions for random initial conditions and its path integral representation -- a mathematical methods approach The Exact Evalution of the Response Function for Wave Propagation in Random Media: A Functional Integral Approach On the Abstract Method of Integration for Wave Equation and some Applications for Wave Physics Feynman Path-Integral Studies for Quantum Particles Propagators Interacting with Environments The Quantum Fractional Dimensional Brownian Motion: A Path-Integral Study Ohm's Law: A Langevin Path-Integral Study at Classical and Quantum W.K.B. Regime On the Statistical of An Ideal Gas with Mass Fluctuations and the Boltzman Ergodic Theorem: The Combined Boltzman-Tsallis Statistics The Suppression of the Quantum Phenomena by an External Strong White-Noise Magnetic Field: A Path Integral Argument Index.
This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.
Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.
Quantum mechanics is one of the most successful theories in science, and is relevant to nearly all modern topics of scientific research. This textbook moves beyond the introductory and intermediate principles of quantum mechanics frequently covered in undergraduate and graduate courses, presenting in-depth coverage of many more exciting and advanced topics. The author provides a clearly structured text for advanced students, graduates and researchers looking to deepen their knowledge of theoretical quantum mechanics. The book opens with a brief introduction covering key concepts and mathematical tools, followed by a detailed description of the Wentzel–Kramers–Brillouin (WKB) method. Two alternative formulations of quantum mechanics are then presented: Wigner's phase space formulation and Feynman's path integral formulation. The text concludes with a chapter examining metastable states and resonances. Step-by-step derivations, worked examples and physical applications are included throughout.